But we also used “generator” to mean “every object is a colimit of copies of the object,” and noted that these conditions are not equivalent: as this MO question discusses, the abelian group satisfies the first condition but not the second. More generally, as Mike Shulman explains here, there are in fact many inequivalent definitions of “generator” in category theory.

The goal of this post is to sort through a few of these definitions, which turn out to be totally ordered in strength, and find additional hypotheses under which they agree. As an application we’ll restate Gabriel’s theorem using weaker definitions of “generator” and give a more explicit description of all of the rings Morita equivalent to a given ring.

**Conventions**

All categories appearing in this post will again be either “ordinary” (-enriched) or “linear” (-enriched). will again denote either presheaves or depending on whether is ordinary or linear.

A **family of objects** in a category is an essentially small full subcategory . This is partly just a way of saying “set of objects” which is invariant under equivalence, although the morphisms in will be important too. At a few points in this post we will take coproducts involving every object in . If is essentially small but not small these coproducts should be understood as ranging over representatives of the isomorphism classes of .

The naming convention for generation conditions is that for some adjective X, a single object is an “X generator” and a family of objects is a “family of X generators.” In both cases this might be shortened to “is X.” We will not use the convention that a family of objects is also an “X generator” because we want to be able to summarize Gabriel’s theorem using the phrase “compact projective generator.”

**Two classical definitions**

In this section we won’t make any explicit use of linearity: everything applies to linear categories thought of as ordinary categories.

**Definition:** A family of objects is a family of **generators** or **faithful generators** (nonstandard) if the following equivalent conditions hold:

- The functors are jointly faithful.
- The restricted Yoneda embedding is faithful.
- (If has coproducts) For all , the natural map is epic.
- (If has coproducts) For all , some map is epic.

Explicitly, these conditions mean that if is a pair of morphisms such that for all , then .

If is linear, #2 does not depend on whether we take set-valued or abelian group-valued presheaves. #3 and #4 are presumably the motivation for the term “generator”: if, for example, is a group and is a subset of , then generates iff the induced map is surjective.

**Definition:** A functor is **conservative** if it reflects isomorphisms: if is an isomorphism, then is an isomorphism.

**Definition:** A family of objects is a family of **strong generators** if the following equivalent conditions hold:

- The functors are jointly faithful and conservative.
- The restricted Yoneda embedding is faithful and conservative.

Explicitly, these conditions mean first that the above condition holds and second that if is a morphism such that is an isomorphism for all , then is an isomorphism.

*Example.* Let be a familiar category of algebraic objects such as groups, rings, or modules over a ring. Then the free object on one generator is a strong generator: is the forgetful functor, which is faithful by definition, and a morphism of groups, rings, or modules over a ring which is a bijection is an isomorphism. This generalizes to the category of models of any Lawvere theory.

*Example.* is a generator but not a strong generator: is the forgetful functor, which is faithful by definition, but a continuous bijection between topological spaces need not be a homeomorphism.

*Example.* (compact Hausdorff spaces) is a strong generator: is the forgetful functor, which is faithful by definition, and a continuous bijection between compact Hausdorff spaces is a homeomorphism. This generalizes to any category which is monadic over .

Recall that a basic property of faithful functors is that they reflect monos and epis. The corresponding property of conservative functors is the following.

**Proposition:** Let be a conservative functor. Then reflects any (shapes of) limits or colimits that it preserves.

*Proof.* By duality it suffices to prove the statement for colimits. Suppose that is a shape of colimit which have and which preserves. To say that reflects colimits of shape is to say the following: suppose is a cocone over a diagram of shape , so that it is equipped with suitable maps . If the induced maps exhibit as the colimit , then the original maps exhibit as the colimit .

Equivalently, the maps and describe maps and , and the condition is that if the latter map is an isomorphism, then so is the former map. But by hypothesis, since preserves colimits of shape , the former map is also the induced map , and if this is an isomorphism then, since is conservative, so is the map .

**Corollary:** Let be a conservative functor. If has either equalizers or coequalizers and preserves them, then is faithful.

*Proof.* Let be a pair of parallel arrows. Then iff is the coequalizer of iff is the equalizer of . If is conservative and preserves either of these, then it reflects them as well.

**Corollary:** If a category has equalizers, then we can drop “jointly faithful” from the definition of strong generators: that is, is a family of strong generators iff the functors are jointly conservative.

*Proof.* The functors preserve any limits that exist in .

*Example.* Let be the homotopy category of pointed connected CW complexes. Then Whitehead’s theorem asserts that the functors are jointly conservative, and these functors are hom functors , so they preserve all limits that exist in . However, they are not jointly faithful: for example, there are interesting homotopy classes of morphisms between Eilenberg-MacLane spaces whose homotopy groups live in distinct degrees, and for other examples see this MO question. It follows that fails to have equalizers.

What can we say about the converse? First, say that a morphism is a **fake isomorphism** (nonstandard) if it is epic and monic, but not an iso. For example, any continuous bijection which is not a homeomorphism is fake.

**Proposition:** Let be a faithful functor. If has no fake isos, then is conservative.

*Proof.* Since is faithful, it reflects epis and monos. If is an iso, then in particular it’s epic and monic, hence so is . By hypothesis, it follows that is an iso.

**Corollary:** If is a family of objects in a category with no fake isos, then is faithful iff it is strong.

In general, a category won’t have fake isos if its epis are well-behaved; for example, whenever epis are regular, a hypothesis we’ll use later as well. This holds in abelian categories but also in and ; however, it does not hold in , since for example is epic but not regular epic, and it does not hold in , since there epis are just surjections but regular epis are quotient maps.

**Three definitions about colimits**

The term “generator” can be interpreted as having something to do with generating a category under colimits. Here are three definitions along those lines. As in the previous section, we won’t make any explicit use of linearity.

**Definition:** Let be a cocomplete category. A family of objects is a family of

**naive generators**(nonstandard) if every is a colimit of objects in ,**presenting generators**(nonstandard) if every is a coequalizer of a pair of morphisms between coproducts of objects in , and**iterated generators**(nonstandard) if every is an iterated colimit of objects in .

Mike Shulman uses the terms **colimit-dense generator** for naive generators and **colimit generator** for iterated generators respectively, but I find it hard to remember which is which (and in fact I mixed them up while writing this sentence). The idea behind the definition of presenting generators is that a coequalizer of the above form is a presentation by “generators” (the ) and “relations” (the ).

*Example.* is not naive, but it is presenting, iterated, strong, and faithful.

The five definitions we’ve introduced are totally ordered in strength as follows. As above, is a family of objects in a cocomplete category.

**Theorem:** Naive presenting iterated strong faithful.

*Proof.*

*Naive presenting:* in a cocomplete category, every colimit can be computed as the coequalizer of a pair of morphisms between two coproducts.

*Presenting iterated:* every coequalizer of a pair of morphisms between coproducts is an iterated colimit (iterated twice: once for the coproducts, once for the coequalizers).

*Iterated faithful:* Suppose is iterated. We want to show that every admits some epi . Every satisfies this condition, and moreover it is closed under coproducts (since coproducts of epis are epis) and coequalizers (since coequalizer projections are epis, and epis are closed under composition), hence it is closed under colimits. So in fact every admits such an epi, and is faithful.

*Iterated strong:* Suppose is iterated. By the above, we know that the functors are jointly faithful, so it suffices to show that they are jointly conservative.

Say that a morphism is an **-isomorphism** if it induces an isomorphism

for all . If denotes the collection of all -isomorphisms, our goal is to show that consists precisely of the isomorphisms.

An object is **-colocal** if every -isomorphism induces an isomorphism . By hypothesis, every is -colocal. If is a colimit of -colocal objects and is an isomorphism, then

is an isomorphism since its components are. Hence the collection of -colocal objects is closed under colimits. Since is iterated, it follows that every is -colocal.

But this means that if is an -isomorphism, then is an isomorphism for all , and by the Yoneda lemma it follows that is an isomorphism. Hence is strong.

*Strong faithful:* by definition.

So far we’ve seen the following counterexamples to the converses of the above implications:

- shows that presenting naive.
- shows that faithful strong.

According to Mike Shulman, iterated strong under mild hypotheses (in addition to being cocomplete, it must have finite limits and satisfy a mild smallness condition), but we won’t use this. I expect that iterated presenting in general but don’t know a counterexample. Under additional hypotheses, we have the following.

**Theorem:** The implications presenting iterated strong faithful can be reversed under the following additional hypotheses:

- Faithful strong if has no fake isos (in particular if is abelian).
- Faithful presenting if every epi in is regular (in particular if is abelian).
- Iterated presenting if every is projective.

*Proof.*

*Faithful sometimes strong:* proven previously.

*Faithful sometimes presenting:* suppose is faithful and let

be an epi. We would like to use this epi to exhibit as a coequalizer.

Now suppose that every epi is regular. Then is regular, hence is the coequalizer of some morphisms . By faithfulness, we can find another epi

and since coequalizers are unchanged by precomposing with an epi, we can replace with , and we conclude that

is a coequalizer diagram. Hence is presenting.

*Iterated sometimes presenting:* suppose is iterated. Let’s try to imitate the argument used in this post for discussing finite presentations in order to reduce any iterated colimit to a coequalizer of coproducts.

First, a colimit is a coequalizer of coproducts in the usual way. Since a coproduct of colimits can be expressed as a single colimit (take the coproduct of the corresponding diagrams), it suffices to show that we can express a coequalizer of colimits as a coequalizer of coproducts.

Hence let

be morphisms, where , whose coequalizer we are trying to express as a coequalizer of coproducts. As before, admits an epi from a coproduct and coequalizers are unchanged by precomposing with an epi, so WLOG the above diagram has the form

.

Now suppose that every is projective. Then projective objects are closed under coproducts (assuming the axiom of choice: pick a lift for each object), so is projective, and the coequalizer projection is an epi, so lift to , and as before we find that we can reduce the above computation to the computation of a single coequalizer. Equivalently, WLOG the above diagram has the form

(although there may be more than there were before). Hence is presenting.

**Corollary:** If is a family of objects in an abelian category, then strong faithful. If is cocomplete, then presenting iterated strong faithful.

Now, in the previous post on tiny objects we proved various facts using naive generators, but an inspection of the proofs involved show that they go through for presenting generators (possibly even iterated generators, but we don’t need this; the point is that if a functor preserves colimits then it preserves iterated colimits), and using the above corollary we deduce the following.

**Corollary (Freyd):** If is a family of objects in a cocomplete abelian category, then the restricted Yoneda embedding is an equivalence of categories iff is a family of compact projective (faithful) generators.

**Corollary (Gabriel):** If is an object in a cocomplete abelian category, then

is an equivalence of categories iff is a compact projective (faithful) generator.

**One last definition for abelian categories**

Let be a linear category. In this setting the condition that is a family of (faithful) generators can be rephrased as the condition that the restricted Yoneda embedding reflects zero morphisms in the sense that if is a morphism in such that

is a zero morphism for all , then .

If is abelian, then a morphism is zero iff its image is zero, which suggests the following definition.

**Definition:** Let be a linear functor between linear categories. is **weakly faithful** (nonstandard) if it reflects zero objects: implies .

**Definition:** Let be a linear category. A family of objects is a family of **weak generators** if the restricted Yoneda embedding is weakly faithful.

More explicitly, this condition means that if is a nonzero object then there is some nonzero morphism .

**Proposition:** Let be a linear functor between linear categories. If is faithful, then it is weakly faithful. The converse holds if is exact and every morphism in has an epi-mono factorization (in particular if is abelian).

By an epi-mono factorization of a morphism we just mean a factorization where is monic and is epic. In an abelian category the image factorization accomplishes this. More generally, in a category with either 1) kernel pairs and coequalizers or 2) cokernel pairs and equalizers the regular coimage or regular image factorizations, respectively, accomplish this, so this is a very mild hypothesis.

*Proof.* : An object is a zero object iff the identity is a zero morphism. Since is faithful, it reflects zero morphisms.

: let be a morphism. By hypothesis, has an epi-mono factorization where fit into a diagram

.

Since is exact, it preserves epis and monos, and hence we get an induced epi-mono factorization .

If , then by cancelling on the left and cancelling on the right we conclude that , so . Since reflects zero objects, it follows that , so . Hence reflects zero morphisms as desired.

I expect that weak faithful in general but don’t know a counterexample.

**Corollary:** If is a family of objects in a cocomplete linear category, then naive presenting iterated strong faithful weak.

**Corollary:** If is a family of projective objects in an abelian category, then strong faithful weak. If is cocomplete, then presenting iterated strong faithful weak.

**Corollary (Freyd):** If is a family of objects in a cocomplete abelian category, then the restricted Yoneda embedding is an equivalence of categories iff is a family of compact projective weak generators.

**Corollary (Gabriel):** If is an object in a cocomplete abelian category, then

is an equivalence of categories iff is a compact projective weak generator.

**More explicit Morita equivalences**

Let be a ring. We now know that every Morita equivalence comes from a module which is a compact projective weak generator, where and the equivalence is given by

.

How explicit can we make this? is compact projective iff it is a finitely presented projective module, or equivalently a retract of a finite free module. This means that we can explicitly describe by writing down an idempotent

such that is the splitting of ; from here it follows that can be explicitly described as

.

This description exhibits as a “non-unital subring” of : the inclusion respects multiplication and addition but not the unit, and in fact the unit in is .

Since we already know that exhibits a Morita equivalence , we may assume WLOG that (by replacing with the Morita equivalent ), so that and . It remains to give an explicit criterion for when is a weak generator.

If , we want to determine when we can always find a nonzero map . The data of such a map is equivalent to the data of an element such that , and it is nonzero iff . So is a weak generator iff acts by a nonzero endomorphism on every module . The action of on any module factors through some quotient of (by a two-sided ideal), and conversely every such quotient appears this way, so the condition is precisely that is not contained in any proper two-sided ideal of .

Undoing the use of the Morita equivalence between and , we conclude the following.

**Theorem:** is a weak generator iff is a **full idempotent**, meaning that it is not contained in any proper two-sided ideal of , or equivalently, that .

More explicitly, the two-sided ideals of are precisely the submodules of regarded as an -bimodule in the usual way, or equivalently as an -module. Submodules are a categorical notion (corresponding to monomorphisms), so they are invariant under Morita equivalence. is Morita equivalent to , with the equivalence sending to regarded as an -module, or equivalently an -module in the usual way. And the two-sided ideals of are precisely the submodules of regarded as an -bimodule. Hence we conclude the following (which can also be proven without knowing anything about Morita equivalence):

**Proposition:** Every two-sided ideal of has the form where is a two-sided ideal of .

**Corollary:** is full iff for all proper two-sided ideals of .

The condition that is very strong: since , it follows by induction that for all , hence that . If is a commutative Noetherian domain and is a proper ideal, then by the Krull intersection theorem, , hence .

**Corollary:** If is a commutative Noetherian domain, then is full iff , so the rings Morita equivalent to are precisely the rings where is a nonzero finitely presented projective module.

Geometrically this has the following interpretation. If is commutative, then defines an algebraic vector bundle over the affine scheme , and the condition that is the condition that the restriction of to affine closed subschemes is nonzero. Idempotence implies that if the coefficients of vanish when restricted to , then they vanish to infinite order. The Krull intersection theorem implies that if this happens when is a Noetherian domain, then , so . (Compare to the statement that if a meromorphic function on a connected open subset of vanishes to infinite order at a point, then it vanishes identically.)

If we don’t assume that is a domain, then what can happen is that has a nontrivial decomposition as a product , so has a nontrivial decomposition as a coproduct , and a vector bundle over may be nonvanishing over one component but not the other. Algebraically this corresponds to the intersection being a nontrivial idempotent ideal in , such as one generated by a nontrivial idempotent.

]]>

.

Determining is equivalent to isolating the module (regarded as a module over via right multiplication), from which we can recover as its endomorphism ring. In some sense what this tells us is that cannot always be isolated in by a categorical property.

The next best thing we can try to do is to classify all of the rings such that by isolating the corresponding modules by some categorical property. The crucial property turns out to be that the hom functor

is faithful and preserves colimits. An object with this second property is called a tiny object, and in this post we’ll discuss how this condition behaves with an eye towards better understanding Morita equivalences. Along the way we’ll prove a theorem due to Gabriel characterizing categories of modules among abelian categories.

**Conventions**

Every limit or colimit appearing in this post is over an essentially small diagram.

Every category appearing in this post will either be “ordinary” (-enriched) or “linear” (-enriched). We will be proving facts for both kinds of categories simultaneously because the proofs are almost identical.

If is an ordinary category then denotes the category of presheaves of sets on , while if is a linear category then denotes the category of presheaves of abelian groups on .

“Module” means “right module.” In particular, if is a ring regarded as a linear category with one object, then is the linear category of -modules.

**Definition and examples**

Let be a cocomplete category (possibly linear).

**Definition:** A **tiny object** is an object such that preserves colimits.

*Warning.* In the previous post about compact objects we didn’t talk about enrichments; for applications to module categories, this is because the forgetful functor preserves filtered colimits, and so in the definition of compactness we did not need to specify whether was regarded as taking values in sets or abelian groups. But the forgetful functor certainly does not preserve all colimits (e.g. it does not preserve coproducts), so here it’s crucial to talk about enrichments.

*Example.* If , then the one-object set is the only tiny object. To see this, if is a set, then in order for to preserve colimits it must in particular preserve finite coproducts, so the map

must be an isomorphism. The RHS can be identified with the set of subsets of , while the inclusion of the LHS can be identified with the inclusion of the empty subset and the subset consisting of all of . Hence if preserves finite coproducts, then must have exactly two subsets, the empty subset and , and this condition is satisfied precisely when .

*Example.* The motivating class of examples is that if is a category (possibly linear), then in its presheaf category , the representable presheaves are tiny by the Yoneda lemma, since colimits of presheaves are computed pointwise. These are the “free modules of rank ” on each . In particular, if is a one-object linear category with endomorphisms , we conclude that the free -module of rank is tiny in .

*Example.* Some categories have the property that the only cocontinuous functor is the constant functor with constant value the empty set (if has an initial object then this is the only possible constant value). Since a hom functor never has this property (it always has at least one nonempty value, on ), it follows that such a category has no tiny objects. Here is a class of examples and an example:

*Subexample.* Suppose has a zero object . Then is empty. Every object admits a map , but in the only set which admits a map to the empty set is the empty set, so is also empty. (In particular, abelian categories have no tiny objects when regarded as enriched over as opposed to enriched over .)

*Subexample.* Let be the category of commutative rings. Then is empty. The zero ring (which is not a zero object) is the coequalizer of the two projections . preserves coequalizers, so is also empty. But every ring admits a morphism , so admits a morphism to the empty set, hence is also empty. (While does not have a zero object, it does have the property that the terminal object is a colimit of copies of the initial object.)

**Closure properties**

Unlike the case of compact objects, which are closed under finite colimits because finite limits commute with filtered colimits (in both and ), at first it might seem like tiny objects have no closure properties under colimits. However, there are some (shapes for) colimits – the **absolute colimits** – which are preserved by any (enriched) functor whatsoever, and hence the corresponding absolute limits commute with arbitrary colimits.

*Example.* We saw in this blog post that zero objects are absolute for categories enriched over pointed sets, and finite coproducts / biproducts are absolute for linear categories.

*Example.* In both ordinary and linear categories, split idempotents are absolute. This is worth going into in some detail.

**The facts of life about idempotents and retracts:** Splitting idempotents is an absolute colimit, in both ordinary and linear categories. Explicitly:

- Let be an object and let be a retract of it, so that there are morphisms and such that . Let be the corresponding
**split idempotent**. Then the retract , equipped with the map , is the coequalizer , and this coequalizer is preserved by any functor whatsoever. Dually, , together with the map , is the equalizer . - Conversely, suppose is an idempotent in a category such that either the equalizer or the coequalizer of and exists. Then they both exist and are canonically isomorphic. The resulting object and the corresponding maps exhibit as a retract of , so and , and is a split idempotent. In particular, the equalizer and coequalizer of and are both preserved by any functor whatsoever.

This makes retracts, like zero objects and biproducts, another rare example of objects satisfying two universal properties, one with respect to maps in and one with respect to maps out.

*Proof.* We want to verify that , together with the map , satisfies the universal property of the above coequalizer. The universal property says that a map out of is the same thing as a map out of such that . So let’s verify this: given a map out of , the pullback is a map out of such that

.

In the other direction, given a map out of such that , the pullback is a map out of , and

.

Since everything we’ve just said involves only morphisms and equations between morphisms, it is preserved by applying an arbitrary functor as desired. Dualizing this argument gives the dual statement. (Working in the opposite category exchanges the roles of and , but it is still true that in the opposite category, is a retract of .)

Conversely, suppose is an idempotent and that the coequalizer of and exists; call it , and write for the corresponding map out of . By construction, a map corresponds, via pulling back along , to a map coequalizing . Since is idempotent, it is itself such a map; write for the corresponding map out of , which by construction satisfies . In particular,

.

Since is a coequalizer it is a regular epimorphism, and in particular an epimorphism, so it is right cancellative. Cancelling it gives

.

So equalizes . In fact exhibits as the equalizer of : if is a map into equalizing , so , then factors through since .

Since is an equalizer it is a regular monomorphism, and in particular a monomorphism, so it is left cancellative. Cancelling it from gives

and hence the maps exhibit as a retract of as desired.

**Proposition:** An absolute colimit of tiny objects is tiny.

*Proof.* Absolute colimits are preserved by all functors, and hence so are their corresponding absolute limits. Maps out of an absolute colimit is an absolute limit, and hence it commutes with colimits.

Explicitly, suppose is an absolute colimit of tiny objects and is an arbitrary colimit . Then by absoluteness, while by tininess, and hence

.

By absoluteness, is an absolute limit in the diagram category (since limits are computed pointwise), and hence is preserved by the colimit functor (or, for linear categories, ), so we can exchange the order of the limit and the colimit to get

.

But by a final application of absoluteness, and the conclusion follows.

**Corollary:** Retracts of tiny objects are tiny. In a linear category, finite biproducts of tiny objects are tiny.

*Example.* In , it follows that the retracts of finite free modules are tiny. These are precisely the finitely presented projective modules. More generally, if is a category (possibly linear), then it follows that the retracts (resp. retracts of finite biproducts) of the representable presheaves are tiny in the presheaf category .

**Compact projective objects**

“Tiny object” is a term that appears to only be used by category theorists. In nice linear situations, however, it is equivalent to a weaker (in general) condition, namely “compact projective object,” and this term is much more widely used.

If is tiny, then because preserves colimits, satisfies any other property which can be expressed in terms of preserving some restricted class of colimits. We’ll highlight three in particular:

- is
**connected**: preserves finite coproducts. (This differs slightly from the nLab’s first definition but is equivalent to it in nice cases.) - is compact: preserves filtered colimits.
- is projective: preserves epimorphisms (since it preserves pushouts).

The terminology of connectedness stems from the fact that in topological spaces, graphs, -sets and other categories where objects have a well-behaved decomposition into “connected components,” connectedness in the above sense is equivalent to connectedness in the usual sense. But this mostly won’t concern us here, although it is a nice way to think about why is the only tiny object in , since we showed above that it’s even the only connected object in .

What can we say about the converse?

**Theorem:** An object in a cocomplete category is tiny iff it is connected, compact, and preserves coequalizers. If is linear, then is tiny iff it is compact and preserves coequalizers, or equivalently cokernels. If is abelian, then is tiny iff it is compact and projective.

*Proof.* In a general cocomplete category (linear or otherwise), for a functor to preserve colimits it suffices that it preserve coproducts and coequalizers. An arbitrary coproduct is the filtered colimit of its constituent finite coproducts (the proof is similar to but simpler than the proof we presented in the previous post regarding arbitrary colimits), so for a functor to preserve colimits it suffices that it preserve finite coproducts, filtered colimits, and coequalizers.

In a linear category, finite coproducts / biproducts are absolute, so every object is connected, and coequalizers can all be computed as cokernels, so preserving one is equivalent to preserving the other. Finally, in a linear abelian category, preserves coequalizers iff is projective (as we showed earlier).

Since we also showed earlier that the compact objects in categories of modules are the finitely presented modules, we also conclude the following.

**Corollary:** In , the tiny objects are precisely the finitely presented projective modules, or equivalently the finitely generated projective modules, or equivalently the retracts of finite free modules.

**Corollary:** If are Morita equivalent rings, then is the endomorphism ring of a finitely presented projective -module.

**Corollary:** If is a ring such that every finitely presented projective -module is free, then the rings Morita equivalent to are precisely the rings .

Examples of such rings include division rings, local rings (by Nakayama’s lemma), and polynomial rings over fields (by the Quillen-Suslin theorem).

This is nice to know, but it relies on a separate characterization of compact objects in which took some work to show. There is in fact a more direct and easier characterization of the tiny objects which we give below.

On the other hand, one reason to factor tininess into compactness and projectivity is that this whole story has a derived version in which, in some sense, every object becomes projective, so only compactness is left to worry about.

*Warning.* Some authors define a compact object in a cocomplete abelian category to be an object such that preserves coproducts. This is implied by compactness in the above sense but is not equivalent to it: for example, in categories of modules, it is satisfied by any finitely generated module (not necessarily finitely presented). It is not even equivalent to finitely generated for modules; see this MO discussion.

However, if is in addition assumed to be projective, so that preserves coequalizers, then preserves coproducts iff it preserves colimits iff is tiny, so with this definition of compact it is still true that “compact projective” is equivalent to “tiny.” And in a derived setting I think we can again pretend that every object is projective.

**Tiny presheaves**

Recall that our strategy for characterizing the compact objects of a nice category was to write every object as a filtered colimit of some readily identifiable class of compact objects, from which it would follow that if is projective then the identity factors through a map to some member of this class, hence must be a retract of a member of this class. We can try this exact strategy for characterizing tiny rather than compact objects, and we get the following.

Suppose is a cocomplete category (possibly linear) and is an essentially small full subcategory of tiny objects of such that every object of is a colimit of objects in .

*Example.* Let be an essentially small category (resp. linear category) and let be its category of presheaves of sets (resp. abelian groups). If is an ordinary category, then by the Yoneda lemma every presheaf is a colimit of representables, and hence we can take to be the representables. If is a linear category, then the statement of the Yoneda lemma needs to be modified, but the correct linear Yoneda lemma implies (with some work) that every presheaf is a colimit of finite biproducts of representables, and hence we can take to be the finite biproducts of representables. This reduces to a familiar fact when has one object, namely that every -module is a colimit of finite free modules.

Unlike the hypotheses we used for studying compact objects, these hypotheses are very restrictive, and we will see later that presheaf categories are the only examples.

**Theorem:** With the above hypotheses, if is ordinary then the tiny objects of are precisely the retracts of the objects in , and if is linear then the tiny objects of are precisely the retracts of finite biproducts of the objects in .

*Proof.* Write a tiny object as a colimit where . By hypothesis, preserves colimits, so

.

If is an ordinary category, so that the above colimit is a colimit of sets, it follows that the identity factors through some , hence that is a retract of some .

If is a linear category, so that the above colimit is a colimit of abelian groups, it follows that the identity is a finite sum of morphisms factoring through some s, hence that it factors through some finite biproduct of the s, hence that is a retract of a finite biproduct of some s.

**Corollary:** If is an essentially small ordinary category, the tiny objects in are precisely the retracts of representable presheaves. If is a linear category, the tiny objects are precisely the retracts of finite biproducts of representable presheaves.

This gives a second and easier proof that the tiny objects in are precisely the retracts of finite free modules, or equivalently the finitely presented projective modules.

*Example.* Let be a monoid; we will use the same symbol to denote the corresponding one-object category. Then the category of presheaves is the category of -modules in set. The unique representable presheaf is the free -module on an element, or regarded as a module over itself. By the Yoneda lemma / Cauchy’s theorem, is the monoid of endomorphisms of (regarded as a module over itself), and hence idempotent endomorphisms of correspond to idempotent elements of . If is such an idempotent, then its retract in is the module generated by .

In particular, if has no nontrivial idempotents (e.g. if it is a group), then is the unique tiny -module.

**Morita equivalences and Cauchy completion**

We are now very close to characterizing Morita equivalences in the following sense.

**Definition:** Two essentially small categories (possibly linear) are **Morita equivalent** if .

When are rings thought of as linear categories with one object this reduces to the usual notion of Morita equivalence of rings, but this notion of Morita equivalence is substantially more general.

*Example.* One way to state the Dold-Kan correspondence is that the linearization of the simplex category (obtained by taking the free abelian group on its homsets; presheaves on this are simplicial abelian groups) is Morita equivalent to the linear category with objects labeled by the non-negative integers and morphisms the free abelian groups on generators satisfying . This is, by construction, the linear category such that presheaves on it are connective chain complexes.

If is a (Morita) equivalence, then it induces an equivalence on tiny presheaves. This motivates the following definition.

**Definition:** If is an essentially small category (possibly linear), then its **Cauchy completion** is the full subcategory of on the tiny presheaves.

For ordinary categories is also known as the Karoubi envelope, the Karoubi completion, or the idempotent completion of . As we more or less showed above, it is obtained from by formally splitting all idempotents, and can be constructed explicitly as follows:

- The objects of are pairs of an object and an idempotent .
- The morphisms are the morphisms such that . Equivalently, they are the morphisms of the form . Composition is as in .

The description of morphisms follows from the fact that the formal splitting of an idempotent has two universal properties: it is both the equalizer and the coequalizer of and . Note that this construction does not depend on being essentially small and makes sense for any category.

*Example.* Lawvere observed that the category of smooth manifolds and smooth maps is the idempotent completion of the category of open subsets of Euclidean spaces and smooth maps.

*Example.* A common step in the construction of categories of motives is to take the idempotent completion of a category of correspondences.

Similarly, as we more or less showed above, for linear categories is obtained from by first formally adjoining finite biproducts and then formally splitting all idempotents. Since we already know how to formally split idempotents it suffices to describe how to formally adjoin finite biproducts, which is done explicitly as follows:

- The objects are tuples of objects in .
- The morphisms are matrices of morphisms in . Composition is given by matrix multiplication as in the end of this blog post.

In general, when formally adjoining a colimit to a category, you know by definition what maps out of the colimit must be, but you have some freedom to decide what maps into the colimit are. In the special case of splitting idempotents and adjoining biproducts there is no such freedom: both of these have two universal properties, one with respect to maps out and one with respect to maps in, and so both are uniquely determined. More generally, there is no such freedom when adjoining absolute colimits because they are preserved by all functors, including hom functors .

Hence there is a unique way to adjoin an absolute colimit to a (possibly enriched) category, and in fact one way to describe the Cauchy completion (in a way that generalizes to other kinds of enriched categories) is that it is obtained by formally adjoining all absolute colimits. We can make this precise as follows.

**Definition:** A category (possibly linear) is **Cauchy complete** if it has all absolute colimits.

Note that we do not require that is essentially small, and in fact we will need to know that and are Cauchy complete (because they are cocomplete; for this statement should be interpreted in the linear sense). More generally, since we showed that tiny objects are closed under absolute colimits, the full subcategory of tiny objects in any cocomplete category is Cauchy complete.

In particular, the Cauchy completion as defined above is Cauchy complete. Let’s show that the Cauchy completion really deserves its name.

**Theorem:** Let be an essentially small (possibly linear) category and let be Cauchy complete (linear if is). Then the restriction functor from the category of (linear if is) functors to the category of functors is an equivalence. In particular, any functor extends essentially uniquely to a functor .

With the right construction of the Cauchy completion we can remove the hypothesis that is essentially small, and then the theorem above states precisely that Cauchy completion is the left (2-)adjoint to the inclusion (2-)functor from Cauchy complete categories to categories (possibly linear).

*Proof.* Using the explicit description of above, we’ll explicitly exhibit an inverse. If is a functor, its extension is uniquely determined by the fact that functors preserve absolute colimits: splits idempotents and, in the linear case, preserves finite biproducts. Similarly, if are functors and is a natural transformation, then its extension is uniquely determined by the fact that functors preserve absolute colimits naturally. All of the isomorphisms and compatibilities needed for this to be an inverse functor again follow from the fact that functors preserve absolute colimits naturally. (This is a lot more tedious to write out than it is to verify.)

**Corollary:** With the above hypotheses, is Cauchy complete iff is an equivalence iff the tiny objects in are precisely the representables iff all idempotents in split and, in the linear case, has finite biproducts.

**Corollary:** With the above hypotheses, the restriction functor is an equivalence; that is, is Morita equivalent to its Cauchy completion .

**Corollary:** Two essentially small categories (possibly linear) are Morita equivalent iff their Cauchy completions are equivalent.

This can be used to organize a proof of the Dold-Kan theorem; see this MO question for details.

**Corollary:** Let be two monoids with no nontrivial idempotents (e.g. groups). Then are Morita equivalent (meaning that their categories of modules in are equivalent) iff they are isomorphic.

This gives a proof, different from the proof given in this previous post, that a group is uniquely determined by the category of -sets.

**Presheaf categories**

The hypotheses on a cocomplete category we used above are very restrictive; the only examples we gave were presheaf categories. In fact, these hypotheses are so restrictive that they characterize presheaf categories.

**Theorem:** Let be a cocomplete (possibly linear) category and be an essentially small full subcategory of tiny objects in such that every object of is a colimit of objects in . Then the natural functor

is an equivalence.

In a sentence: presheaf categories are precisely the cocomplete categories with a family of tiny generators. (Here “family of generators” means the colimit condition above.) In the linear case this is due to Freyd.

*Proof.* We want to show that is fully faithful and essentially surjective. Let be two objects in , and write them as colimits of objects in . Since the are tiny,

.

Since every object of is tiny, the functor is cocontinuous, and it is automatically continuous, so applying this functor (and using the fact that is a full subcategory of and that the Yoneda embedding is full) preserves the above isomorphism. Hence is fully faithful.

Now let be a presheaf. By the Yoneda lemma, is a colimit of (in the linear case, finite biproducts of) representables. We can take the same colimit in , and because is cocontinuous, it gets preserved. Hence is essentially surjective, and the conclusion follows.

**Corollary (Gabriel):** Let be a cocomplete abelian category and be a compact projective object such that every object in is a colimit of finite biproducts of copies of . Then

is an equivalence.

Note that every equivalence from a linear category to a category of modules must have this form.

*Proof.* Apply the theorem to (rather than to ), then use the fact that is equivalent to the category of -modules because finite biproducts are absolute.

The hypothesis that is abelian is somewhat misleading: if “projective” is interpreted to mean “ preserves coequalizers” or “compact projective” is interpreted to mean tiny, then we only need to know that is a cocomplete linear category, and in particular we do not need to assume anything about limits in .

This theorem characterizes categories of modules among abelian categories by a categorical property, usually stated as “cocomplete and admits a compact projective generator,” and often misstated without the cocompleteness hypothesis. Note that “generator” in this statement means something slightly different than it did above; see this MO question.

This theorem also characterizes Morita equivalences between rings in the following sense: if is a ring, the rings Morita equivalent to are precisely the endomorphism rings of finitely presented projective modules which generate in the above sense.

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It’s therefore something of a surprise that “finitely presented” *is* a categorical property of modules, and hence that it does not depend on a choice of ring . The reason is that being finitely presented is equivalent to a categorical property called compactness.

**Definition and examples**

Let be a category which has filtered colimits.

**Definition:** An object in is **compact** if preserves filtered colimits.

The terminology stems from the fact that a topological space is compact in the usual sense iff it is compact as an object in the category of open subsets of . In general, compactness can be thought of as a finiteness condition on an object : it means that cannot “spread itself out” too much. One way to make this precise is that if is compact, then any morphism from to a filtered colimit factors through some .

As expected of a finiteness condition, compactness is preserved by finite constructions.

**Proposition:** A finite colimit of compact objects is compact.

*Proof.* Maps out of a finite colimit is a finite limit, and finite limits commute with filtered colimits in .

Since infinite limits do not in general commute with filtered colimits in , there is no reason to expect that compactness is preserved by infinite colimits. On the other hand, since filtered colimits are precisely the colimits which commute with finite limits in , compactness is the strongest colimit-preserving condition on that is itself preserved by taking finite colimits.

*Example.* In , the one-point set is compact since is the identity functor. The finite colimits of copies of are precisely the finite sets, so finite sets are also compact. These are in fact all of the compact objects. The reason is that every set is the filtered colimit of its finite subsets, so if is a compact set then the identity factors through a finite subset of . It follows that is a retract of a finite set, hence is itself finite.

*Example.* In , the free abelian group is compact since represents the forgetful functor from groups to sets, which preserves filtered colimits. Starting from and iterating finite colimits, we get precisely the finitely presented abelian groups, so these are also compact. These are in fact all of the compact objects. The argument is nearly identical to the previous one: every abelian group is the filtered colimit of its finitely generated (hence finitely presented, since is Noetherian) subgroups, so if is a abelian group then the identity factors through a finitely presented subgroup of . It follows that is a retract of a finitely presented abelian group, hence is itself finitely presented.

**Finitely presented objects**

The argument given above is clearly very general; here is a very general setting in which to put it.

Let be a cocomplete category for which there exists an essentially small full subcategory of compact objects in such that every object in is a colimit of objects in . One might call a **compactly generated** category if it admits such an , although this isn’t standard and is used to refer to a different but related condition in the setting of model categories. I believe this condition is equivalent to the condition that is **locally finitely presentable**, although we won’t need this.

*Example.* Let be a small category. Then the category of presheaves on is compactly generated. can be taken to consist of the objects , or more precisely the corresponding representable presheaves; by the Yoneda lemma, morphisms out of a representable presheaf correspond to evaluation of presheaves on objects, and since colimits in functor categories are computed pointwise preserves not only filtered colimits but all colimits. And by the universal property of the Yoneda embedding, every presheaf is a colimit of representable presheaves.

*Example.* Let be the category of models of a Lawvere theory (so could be the category of groups, rings, or modules over a ring). Then is compactly generated. can be taken to consist of the finitely generated free objects . These are compact because they represent the finite powers of the forgetful functor , which preserve filtered colimits (see this previous post). And every object is the colimit of the canonical diagram of finitely generated free objects mapping into it; this diagram produces a presentation of in which every element of is used as a generator and every relation between elements of is used as a relation. Intuitively, the reason we can restrict our attention to finitely generated free objects rather than all free objects is that the operations in a Lawvere theory only involve finitely many elements at a time.

Fix a particular choice of .

**Definition:** An object of is **finitely presented** if it is an iterated finite colimit of objects in .

This terminology requires some justification.

**Proposition:** When is the category of models of a Lawvere theory and is the finitely generated free objects, then the finitely presentable objects in the above sense are precisely the finitely presentable objects in the usual sense.

*Proof.* “The usual sense” means an object which can be written as a coequalizer

where denotes the free object on generators. Here the are the generators and the pair of maps from are the relations between generators. By definition, such an object is therefore finitely presentable in the above sense. Using the coequalizer-of-coproducts construction of colimits, being finitely presentable in the usual sense is equivalent to being expressible as a single (not iterated) finite colimit of s.

From here it remains to show that the condition of being finitely presented in the usual sense is closed under finite colimits. Since the coproduct of coequalizers is the coequalizer of coproducts, being finitely presented in the usual sense is closed under finite coproducts. It remains to show that it is closed under coequalizers, and this reduces to two observations.

First, recall that when computing the coequalizer of two maps we can precompose with an epimorphism and get the same coequalizer. In the category of models of a Lawvere theory, this tells us that when computing a coequalizer where the source is finitely presented in the usual sense, we can precompose with an epimorphism from an , and hence we may assume WLOG that the source is an .

Second, suppose are a pair of maps and is finitely presented in the usual sense, hence is itself the coequalizer of a pair of maps . Because free objects are projective (in the sense of this previous post, and because the forgetful functor preserves epimorphisms), lift to a pair of maps . Now the claim is that can be computed as the coequalizer of the maps

.

This is just a matter of verifying that the two coequalizers have the same universal property. A map out of this coequalizer is a map out of whose pullback to resp. coequalizes the maps resp. . The second condition implies that this map factors through , and since are lifts of , the first condition implies that this map, when regarded as a map out of , coequalizes . But this new coequalizer, unlike the original one, is finitely presented in the usual sense, and the conclusion follows.

By construction, the finitely presented objects are compact and closed under finite colimits. We’d like to show that in fact, with the above hypotheses, they are precisely the compact objects. The key result will be the following.

**Theorem:** With the above hypotheses, every object of is a filtered colimit of finitely presented objects.

*Proof.* Let be written as a colimit where is a (small) diagram category and is the corresponding diagram, and moreover each . The coequalizer-of-coproducts description of this colimit is

.

This captures exactly the universal property that a map out of is a collection of maps out of each such that for every morphism in we have

.

The point of the proof is that this condition can be checked one morphism at a time, and so it can be checked on the finite subgraphs of . More precisely, let be the diagram of inclusions among finite subgraphs of . Then is not only a filtered category but in fact a directed poset, since the union of two finite subgraphs is another finite subgraph. (Note that we can’t replace “finite subgraph” with “finite subcategory” here, since some categories may have no interesting finite subcategories.)

If is a finite subgraph, write for the same coequalizer as above, but only involving objects and morphisms in . This is a slight abuse of notation since is not a category, but it is still true that this is a coequalizer of a pair of maps between finite coproducts of objects in , and hence is finitely presented. We want to show that is canonically isomorphic to

.

But this is clear: a morphism out of this colimit is a compatible family of morphisms out of for each finite subgraph . Compatibility implies that this family of morphisms assigns a well-defined morphism out of each (that is, that does not depend on which finite subgraph we regard as living in), and also that for every morphism in (which occurs in some finite subgraph containing and ) we have

and no other conditions are imposed, so this colimit has the same universal property as the original colimit.

Note that in groups, rings, and modules there is a more obvious version of this result, which is that every object is a filtered colimit of finitely generated objects: in fact every object is the filtered (even directed) colimit of its finitely generated subobjects. A finitely generated subobject need not be finitely presented, e.g. in the case of modules over non-Noetherian rings, but intuitively, in the above proof we get around this by imposing the necessary relations one at a time.

**Corollary:** With the above hypotheses, the compact objects of are precisely the finitely presented objects. In particular, with the above hypotheses, the notion of a finitely presented object does not depend on the choice of .

*Proof.* Let be a compact object. Write as a filtered colimit of finitely presented objects . Then the identity morphism factors through some . It follows that is a retract of a finitely presented object, hence is itself finitely presented.

**Corollary:** Let be a ring. Then the compact objects of are precisely the finitely presented modules. In particular, the notion of a finitely presented module does not depend on the choice of .

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In particular, several times below we’ll give a list of conditions and a hypothesis under which they’ll be equivalent, and these conditions won’t all be equivalent in general. In these lists we’ll adopt the following convention: whenever we give a list of conditions and prove implications between them, the list will be organized so that proofs downward are easier and require fewer hypotheses, while proofs upward are harder and require more hypotheses. We’ll also prove more implications than we strictly need in order to see this more explicitly.

**Exact functors**

Here is some slightly nonstandard terminology: we’ll call an -enriched category a **linear category** for better agreement with the terminology that a -enriched category ( a commutative ring) is a -linear category, and similarly we’ll call an -enriched functor between such categories a **linear functor**. Recall that for a functor between additive categories this condition is equivalent to preserving finite biproducts and is often called being an additive functor.

For us, a functor is **right exact** if it preserves finite colimits. Dually, it is **left exact** if it preserves finite limits, and **exact** if it preserves both finite colimits and finite limits. This definition may not look familiar, but it’s equivalent to more familiar definitions in linear categories; on the other hand, unlike the more familiar definition in terms of short exact sequences, it continues to make sense in “nonlinear” categories.

**Proposition:** Let be a linear functor between linear categories with finite colimits. The following conditions are equivalent:

- is right exact.
- preserves coequalizers.
- preserves pushouts.
- preserves cokernels.

Dually, is left exact iff it preserves equalizers iff it preserves pullbacks iff it preserves kernels.

*Proof.* : by definition. (This makes no use of linearity except that for we need that have zero morphisms and that preserves them.)

: recall that in a category with finite colimits, all finite colimits can be built out of finite coproducts and coequalizers, and hence a functor between two such categories is right exact iff it preserves finite coproducts and coequalizers. (This again makes no use of linearity.)

Since a linear functor automatically preserves finite coproducts, is right exact iff it preserves coequalizers.

: we already know that finite colimits can be built out of finite coproducts and coequalizers, but it’s worth being explicit for pushouts. The pushout of two maps is equivalent to the coequalizer of

where the two arrows are the composites and . Hence if preserves coequalizers and finite coproducts, which as usual is automatic when is linear, then preserves pushouts.

: a cokernel is just a coequalizer where one of the maps is zero, so for a functor which preserves coequalizers to preserve cokernels it suffices that preserve zero morphisms. (This makes no use of linearity except for having and preserving zero morphisms.)

: a cokernel is also just a pushout where one of the objects is zero, so for a functor which preserves pushouts to preserve cokernels it suffices that preserves zero objects. (This makes no use of linearity except for having and preserving zero objects.)

: in a linear category, the coequalizer of two maps is the cokernel of the map , and hence a linear functor between linear categories which preserves cokernels also preserves coequalizers. (This is the first and only place where we’ve needed to subtract morphisms.)

These implications are not reversible in general. For example, the underlying set functor preserves coequalizers but not coproducts. It does not preserve coproducts since, for example, the natural map is not an isomorphism. But it does preserve coequalizers: if is a parallel pair of morphisms between two abelian groups, then their coequalizer is the quotient of by the relation in both sets and abelian groups.

(Note that restricting attention to abelian groups is crucial here: for groups, the coequalizer of a pair of maps is the quotient of by the normal closure of the subgroup generated by , and so the above argument fails for two reasons: first, the elements don’t form a subgroup in general, and second, they don’t form a normal subgroup in general.)

For functors between abelian categories there are some other equivalent conditions which are perhaps even more familiar, and which are the origin of the “right” part of “right exact.”

**Proposition:** Let be a linear functor between abelian categories. The following conditions are equivalent:

- is right exact.
- preserves epimorphisms.
- If is an exact sequence, then so is .

A subtlety in what follows is that there are various equivalent conditions for a sequence to be exact at an object which are equivalent in abelian categories but not in general. We won’t spend time on this, though; exact sequences are really only reasonable to talk about in abelian categories anyway.

*Proof.* : Recall that a morphism is an epimorphism if and only if its cokernel pair is trivial, or explicitly if it is equipped with the identity maps . This condition is preserved by any functor which preserves pushouts, hence by any functor which preserves finite colimits.

: exactness at means that the cokernel of is trivial, and this follows since the cokernel of is trivial by exactness at and is right exact. Exactness at is equivalent to the condition that the cokernel of is , and this again follows since the cokernel of is by exactness at and is right exact.

: if is an epimorphism, then taking its kernel we have a short exact sequence . By hypothesis, is exact, so is also an epimorphism.

: If is a morphism, then taking its cokernel gives an exact sequence

and by hypothesis, applying to this exact sequence gives an exact sequence

.

By exactness at , , and hence preserves cokernels.

**Projective objects**

There are several equivalent ways to define projective objects in an abelian category; six of them are provided by the above, while a seventh is new.

**Definition-Theorem:** An object in an abelian category is a **projective object** if it satisfies any of the following equivalent conditions:

- is right exact (hence exact).
- preserves coequalizers.
- preserves pushouts.
- preserves cokernels.
- preserves epimorphisms; explicitly, if is a morphism and is an epimorphism, then factors through .
- If is a short exact sequence, then so is .
- Every short exact sequence splits.

Dually, an object in an abelian category is an **injective object** if it satisfies any of the above equivalent conditions in the opposite category.

(It’s crucial for this theorem that is thought of as taking values in abelian groups rather than sets.)

*Proof.* is a linear functor between linear categories, so we know that the first four conditions are equivalent by the above. It is even a linear functor between abelian categories, so we know that the first six conditions are equivalent by the above. It remains to show that the seventh condition is equivalent to the others. There are a few ways to do this.

: if is exact, then in particular is an epimorphism. By hypothesis, the identity factors through it, and so the short exact sequence splits.

: if is exact, then by hypothesis so is

and in particular is an epimorphism, so there exists some mapping to . This condition is precisely the condition that is a splitting of the short exact sequence.

: let be a map and be an epimorphism. Then factors through iff the projection map

from the pullback of and to has a section (by the universal property). In an abelian category, epimorphisms are stable under pullback, so is also an epimorphism, and hence it fits into a short exact sequence

which, by hypothesis, splits.

*Example.* In the abelian category of modules over a ring , every free module is a projective module. It’s easy to verify, for example, that every short exact sequence ending in a free module splits since one can choose a splitting on each generator using freeness (although for arbitrary free modules we need the axiom of choice to do this).

There’s a sense in which the above argument mostly does not take place in , but in . To formalize this, we’ll repackage this argument as follows:

- First, find a definition of projective object in an arbitrary category, not necessarily abelian, with the property that every set is a projective object (assuming the axiom of choice).
- Second, show that the free module functor sends projective objects to projective objects.

We have seven equivalent definitions to choose from. Of them, only definitions 1, 2, 3, and 5 make sense in an arbitrary category. The first three can be ruled out because they don’t have the correct behavior in : for example, the only set such that preserves even finite coproducts (let alone pushouts or finite colimits) is the one-element set. The definition we’ll use is Definition 5.

**Definition:** An object in a category is a **projective object** if preserves epimorphisms.

(Here it’s not crucial, in an abelian category, that is thought of as taking values in abelian groups; we can think of it as taking values in sets for the purposes of this definition. In general these two conditions may behave differently in an enriched category depending on whether one takes the ordinary hom or the enriched hom in the first condition, but we won’t deal with any such cases.)

It now follows, assuming the axiom of choice, that every set is projective. In fact this statement can be shown to be equivalent to the axiom of choice!

The statement that free modules are projective now follows from the following observation.

**Proposition:** Let be a functor with a right adjoint . If preserves epimorphisms, then preserves projective objects.

*Proof.* Let be a projective object. Then, by hypothesis,

.

The RHS is a composite of two functors that preserve epimorphisms, so it also preserves epimorphisms; hence the LHS also preserves epimorphisms.

**Corollary:** The free -module functor preserves projective objects; hence, assuming the axiom of choice, every free -module is projective.

*Proof.* has right adjoint the forgetful functor , so it suffices to show that the forgetful functor preserves epimorphisms. But the epimorphisms of -modules are precisely the morphisms with trivial cokernel, which in turn are precisely the morphisms which are surjective on underlying sets. One way to see this is to show, as we did above for abelian groups, that the forgetful functor preserves coequalizers.

**Corollary:** Let be a commutative ring and let be projective modules over . Then the tensor product is also projective.

*Proof.* The tensor-hom adjunction here reads

.

By hypothesis, is projective, so preserves epimorphisms. It follows that preserves projective objects.

In fact, we are surprisingly close to describing all projective modules. Next we need the following observation. Recall that an object is a retract of an object if there are maps such that (so is a split epimorphism and is a split monomorphism). In an abelian category, this condition just means that is a direct summand of .

**Proposition:** A retract of a projective object is projective.

*Proof.* Let be a projective object, be an object, and be maps satisfying , so that is a retract of . Let be an epimorphism. By hypothesis, every map factors through , and we want to show that every map similarly factors through .

Given a map , by hypothesis the composite

factors through . Since , it follows that the composite

also factors through , as desired.

**Corollary:** A retract / direct summand of a free module is a projective module.

*Example.* Let be the ring of continuous functions on a compact Hausdorff space . Any vector bundle on is a direct summand of a trivial vector bundle, so the -module of continuous sections of a vector bundle is a finitely generated projective -module, and in fact by the Serre-Swan theorem this is an equivalence of categories.

This construction already exhausts all projective modules.

**Theorem:** The projective modules over a ring are precisely the retracts of free modules. If is finitely generated, then the free module can also be taken to be finitely generated.

*Proof.* Let be a projective module and let be a free module equipped with an epimorphism (so that is sent to a set of generators of ). Then fits into a short exact sequence

and by hypothesis this short exact sequence splits, so is a retract of as desired.

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.

Then it’s not hard to see that this problem is equivalent to the problem of finding -algebra homomorphisms from to . This is equivalent to the problem of finding left inverses to the morphism

of commutative rings making an -algebra, or more geometrically equivalent to the problem of finding right inverses, or **sections**, of the corresponding map

of affine schemes. Allowing to be a more general scheme over can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the **section problem** – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.

**Preliminaries: morphisms as families**

It will be useful to keep the following intuition in mind throughout this post: in a category of some sort of spaces, a morphism should be thought of as a family or bundle of spaces varying over the base space , and anything one does for spaces one can try to do for families of spaces. Here by I mean a suitable pullback, namely the pullback of a diagram of the form , where is any object in the category in question serving as a point. From this perspective, trying to find a section of can be thought of as finding a “continuous” choice of point in each space in this family: it can be thought of as the families version of the problem of finding a point in a space.

This is Grothendieck’s relative point of view, which was perhaps first made famous via the Grothendieck-Riemann-Roch theorem in algebraic geometry. This is the families version of the Hirzebruch-Riemann-Roch theorem. But there are simpler examples of the relative point of view.

*Example.* A covering map should be thought of as a locally constant family of sets varying over . This idea can be made precise as the following restatement of the classification of covering spaces: for reasonable base spaces (locally contractible should be enough; there’s no need to require that or its covers be path-connected), there is an equivalence of categories between

- covering spaces of and covering maps, and
- functors from the fundamental groupoid of to and natural transformations

where the equivalence is given in one direction by monodromy: we send a covering space to the functor sending a point to the fiber and we send a path between points to map of sets given by taking the unique lift of to a path in starting at a point in and then evaluating it at to obtain a point in .

This expanded version of the classification of covering spaces, where we do not restrict to path-connected bases or path-connected covers, results in a category with much better formal properties than the category of path-connected covers of a path-connected base; for example, we can now take coproducts and products of covering spaces, which correspond to taking disjoint unions and fiber products respectively. In fact, the equivalence above makes it clear that anything we can do to a family of sets we can do fiberwise to a family of covering spaces.

(This is a good place to really see fiber products earn their names: thinking of a morphism in terms of its fibers makes it clear that taking the fiber product of two such morphisms amounts, on fibers, to literally taking the products of the fibers, thanks to the fact that limits commute with limits.)

**Preliminaries: the function field analogy**

The analogy we’re implicitly trying to make here can be thought of as a relative of the function field analogy. Already it’s interesting to think about the function field analogue of solving Diophantine equations, e.g. finding solutions in to systems of polynomial equations with coefficients in , where is a field. Geometrically such a thing defines a map

which, from the relative point of view, we should think of as a family of affine varieties over the affine line, and finding a solution to the corresponding Diophantine equation amounts to “continuously” choosing a point in each of these varieties. When , we can even equip the corresponding complex affine varieties with their analytic topologies, and then ask for topological obstructions for the corresponding map of topological spaces to admit a continuous section; such obstructions also obstruct the original map of varieties having an algebraic section.

*Example.* Sections of the map

encode solutions to the Diophantine equation , where we want to find solutions . This equation of course fails to have a solution, but it fails to have a solution for several reasons which generalize to more complicated situations.

First, the fiber over each -point of the affine line is the affine scheme whose points over are the solutions to in , so if there are any with no square roots then there is a local obstruction to the existence of a solution to in . In more number-theoretic language, an obstruction to the existence of a solution is the existence of a solution .

But this is not enough: even if is algebraically closed, there are still no solutions. There is a further problem that is necessarily divisible by an even number of times, while is divisible by once (which is odd). Equivalently, the problem is that there is no solution , or more geometrically, that the map above, which can equivalently be described as the squaring map

,

fails to be surjective on Zariski tangent spaces at . Yet another description of the problem is that although there exists a solution locally at , that local solution cannot be extended to a formal neighborhood of .

Moreover, even if we delete by localizing away from it to get a map

then there is still no section / solution. One way to describe the problem is that although we can no longer talk about solutions , we can still talk about solutions in formal Laurent series in , and looking at -adic valuations we see that there aren’t any such solutions. Equivalently, we are looking at solutions in a formal neighborhood of the deleted point even though we can no longer look at the deleted point itself.

There is a related global topological obstruction in the case that , which is that we get an induced map on the punctured complex line

which induces multiplication by on , and this map has no section (in particular, is not surjective) so the original map cannot either.

**Examples: associated bundles of vector bundles**

In this section we’ll describe a large source of interesting examples of section problems in topology coming from vector bundles.

Let be a vector bundle on a base , for example the tangent bundle of a smooth manifold. From we can construct various associated bundles whose sections, if they exist, have interesting meanings in terms of . (The problem of classifying sections of itself is also interesting, but the problem of determining whether they exist is not, since the zero section always exists.)

*Example.* If denotes the fiber over , then removing the zero section from gives a bundle over whose fiber over is and whose sections are precisely nonvanishing sections of . More generally, there is an associated bundle whose fiber over is linearly independent -tuples and whose sections are precisely -tuples of (pointwise) linearly independent sections of . Already the problem of describing the largest for which this is possible for the tangent bundles of spheres is an extremely interesting problem, solved by Adams in 1962 using topological K-theory. For example, the only spheres for which it is possible to construct the maximum possible number of linearly independent vector fields, namely , occur when ; they can be constructed using the fact that these are precisely the unit spheres in the complex numbers, the quaternions, and the octonions respectively.

Characteristic classes give obstructions to finding such sections: using the fact that the total Stiefel-Whitney resp. Chern class is multiplicative under direct sum, it’s not hard to show that if a real resp. complex vector bundle of dimension admits linearly independent sections then its top Stiefel-Whitney classes resp. Chern classes vanish. Similarly, if is a real oriented vector bundle and it admits a single nonvanishing section then its Euler class vanishes.

*Subexample.* For the tangent bundles of oriented smooth closed manifolds, where the Euler class evaluates to the Euler characteristic, the last observation above reproduces the PoincarÃ©â€“Hopf theorem and shows that the even-dimensional spheres don’t admit nonvanishing vector fields. Applied to , we reproduce the hairy ball theorem.

*Example.* If is a real vector bundle of even dimension , then there is an associated bundle whose fiber over is the space of complex structures on (that is, the space of ways to equip with the structure of a complex vector space). Explicitly, this is the space of automorphisms such that , topologized as a subspace of with the usual Euclidean topology. acts transitively on the space of complex structures on , with the stabilizer of a fixed complex structure (coming from a fixed identification isomorphic to . Similar remarks apply in the presence of a Riemannian metric on and hence the space of complex structures can be identified as a homogeneous space

.

Sections of the corresponding bundle then correspond, unsurprisingly, to complex structures on (that is, ways to equip with the structure of an -dimensional complex vector bundle). When is the tangent bundle of , equipping with a complex structure is in turn an obstruction to equipping with the structure of a complex manifold; a manifold which has the weaker structure of a complex structure on its tangent bundle is called an almost complex manifold, and the distinction between the two is given by the Newlander-Nirenberg theorem.

Characteristic classes also give obstructions to finding complex structures: as we saw earlier, if a real vector bundle has a complex structure then the odd Stiefel-Whitney classes vanish and the even Stiefel-Whitney classes are reductions of Chern classes ; equivalently, after applying the Bockstein homomorphism , the odd integral Stiefel-Whitney classes vanish. The Pontryagin classes must also satisfy some identities determining them in terms of Chern classes.

Moreover, since any symplectic manifold admits a compatible almost complex structure, any obstruction to having an almost complex structure is also an obstruction to having a symplectic structure.

*Subexample.* This is another problem that is already interesting for spheres. First, using Pontryagin classes we can show that the spheres don’t admit almost complex structures, as follows. If admitted an almost complex structure, then it would have Chern classes, although all of them except automatically vanish. This last Chern class does not vanish since it must be equal to the Euler characteristic , where we identify via an orientation. We know that we can express the Pontryagin classes of a complex vector bundle in terms of its Chern classes, and here that gives us a top Pontryagin class of

.

On the other hand, since all spheres are stably parallelizable, all of their Pontryagin classes must vanish; contradiction.

Next, an argument relying on stronger tools in fact shows that doesn’t admit an almost complex structure for . Namely, the following version of the Hirzebruch-Riemann-Roch theorem can be deduced from the Atiyah-Singer index theorem: if is a closed almost complex manifold and is a complex vector bundle on , then

is the index of a certain Dirac operator, and hence is an integer. Here is the Chern character of while denotes the Todd class. If admits an almost complex structure, then and are nonvanishing only in bottom and top degrees, since in all other degrees the relevant cohomology groups vanish, and so the above expression reduces to

where and denote the components of the Chern character and Todd class respectively in . Applying the index theorem twice, first with a trivial bundle, we conclude that the Todd genus is an integer and hence that

for all complex vector bundles . Now taking to be the tangent bundle of itself, and using the fact that we know that all of the Chern classes vanish except the top class , which as above must be twice a generator of , we compute (e.g. using the splitting principle) that and hence that

from which it follows that , so as desired.

In fact the intermediate result above, that for a -dimensional complex vector bundle on the top Chern class is divisible by , is true for all and is due to Bott; see this blog post by Akhil Mathew for an alternate proof using K-theory. The proof above can be salvaged using a stronger version of the Hirzebruch-Riemann-Roch theorem: it suffices for to have a -structure, which unlike an almost complex structure every sphere possesses.

Since odd-dimensional manifolds can’t admit almost complex structures, the only spheres we haven’t ruled out at this point are and . has a complex structure coming from its identification with the complex projective line , while has an almost complex structure coming from its identification with the unit imaginary octonions. It is a major open problem to determine whether admits a complex structure; see, for example, this MO question.

**Some categorical remarks**

Recall that if a morphism has a section, or equivalently a right inverse, then it is called a split epimorphism, and in particular it is an epimorphism. Recall also the following two equivalent alternative definitions of a split epimorphism:

- A split epimorphism is a morphism which is an absolute epimorphism in the sense that if is any functor, then is an epimorphism;
- A split epimorphism is a morphism which is surjective on generalized points in the sense that for any other object , the induced map is surjective.

Another way of restating the second definition which is particularly amenable to topological thinking is that a split epimorphism is a map such that any map lifts to a map along .

Both of these equivalent definitions give several straightforward obstructions for a map of spaces to admit a section. For example, applying homology functors, we get that the induced maps on homology must admit sections: this happens iff is surjective and the short exact sequence

splits. Similarly, the induced maps on fundamental groups (with some choice of basepoint) must admit sections, and again this happens iff is surjective and the short exact sequence

splits. If is a smooth map between smooth manifolds, then must be a submersion. And so forth.

**Two Hasse principles**

In this section the term “Hasse principle” will mean a necessary condition for a section to exist which is roughly of the form “in order for a section to exist, it must exist locally,” analogous to the statement that in order for a Diophantine equation to have a solution over it must have a solution over all completions . The term “Hasse principle holds” means the stronger statement that this condition is also sufficient, which won’t hold for most of our examples (much as the condition in the Hasse principle itself isn’t sufficient for most Diophantine equations).

The simplest thing that could be called a topological Hasse principle is the **pointwise Hasse principle**: in order for a map to have a section , the fiber over every point must be nonempty, since for all . Equivalently, must be surjective. Intuitively, for a section to exist, it must first exist locally in the most local possible sense, namely pointwise. The number-theoretic analogue is that in order for a Diophantine equation with integer coefficients to have solutions over it must have solutions over for all .

The pointwise Hasse principle is very weak. Its hypothesis is always satisfied for fiber bundles, and in particular is always satisfied for covering maps. But a nontrivial covering map (say path-connected, with a path-connected base) never has a section because the induced map on fundamental groups is not surjective with any choice of basepoints, and so cannot have a section.

(Note, however, that “a map of sets has a section iff it’s surjective” is equivalent to the axiom of choice, and hence we can think of the axiom of choice as asserting that the pointwise Hasse principle holds for sets.)

But there are even simpler examples involving no algebraic topology: consider the map

where denotes the disjoint union and so there are two copies of in the codomain, and where restricts to the obvious inclusion on each connected component of the codomain. This map has no section despite the fact that the base is contractible and the induced map on is surjective, so no homotopy-invariant argument can detect this fact.

In the above example not only fails to have a section defined on all of , but in fact it fails to have a section defined on any neighborhood of . This suggests the following construction. Starting from a map we can build a **sheaf** on whose sections over an open subset (in the sheaf sense) consist of sections of (in the right-inverse sense) over :

.

The problem of finding a section of is then equivalent to the problem of finding a **global section** of the sheaf . This sheaf is a convenient way of encoding the local-to-global aspects of this problem.

does not allow us to recover the data of the fibers of . The next best thing we can do is to look at the **stalks** at each point , defined as a cofiltered limit

over for all containing . Equivalently, consists of equivalence classes of sections of over an open neighborhood of modulo the equivalence relation of being equal in some possibly smaller open neighborhood of ; these are the **germs** of sections of at .

Looking at stalks gives us a **stalkwise Hasse principle**: in order for to have a section, each stalk must be non-empty. Equivalently, for every there must be a section of defined on some open neighborhood of . A number-theoretic analogue is looking at solutions over the -adics rather than just looking at solutions (although this involves looking at formal neighborhoods rather than, say, open neighborhoods in the Zariski topology), so we’re getting closer to the actual Hasse principle.

The stalkwise Hasse principle successfully detects that the map

has no section, since the stalk at is empty: equivalently, has no section defined in a neighborhood of . But a slight modification of this example defeats even the stalkwise Hasse principle: consider now the map

.

Here the problem is that there is a unique section over , and similarly a unique section over , but these two sections don’t agree on their intersection . And again the base is contractible.

Like the pointwise Hasse principle, the hypothesis of the stalkwise Hasse principle is also satisfied for all fiber bundles. So even in fairly straightforward examples we see that there are many global obstructions for sections to exist. In that light the fact that the usual Hasse principle holds for quadratic forms is quite surprising.

**A Brauer-Manin obstruction**

Suppose that is a map and is an open cover of the base for which we’ve found, on each open , a local section . (This is equivalent to the hypothesis of the stalkwise Hasse principle.) Then to check whether the glue to a section it remains to check whether they agree on intersections in the sense that

where .

Now suppose that for whatever reason we don’t want to or can’t do this, but that we understand the cohomology of all of the spaces involved fairly well. Then we can do the following instead: each induces a map

which gives us a pairing

from which we can build a pairing

.

In other words, given a family of local sections, we can pull a cohomology class in back along all of the local sections to get a family of cohomology classes in . But there are restrictions on what families of cohomology classes we can get in this way: if is a global section, then it induces a map

which lets us construct a pairing

and this pairing and the above pairing fit into a commutative square

expressing the following restriction: if the glue together to (equivalently, are induced by restriction from) a global section , then pairing the with a cohomology class in gives a family of cohomology classes in which glue together to (equivalently, are induced by restriction from) a cohomology class in . In other words, there is a pairing

and a necessary condition for a family of local sections to glue to a global section is that the family must pair to zero with every class in .

This is what might be called a topological **Brauer-Manin obstruction**. The number-theoretic analogue, namely the usual Brauer-Manin obsruction, comes from making the following substitutions to the above picture.

First, is (for simplicity), is a variety over (for example, a smooth projective algebraic curve defined by equations with rational coefficients), and the are where runs over all primes, including the “infinite prime” , where . So the situation is that we want to find rational points on the variety , we’ve found points over for all primes , and we’d like to write down a cohomological obstruction to them gluing together to a rational point.

Next, is the Brauer group of a scheme ; for this is the Brauer group of in the usual sense, and is equivalently the Galois cohomology group

or the etale cohomology group

.

In general the Brauer group is some torsion subgroup of . In particular, , like , is a contravariant functor in .

This wouldn’t be a useful thing to write down if we didn’t know the Brauer groups of the relevant fields, but in fact we do (this is part of class field theory): are the Brauer groups , which are equal to when is finite and when , and is the Brauer group , which fits into a short exact sequence

.

(In particular, the pullback of an element of to is nontrivial for only finitely many primes .) Letting denote the -points of the variety , the number-theoretic analogue of the pairing we constructed above is a pairing

and the Brauer-Manin obstruction is the necessary condition for a collection of points in to lift to a point in that this pairing must be zero for every element of . I am told that there are examples of curves for which each is non-empty but where the Brauer-Manin obstruction does not vanish, and examples of higher-dimensional varieties for which each is non-empty and the Brauer-Manin obstruction does not vanish but there are still no rational points.

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**Definition:** The **Picard group** of is the group of isomorphism classes of -modules which are invertible with respect to the tensor product.

By invertible we mean the following: for there exists some such that the tensor product is isomorphic to the identity for the tensor product, namely .

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme ).

**Some propositions**

**Proposition:** An invertible module is finitely presented and projective.

*Proof.* If is invertible, then the functor is an autoequivalence of categories with inverse , and consequently it preserves all categorical properties; in addition, it sends to , so it follows that any categorical property of is also a categorical property of . In particular, since being projective is a categorical property (namely the property that is exact) and is projective, is also projective.

Less obviously, being finitely presented is also a categorical property: an -module is finitely presented iff preserves filtered colimits (that is, is a compact object).

We can avoid appealing to this fact with the following more hands-on argument. By assumption, . Let

be the element of representing . Then the map

is surjective, where denotes , as can be seen by setting . Surjective morphisms of -modules are precisely the epimorphisms, and this is a categorical property, so tensoring with we get an epimorphism , from which it follows that is finitely generated (by the elements ), and for a projective module this is equivalent to being finitely presented.

**Proposition:** Being invertible is preserved under extension of scalars. More precisely, if is a morphism of commutative rings, then the extension of scalars functor

sends invertible modules to invertible modules, and in fact induces a homomorphism .

*Proof.* It suffices to show that extension of scalars is a monoidal functor. More or less this boils down to having a natural isomorphism

.

But by the associativity of the tensor product, the RHS is just

and , so the conclusion follows using the commutativity of the tensor product of modules over commutative rings.

**Theorem:** The following conditions on an -module are equivalent.

- is invertible.
- is locally free of rank : that is, for every prime ideal of , the localization is a free -module of rank .
- The natural map is an isomorphism, where is the dual module. In this case .

*Proof.* 1) 2): Since localization is a special case of extension by scalars, remains invertible, hence is in particular finitely presented projective. We can give an independent argument that this is true as follows: since localization is exact, it preserves finite presentations, so is finitely presented. Since we have a tensor-hom adjunction

it follows that if is exact then so is , hence localization preserves projectivity.

But since is a local ring, it follows that must be free, and since rank is multiplicative under tensor product, must be free of rank : that is, we must have .

2) 3): being an isomorphism is a local property, so the natural map is an isomorphism iff its localizations are. But if is locally free of rank then for all .

3) 1): by definition.

The proposition shows in particular that invertibility is a local condition: is invertible iff is invertible for all . We still haven’t given any interesting examples of invertible modules, though.

**The ideal class group**

**Proposition:** Let be an integral domain with fraction field . Then every invertible module over is isomorphic to a fractional ideal of .

*Proof.* is projective, hence in particular torsion-free, so the natural inclusion is an embedding. Since is finitely generated, we can multiply the image of this embedding by the product of the denominators of the images of its generators in , and we conclude that an -multiple of the image of lands in as desired.

**Proposition:** Let be a Dedekind domain. Then a finitely generated module over is projective iff it is torsion-free.

*Proof.* Projectivity is a local property, so is projective iff is projective for all . If is torsion-free, then so is . Since the localizations are all DVRs, hence in particular PIDs, it follows by the structure theorem for finitely generated modules over a PID that each is free, hence in particular projective.

**Proposition:** Let be two fractional ideals of a Dedekind domain . Then .

*Proof.* WLOG are ideals. We want to show that the natural surjection

is an isomorphism. Since are fractional ideals, they are torsion-free, hence projective, hence flat, so embeds into . Any element in the kernel of the natural surjection above must therefore also be in the kernel of the natural map , but this natural map is an isomorphism.

**Theorem:** The Picard group of a Dedekind domain is canonically isomorphic to its **ideal class group** of invertible fractional ideals modulo principal invertible fractional ideals. In particular, is nontrivial iff is not a UFD.

*Proof.* By the above proposition, any invertible fractional ideal gives rise to an invertible module, and moreover multiplication of fractional ideals corresponds to tensor product of ideals. The kernel of this map consists of invertible fractional ideals which are isomorphic to the trivial module, which is precisely the principal invertible fractional ideals; hence we get an injection from the ideal class group to . We also showed that every invertible module comes from a fractional ideal, necessarily also invertible, so this injection is a surjection and hence a bijection.

*Example.* Let be the ring of integers of the number field . This is a Dedekind domain which is not a UFD because of the non-unique factorization

.

Here have norms respectively. An examination of the norm form reveals that has no elements of norm or , hence all four of these elements are irreducible. The factorization above refines to unique prime ideal factorizations

which gives and in the ideal class group. Because has norm and there exist no elements of of norm , we also know that in the ideal class group.

By the Minkowski bound, the ideal class group is generated by ideals of norm at most

and since is the only prime ideal lying over , the ideal class group must be generated by . Hence we compute the Picard group to be

.

This example turns out to be minimal in the sense that is the number field of smallest discriminant (in absolute value) whose ideal class group is nontrivial.

*Example.* Let be the ring of functions on a smooth affine curve

in the complex plane, and let denote its projective closure in the complex projective plane. Then ideals of can be identified with effective divisors on , and principal ideals of can be identified with effective principal divisors. Since meromorphic functions on are quotients of functions in , it follows that is canonically isomorphic to the divisor class group of , which is closely related to the divisor class group of , which is in turn very well-understood and which we will turn to later. The relationship is the following: restriction of divisors gives a surjection

(a priori it only gives a surjection on divisors, but since and have the same meromorphic functions, the natural map on divisors respects the quotient by principal divisors). The kernel of this map clearly contains the subgroup of generated by the points in , and in fact it must be precisely this subgroup: if is a divisor in the kernel, then is the divisor of some function on (that is, some element of ), but extends to a meromorphic function on and hence has a principal divisor whose restriction to is precisely . If there are points in , then we have an exact sequence

.

As we’ll see later, if has genus then

where denotes the Jacobian and denotes a torus of (real) dimension . In particular, is uncountable as soon as , and hence its quotient by the image of is nontrivial as soon as .

**The topological Picard groups**

The characterization of invertible modules as locally free modules of rank suggests that invertible modules over a commutative ring should be thought of as (modules of sections of) **line bundles** on . This idea is strongly supported by variants of the Serre-Swan theorem, such as the following.

**Theorem:** Let be a compact Hausdorff space and let resp. be the ring of continuous real-valued resp. complex-valued functions on . Assigning a real resp. complex vector bundle on its module of continuous sections gives an equivalence of monoidal categories between real resp. complex vector bundles on and finitely presented projective modules over resp. .

**Corollary:** resp. is canonically isomorphic to the abelian group of topological real resp. complex line bundles on , which is in turn canonically isomorphic to resp. .

This theorem suggests a natural definition of the real resp. complex Picard groups of an arbitrary space , not necessarily compact Hausdorff, namely the group of isomorphism classes of real resp. complex line bundles on .

*Example.* Let ; this is arguably the simplest example of a space with a nontrivial real line bundle over it. Since , there are exactly two (isomorphism classes of) real line bundles over , one trivial and one nontrivial. The nontrivial line bundle is the MÃ¶bius bundle. Its -module of continuous sections can be identified with the function space

where itself is thought of as the function space

and the module structure is given by pointwise multiplication. (So here we are thinking of as the quotient .)

*Example.* Let ; this is arguably the simplest space with a nontrivial complex line bundle over it. Since , there are countably many (isomorphism classes of) complex line bundles over , all of which are powers of a single generator.

Thinking of as the complex projective line , there are two choices for such a generator, one given by the tautological bundle which assigns to a point in the complex line in it represents, and the other given by its dual ; the other line bundles are given by , where if is negative then as expected.

The bundles are important in algebraic geometry because their spaces of algebraic (or equivalently, holomorphic) sections are precisely the homogeneous polynomials of degree . Their -modules of continuous sections can be identified with the function spaces

where itself is thought of as the function space

and, as above, the module structure is given by pointwise multiplication. (So here we are thinking of as the quotient .)

We can make the construction for look more like the construction for by thinking of as the real projective line and exhibiting it as the quotient of by the action of .

**The algebraic and analytic Picard groups**

The discussion of above blurred the distinction between topological, holomorphic, and algebraic line bundles, so it’s worth making the distinction in general.

To talk about topological vector bundles on a space only requires that it be equipped with a topology. To talk about holomorphic vector bundles requires that be equipped with the structure of a complex manifold. Finally, to talk about algebraic vector bundles requires that be equipped with the structure of a scheme, e.g. might be a complex variety. We can talk about all three on a smooth complex variety. On all three notions coincide, but in general they all differ.

The GAGA principle implies that the classifications of holomorphic and algebraic vector bundles on a smooth projective complex variety coincide; in particular, the classifications of holomorphic and algebraic line bundles on coincide. However, it is not true in general that these classifications also coincide with the classification of topological line bundles, although it is true for the complex projective spaces . In general, if is a complex manifold, then the exponential sheaf sequence

gives rise to a long exact sequence in sheaf cohomology

where turns out to be the Picard group of holomorphic line bundles on and the connecting homomorphism to sends such a line bundle to its first Chern class , which completely determines the underlying topological line bundle but not its holomorphic structure.

The classifications of holomorphic and topological line bundles on coincide iff this connecting homomorphism is an isomorphism. By exactness, this is guaranteed if , which in particular holds if is a Stein manifold (e.g. a smooth affine variety) by Cartan’s theorem B. More generally, the Oka-Grauert principle asserts that the classifications of holomorphic and topological vector bundles on a Stein manifold coincide.

But and are both nontrivial in general, so we can’t expect the holomorphic and topological classifications to agree in general. And because the holomorphic and topological classifications agree on smooth affine varieties, a smooth affine variety on which the algebraic and topological classifications disagree also shows that the algebraic and holomorphic classifications disagree in general.

*Example.* On a smooth projective curve of genus , the divisor class group turns out to be naturally isomorphic to the Picard group, with the first Chern class map corresponding to the degree map

under the isomorphism given by pairing with the fundamental class. Hence the degree zero divisor class group is naturally isomorphic to the Picard group of line bundles with vanishing first Chern class. This group measures the difference between the holomorphic / algebraic and the topological classifications of line bundles on .

An inspection of the long exact sequence associated to the exponential sequence shows that this group is in turn isomorphic to the quotient

which is one description of the Jacobian. As we saw previously, , and we know that . This exhibits as a complex torus (at least provided that we show that the image of in is a lattice). In particular, once , the holomorphic / algebraic and topological classifications of line bundles on disagree.

*Example.* As when we were discussing Dedekind domains, let be an elliptic curve minus a point. Then is a smooth affine variety, hence in particular a Stein manifold, so the classifications of holomorphic and topological line bundles agree: both Picard groups are isomorphic to , which vanishes since , being topologically a torus minus a point, deformation retracts onto its -skeleton. But the classification of algebraic line bundles is given by the divisor class group, which we saw earlier was uncountable. In particular, the holomorphic and algebraic classifications of line bundles on disagree.

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**Proof 1: PoincarÃ© duality**

A corollary of PoincarÃ© duality is that if is a closed orientable manifold of dimension , then the Betti numbers satisfy . When is odd, this implies that the Euler characteristic

is equal to zero, since . In fact slightly more is true.

**Proposition:** Let be a closed manifold of dimension , not necessarily orientable. If is odd, then . If is even and is a boundary, then .

*Proof.* When is odd, let be the orientable double cover of , so that . By PoincarÃ© duality, , so the same is true for . Alternatively, because the Euler characteristic can also be calculated using the cohomology over , we can also use PoincarÃ© duality over , which holds for all closed manifolds since all closed manifolds have fundamental classes over .

When is even and is the boundary of a compact manifold , let be the manifold obtained from two copies of by gluing along their common boundary. Then is a closed odd-dimensional manifold, hence . But

(e.g. by an application of Mayer-Vietoris), from which it follows that .

**Corollary:** The Euler characteristic of is even.

*Proof.* is the boundary of the solid -holed torus.

**Corollary:** No product of the even-dimensional real projective spaces is a boundary.

*Proof.* Since and is double covered by , we have , hence any product of even-dimensional real projective spaces also has Euler characteristic , which in particular is odd.

**Corollary:** The Euler characteristic is a cobordism invariant.

*Proof.* Let be two closed manifolds which are cobordant, so that there exists a closed manifold such that . Then , hence .

In addition to satisfying , the Euler characteristic also satisfies (e.g. by the KÃ¼nneth theorem). It follows that the Euler characteristic is a genus of unoriented manifolds, or equivalently that it defines a ring homomorphism

where is the unoriented cobordism ring and is the Thom spectrum for unoriented cobordism. This is arguably the simplest example of a genus.

*Warning.* The Euler characteristic itself is not a genus because it is not a cobordism invariant. For example, is a boundary, hence cobordant to the empty manifold, but . There is an integer-valued genus lifting the Euler characteristic on oriented manifolds, although it is not the Euler characteristic but the signature

where is the oriented cobordism ring and is the Thom spectrum for oriented cobordism.

**Proof 2: PoincarÃ© duality again**

Let be a closed oriented manifold of even dimension . Then the cup product defines a pairing

on middle cohomology which is nondegenerate by PoincarÃ© duality, symmetric if is even, and skew-symmetric if is odd. Previously we used this pairing when and over to understand 4-manifolds. When is odd we can say the following.

**Proposition:** With hypotheses as above (in particular, odd), the Betti number is even.

*Proof.* On the cup product pairing is a symplectic form, and symplectic vector spaces are even-dimensional. (This follows from the fact that by induction on the dimension, every symplectic vector space has a symplectic basis, namely a basis such that and . This is a pointwise form of Darboux’s theorem.)

**Corollary:** The Euler characteristic of a closed orientable manifold of dimension is even. In particular, the Euler characteristic of is even.

*Proof.* As above, let . In the sum

every term is canceled by the corresponding term by PoincarÃ© duality, except for the middle term , which we now know is even.

*Remark.* Although this proof also uses PoincarÃ© duality and has the same conclusion as the previous proof, it proves a genuinely different fact about manifolds: on the one hand, it only applies to manifolds of dimension and requires orientability over and not just over , but on the other hand it applies in principle to manifolds which are not boundaries.

Going back to the particular case of surfaces , we can even write down a fairly explicit choice of symplectic basis for as follows: thinking of as a -holed torus, hence equivalently as the connected sum of tori, we can write down the usual basis of the first homology of the torus. Together these give the standard choice of generators of the fundamental group , as well as of the first homology , and their PoincarÃ© duals in form the symplectic basis we want by the standard relationship between intersections and cup products.

The symplectic structure on is a shadow of a more general construction of symplectic structures on character varieties of surfaces; these are moduli spaces of flat -bundles with connection on . The connection is that is the tangent space at the identity of the moduli space of flat (unitary, complex) line bundles on . These moduli spaces are what classical Chern-Simons theory assigns to , and applying geometric quantization to these moduli spaces is one way to rigorously construct quantum Chern-Simons theory.

**Proof 3: characteristic classes (and PoincarÃ© duality)**

For a closed surface , the Euler characteristic is equivalently the Stiefel-Whitney number , where is the second Stiefel-Whitney class and is the -fundamental class, which, as above, exists whether or not is orientable. In general, the top Stiefel-Whitney class of an -dimensional real vector bundle is its Euler class.

Proof 1 showed that this Stiefel-Whitney number is a cobordism invariant; in fact every Stiefel-Whitney number is a cobordism invariant, although we will not use this. In any case, to show that the Euler characteristic of is even when is orientable it suffices to show that .

**Proposition:** Let be a closed orientable surface. Then .

*Proof.* We will again appeal to the relationship between the Stiefel-Whitney classes and the Wu classes . Since is orientable, , so , where represents the second Steenrod square in the sense that

where denotes the cup product of and . But vanishes on classes of degree less than , so above, hence (by PoincarÃ© duality ) as well.

**Corollary:** The Euler characteristic of is even.

**Corollary:** admits a spin structure.

*Remark.* Atiyah observed that spin structures on turn out to be equivalent to theta characteristics, after picking a complex structure. See Akhil Mathew’s blog post on this topic for more.

So we’ve shown that all of the Stiefel-Whitney classes of vanish. It follows that all of the Stiefel-Whitney numbers of vanish, and this is known to be a necessary and sufficient criterion for to be a boundary, a fact which we used in Proof 1. Essentially the same argument shows that all of the Stiefel-Whitney classes of a closed orientable -manifold vanish, so all of the Stiefel-Whitney numbers vanish, and we get the less trivial fact that all closed orientable -manifolds are boundaries. We also get that they all admit spin structures.

In the next two proofs we’ll finally stop using PoincarÃ© duality, but now we’ll start using the fact that admits not only an orientation but a complex structure.

**Proof 4: the Hodge decomposition**

Any compact orientable surface can be given the structure of a compact Riemann surface, and so in particular the structure of a compact KÃ¤hler manifold, with KÃ¤hler metric inherited from any embedding into with the Fubini-Study metric. For any compact KÃ¤hler manifold , its complex cohomology has a Hodge decomposition

where is equivalently either the subspace of represented by complex differential forms of type or the Dolbeault cohomology group

.

Here is the sheaf of holomorphic -forms and the cohomology being taken is sheaf cohomology. Moreover, since , the LHS has a notion of complex conjugate, hence we can define the complex conjugate of a subspace, and with respect to this complex structure we have Hodge symmetry: . This implies the following.

**Proposition:** Let be a compact KÃ¤hler manifold (e.g. a smooth projective algebraic variety over ). If is odd, then the Betti number is even.

*Proof.* Let be the Hodge number of . The Hodge decomposition implies that

and Hodge symmetry implies that . When is odd, every term in the above sum is paired with a different term equal to it, hence as desired.

**Corollary:** The Euler characteristic of is even.

*Proof.* As before, we have , and is even by the above.

**Corollary:** Let be a finitely presented group. If has a finite index subgroup such that the first Betti number

of is odd, then cannot be the fundamental group of a compact KÃ¤hler manifold, and in particular cannot be the fundamental group of a smooth projective complex variety.

Fundamental groups of compact KÃ¤hler manifolds are called KÃ¤hler groups; see these two blog posts by Danny Calegari for more.

*Proof.* Since a finite cover of a compact KÃ¤hler manifold is naturally a compact KÃ¤hler manifold, if is a KÃ¤hler group then so are all of its finite index subgroups; taking the contrapositive, if any of the finite index subgroups of are not KÃ¤hler, then neither is . If is any space with , then , hence the former is odd iff the latter is. It follows that if is the fundamental group of a compact KÃ¤hler manifold then is even; taking the contrapositive, we get the desired result.

*Example.* The free abelian groups of odd rank have first Betti number and hence are not KÃ¤hler groups. On the other hand, the free abelian groups of even rank are the fundamental groups of complex tori (e.g. products of elliptic curves).

*Example.* The free groups of odd rank have first Betti number and hence are not KÃ¤hler groups. The free groups of even rank turn out to have free groups of odd rank as finite index subgroups and hence are also not KÃ¤hler.

To see this, first note that if is any free group, then admits finite index subgroups of every possible index because it is possible to write down surjections from into finite groups of every possible size (e.g. cyclic groups). Second, by the standard topological argument every finite index subgroup of is again free because every finite cover of the wedge of circles is a graph and hence homotopy equivalent to a wedge of circles; moreover, by the multiplicativity of Euler characteristics under coverings, if is an index subgroup of then

and hence has subgroups of index and first Betti number

for all . This is odd whenever is even, and in particular when . More explicitly, if is free on generators , then

is a surjection onto a finite group of order , and hence its kernel must be free on generators. One possible choice of generators is

.

**Corollary:** The fundamental groups of compact Riemann surfaces are not free.

There is a great MO question on the topic of why is not free in which this argument is given in the comments. As it happens, that MO question loosely inspired this post.

Above, instead of using Hodge symmetry, we can also do the following. In the particular case of surfaces , we in fact have , hence the two interesting Hodge numbers are

.

In terms of Dolbeault cohomology, this gives

.

Here is the sheaf of holomorphic -forms, or equivalently the structure sheaf of holomorphic functions.

The identity gives us one possible definition of the genus of a compact Riemann surface, namely the dimension of the space of holomorphic forms. In general, if is a complex manifold we can define its geometric genus to be the Hodge number , where is the canonical bundle, hence the dimension of the space of top forms.

The identity can be thought of in terms of Hodge symmetry, but it can also be thought of in terms of Serre duality. On the Dolbeault cohomology groups of a compact complex manifold of complex dimension , Serre duality gives an identification

and hence , which is a different symmetry of the Hodge numbers than Hodge symmetry gives. When is KÃ¤hler, in terms of the Hodge decomposition Serre duality refines PoincarÃ© duality, which only gives

.

In particular, we have

which gives a second proof, independent of Hodge symmetry but still depending on the Hodge decomposition, that is even.

Moreover, since Serre duality is a refinement of PoincarÃ© duality we conclude that is, as a symplectic vector space (as in Proof 2), isomorphic (possibly up to a scalar) to with its standard symplectic structure

where is either or . Hence a complex structure on equips the symplectic vector space with a Lagrangian subspace.

**Digression: the Riemann-Roch theorem**

The motivation for the fifth proof starts from the observation that one way to write down the Riemann-Roch theorem for compact Riemann surfaces is

.

If we can write down a proof of the Riemann-Roch theorem with the genus appearing directly in this form, in terms of half the Euler characteristic, as opposed to the other ways the genus can appear in a formula involving Riemann surfaces (e.g. as the dimension of the space of holomorphic forms), then since all of the other terms are manifestly integers we would get a proof that is even.

Here is a proof which does *not* accomplish this. Let denote the line bundle associated to the divisor . Then

and

since is the divisor corresponding to the canonical bundle and . Now Serre duality gives

and hence we can rewrite the LHS as an Euler characteristic

where we are using that the cohomology of sheaves on vanishes above its complex dimension, namely . This lets us rewrite Riemann-Roch in the form

.

Let and let be a point, so that the meromorphic functions in can have poles of order at most at . Then there is an evaluation map

given by taking the coefficient of where is a local coordinate at ; here denotes the skyscraper sheaf supported at with stalk . The kernel of this evaluation map consists of functions in which have poles of order at most at , which are precisely the sections of the sheaf . Hence we have a short exact sequence of sheaves

.

Since the Euler characteristic of sheaf cohomology is additive in short exact sequences, it follows that

.

Since and, being a skyscraper sheaf, has no higher sheaf cohomology, we have , hence

.

Noting that we also have , by adding and removing points suitably we conclude that if are any two divisors, then

or equivalently that there is a constant such that

for all divisors . To determine it suffices to determine the Euler characteristic of any of the sheaves , which we can do with a second application of Serre duality: for , so that is the structure sheaf, we have

since the holomorphic functions on a compact Riemann surface are constant, and

by Serre duality and the definition of in terms of holomorphic forms. Hence

from which it follows that . This proves Riemann-Roch, but appears as the holomorphic Euler characteristic of rather than as half the topological Euler characteristic like we wanted. The two can be related using the Hodge decomposition, which shows more generally that for a compact KÃ¤hler manifold of complex dimension ,

can be written in terms of Hodge numbers as

which we can further rewrite as an alternating sum of Euler characteristics

Abstractly this identity reflects the fact that the sheaves together form a resolution of the constant sheaf , just as in the case of smooth differential forms on a smooth manifold. However, in the smooth case, the sheaves of smooth differential forms do not themselves have any higher sheaf cohomology, whereas in the complex case, the sheaves of holomorphic differential forms do in general have higher cohomology. This resolution also exists on any complex manifold, not necessarily compact or KÃ¤hler. It gives rise to the Hodge-to-de Rham (or FrÃ¶licher) spectral sequence in general, and the existence of the Hodge decomposition reflects the fact that on compact KÃ¤hler manifolds this spectral sequence degenerates.

Returning to the case of a compact Riemann surface , we get that

but by Serre duality , hence

.

Hence the topological Euler characteristic of is twice its holomorphic Euler characteristic. This argument not only shows that the topological Euler characteristic is even but gives an interpretation of the number obtained by dividing it by .

But we used the Hodge decomposition and Serre duality already, so let’s do something else.

**Proof 5: the Hirzebruch-Riemann-Roch theorem**

The Riemann-Roch theorem has the following more general form. Let be a holomorphic vector bundle on a compact complex manifold of complex dimension . Let

denote the Euler characteristic of the sheaf of holomorphic sections of , as we did above for line bundles. Let denote the Chern character of , which is defined via the splitting principle as

for a direct sum of complex line bundles. can be written in terms of the Chern classes using the fact that the total Chern class

can be defined via the splitting principle as

.

Equivalently, is the elementary symmetric function in the Chern roots . Expanding out the definition of gives power symmetric functions of the Chern roots which we can write as a polynomial in the elementary symmetric functions, e.g. using Newton’s identities, hence as a polynomial in the Chern classes. The first three terms are

.

Similarly, let denote the Todd class of (the tangent bundle of) , which is defined via the splitting principle as

for a direct sum of complex line bundles. Again we can use symmetric function identities to write in terms of the Chern classes of (the tangent bundle of) . The first three terms are

.

Finally, suppose that

is a (mixed) cohomology class, and let

denote the pairing of the degree part of with the fundamental class .

**Theorem (Hirzebruch-Riemann-Roch):** With hypotheses as above, the Euler characteristic satisfies

.

We’ll make no attempt to prove this, but here are some notable features of this theorem.

First, 1) only depends on the isomorphism class of as a topological, rather than holomorphic, complex vector bundle, 2) only depends on the isomorphism class of the tangent bundle of as a topological complex vector bundle, and 3) only depends on the orientation of coming from the complex structure on its tangent bundle. In other words, the RHS consists of topological, rather than holomorphic, data. This reflects the way the Hirzebruch-Riemann-Roch theorem is a special case of the Atiyah-Singer index theorem.

In addition, the RHS is a rational linear combination of certain characteristic numbers, hence is a priori rational, but Hirzebruch-Riemann-Roch tells us that it is in fact an integer. This implies divisibility relations which substantially generalize the divisibility relation we’re looking for, namely that .

**Corollary (Riemann-Roch):** Let be a holomorphic line bundle on a compact Riemann surface . Then

.

In particular, the holomorphic Euler characteristic satisfies .

*Proof.* In general, the top Chern class of an -dimensional complex vector bundle is its Euler class . In particular, is the Euler class , hence .

It remains to show that . Morally speaking this is because if then is PoincarÃ© dual to , which is morally the vanishing locus of a generic section of . But I am not sure how to make this precise easily. An unsatisfying proof that gets the job done is to use the same additivity argument involving skyscraper sheaves as in the previous proof of Riemann-Roch to conclude that for some constant and then to note that, since is topologically the trivial line bundle, , hence .

**Corollary:** The holomorphic Euler characteristic is equal to the Todd genus of :

.

*Proof.* The underlying topological line bundle of the structure sheaf is the trivial line bundle, and hence has trivial Chern character.

In particular, only depends on the Chern numbers of . These are known to be complex cobordism invariants, and in fact the Todd genus is a genus: it gives a ring homomorphism

where is the complex cobordism ring and is the Thom spectrum for complex cobordism.

In the next dimension up (complex dimension , real dimension ), the Hirzebruch-Riemann-Roch theorem gives the following divisibility relation.

**Corollary (Noether’s formula):** The holomorphic Euler characteristic of a compact complex surface satisfies

.

In particular, the RHS is an integer.

**Corollary:** If is a compact complex surface with (in particular if is Calabi-Yau; the converse holds if is KÃ¤hler), then .

Examples include the hypersurface of degree in (as we saw previously), and more generally any K3 surface, with Euler characteristic .

]]>

Our route towards this result will turn out to pass through all of the most common types of characteristic classes: we’ll invoke, in order, Euler classes, Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes.

**Examples in the plane**

Recall that a smooth projective hypersurface of degree is a projective variety cut out by a single homogeneous polynomial of degree which is smooth. This is the case if and only if the partial derivatives have no zeroes in common with in . Such a variety has complex dimension , hence real dimension .

*Example.* When we are considering smooth projective curves in the projective plane . Examples are given by the Fermat curves

.

Topologically, these are compact oriented surfaces, and hence their homeomorphism and even diffeomorphism type is completely determined by the rank of their first homology, or equivalently by their genus . The genus-degree formula asserts that the genus of a plane curve of degree is .

*Subexample.* When or the genus is , so we just get projective lines , or topologically we get -spheres . When the genus is , so we get elliptic curves (after choosing identities), or topologically we get tori .

There is a nice heuristic proof of the genus-degree formula (which can be made rigorous; see this MO discussion) which goes as follows. First consider the singular curve of degree given by lines in general position, so that every pair of lines intersects exactly once but otherwise there are no intersections. Topologically this gives a collection of spheres each pairwise intersecting in a point. If we perturb the coefficients of the singular curve, it will become smooth; topologically the spheres become pairwise connected by tubes. After using of these tubes to connect the spheres in a line, to obtain a sphere, the remaining tubes each increase the genus of the resulting surface by .

**An aside**

The following is not necessary for the computation to come but is nevertheless a nice explanation of a particular aspect of how it turns out. Eventually we’ll show that the cohomology of a smooth projective hypersurface depends only on the degree and the dimension of the ambient projective space, and this is explained by the fact that an even stronger statement than this holds.

**Theorem:** The diffeomorphism type of a smooth projective hypersurface of degree in depends only on and .

*Remark.* This statement cannot be strengthened to a statement about isomorphism in the holomorphic / algebraic category, as the example of cubic curves in already shows.

*Rough sketch.* The idea is that slightly perturbing the coefficients of a homogeneous polynomial does not affect the diffeomorphism type of the hypersurface it cuts out, and moreover that the space of homogeneous polynomials defining a smooth hypersurface is the complement of a subvariety (the subvariety of polynomials sharing at least one zero with its partial derivatives), hence has real codimension and in particular is path-connected, so we can perturb the coefficients of any such polynomial to get any other such polynomial.

*Proof.* Let be a complex vector space of dimension , so that we can identify with . A homogeneous polynomial of degree on is an element of , but since we’re only looking at the hypersurface cut out by such a polynomial we can ignore the zero polynomial and scaling, so we are really looking at an element of . Now let

be the complement in of the singular locus of polynomials having a zero in common with their partial derivatives, and let

.

The space admits a projection map onto the second coordinate , and the hypersurface cut out by is precisely the fiber .

Our goal is to show that the fibers of this map are diffeomorphic by applying Ehresmann’s theorem to it, which tells us that is a locally trivial smooth fibration provided that it is a proper surjective submersion. This implies in particular that the fibers are all diffeomorphic if is path-connected.

We’ll divide up the rest of the proof into the following steps.

**Step 1:** is a path-connected smooth manifold. More generally the following is true.

**Proposition:** Let be a Zariski-closed subset. Then is a path-connected smooth manifold.

*Proof.* A Zariski-closed subset is in particular closed, so is an open subset of a smooth manifold and hence a smooth manifold. The key point for path-connectedness is that has codimension at least , but we can avoid explicitly using this fact as follows. Any two distinct points determine a complex line passing through them. The intersection of this complex line with is finite, since it is a Zariski-closed subset of but not the whole thing. Now minus a finite set of points is path-connected, so can be connected by a path lying inside as desired.

It remains to show that the singular locus of polynomials having a zero in common with their partial derivatives is Zariski-closed, but this is a corollary of the existence of the multivariate resultant of the polynomials , which is a polynomial in the coefficients vanishing iff the polynomials have a common zero, together with the identity

showing that if all of the vanish at a point then so does .

**Step 2:** is a smooth manifold. To start with, we’ll work locally. On the open subset where and the coefficient of in is also nonzero, is locally the zero locus of the function

where and is the dehomogenization of , scaled so that the constant coefficient (the coefficient of in ) is . Fixing , the differential of this map in the has coefficients the partial derivatives , and since by assumption we’ve removed the singular hypersurfaces, at least one of these partial derivatives must be nonzero, so by the regular value theorem the zero locus is locally a smooth manifold. Running this argument with replaced by any and replaced by any monomial of degree , we get that is a smooth manifold as desired.

**Step 3:** is a submersion. is clearly surjective and proper (since hypersurfaces are compact), so this is the only interesting step remaining. Again working locally and on the open subset where and the coefficient of in is nonzero, locally takes the form

where again is the dehomogenization scaled to have constant coefficient , and . To show that is surjective on tangent spaces it suffices to show that any infinitesimal deformation in the coefficients of can be canceled out by a corresponding deformation in the so that the relation continues to hold (this is what it means to lift a tangent vector from our target to our source). But this is precisely guaranteed by the condition that at least one of the partial derivatives is nonzero. Again, running this argument with all of the other coordinates and monomials we get the result.

*Remark.* A simpler version of this argument can be used to give a proof of the fundamental theorem of algebra. The rough sketch here is to argue 1) that the space of polynomials with nonzero discriminant is connected, 2) that the number of roots of a polynomial with nonzero discriminant does not change when you perturb its coefficients, and 3) that establishing the fundamental theorem for polynomials with nonzero discriminant establishes it for all polynomials, since if is any polynomial then has nonzero discriminant, or equivalently is squarefree.

**Most of the cohomology**

Below all cohomologies are with integer coefficients unless otherwise stated.

Let be a smooth projective hypersurface of degree in . Most of the cohomology of is determined by the Lefschetz hyperplane theorem, as follows. Thinking again of as where , we have a Veronese embedding

and, essentially by definition, is the intersection of the image of the Veronese embedding with a hyperplane in . The Lefschetz hyperplane theorem then guarantees that the natural map is an isomorphism for and an injection for . Recalling that

we conclude that is if is even and otherwise for all . Moreover, since , by virtue of being a compact complex manifold, is in particular a compact oriented manifold, we can apply PoincarÃ© duality to conclude that the same is true of . That is,

and so the only remaining question is what the middle cohomology looks like. So far all we know is that injects into it; this is if is odd but if is even.

**Reduction to the Euler characteristic**

We claim that to compute the middle cohomology of it suffices to compute its Euler characteristic . First, recall that a compact manifold has finitely generated cohomology. It follows that has a well-defined Euler characteristic. Since we know all of the Betti numbers except one, computing the Euler characteristic will tell us the remaining Betti number. Explicitly, our computations above give

.

However, we still need to rule out the possibility of torsion in the middle cohomology in order to be confident that knowing the Betti number is enough. We can do this using the universal coefficient theorem, which gives a short exact sequence

.

The group on the right is torsion-free because it is given by homomorphisms into a torsion-free group, and the group on the left is torsion-free because it vanishes: is free by another part of the Lefschetz hyperplane theorem, hence has no nontrivial extensions. It follows that is free abelian, so is determined by its rank .

**The Euler characteristic via Chern classes**

Recall that the Euler characteristic of a compact oriented smooth manifold can be computed as the evaluation of the Euler class of its tangent bundle on the fundamental class . (Since the Euler class of a vector bundle can be thought of as PoincarÃ© dual to the zero locus of a generic section, this can be thought of as a restatement of the PoincarÃ©-Hopf theorem.)

On a compact complex manifold, the tangent bundle has a complex structure and hence Chern classes . It is common to refer and to notate these as the Chern classes of itself. Moreover, the top Chern class is the Euler class. Hence one way to compute the Euler characteristic of a compact complex manifold is to compute its top Chern class, which is the approach we will take: in fact we will compute all Chern classes.

We will first need to compute the Chern classes of . The key tool is the Euler sequence

where is the trivial line bundle and is the dual of the tautological line bundle whose fiber at a point in is the line in it represents; equivalently, is the line bundle whose holomorphic sections are homogeneous polynomials of degree . Since the total Chern class is multiplicative in exact sequences, we get

where is a generator of the cohomology ring . It follows that the Chern classes of are given by

.

(In particular, the top Chern class is , which when evaluated on the fundamental class gives the Euler characteristic as expected.)

To get from here to the Chern classes of a hypersurface we need to relate the two tangent bundles, which we do via the short exact sequence

of vector bundles on , where is the normal bundle.

Now, it turns out that the normal bundle is the restriction to of the line bundle whose holomorphic sections are homogeneous polynomials of degree ; this is essentially the content of the adjunction formula. Roughly speaking this is because is defined as the zero locus of a nonvanishing section of , and the actual map can be thought of as the derivative of this section, although I’m not sure how to make this precise.

In particular, since , the total Chern class of is given by , and hence the total Chern class of is

where by abuse of notation denotes the pullback of our previously chosen generator of to . We can now compute that the top Chern class of is

.

It remains to evaluate on the fundamental class of . Now, is PoincarÃ© dual to the intersection of generic hyperplanes in , which give a copy of , and since is cut out by a hypersurface of degree intersecting it with a generic line gives points, so we conclude that and hence that

which gives our desired computation of the rank of the middle cohomology:

.

*Example.* Let . As mentioned above, in this case is topologically a compact oriented surface The Betti numbers of are , and

and we recover the genus-degree formula.

*Example.* Rewriting the formula above as

makes it more convenient to do some kinds of computations with. In particular, for we get

as expected since in this case is just and we know its middle cohomology already. For we get

which is a little more interesting; the resulting hypersurfaces, namely the quadric hypersurfaces, are birational to but not necessarily homotopy equivalent. We’ll identify the quadric hypersurface when below; when it turns out to be the Grassmannian of complex planes in , with the embedding into being given up to projective change of coordinates by the PlÃ¼cker embedding.

For by inspection the Betti number grows exponentially in .

**Complex surfaces as 4-manifolds**

Now let . In this case is topologically a compact oriented 4-manifold. The Betti numbers of are , and

.

*Example.* When , so that is , we get as expected.

*Example.* When , so that is a quadric surface, we get ; here is , so diffeomorphic to , with the embedding into being given up to projective change of coordinates by the Segre embedding.

*Example.* When , so that is a cubic surface, we get .

*Example.* When , so that is a quartic surface, we get ; in this case is also a K3 surface.

(When , is a surface of general type.)

For , the homotopy group version of the Lefschetz hyperplane theorem implies that the natural map is an isomorphism; since the latter is trivial, so is the former. Hence as 4-manifolds, our complex surfaces are compact, oriented, and simply connected.

For such a 4-manifold, once we know its cohomology groups the only additional data of the sort that one usually calculates in a first course in algebraic topology is the cup product, which is completely determined by the **intersection form**

where is the fundamental class in . Since is even, the intersection form is symmetric, so gives the structure of an integral **lattice** (that is, a free abelian group equipped with a symmetric bilinear -valued form), and by PoincarÃ© duality this lattice is **unimodular**.

Thus identifying invariants of lattices immediately gives us (oriented homotopy) invariants of compact oriented 4-manifolds, and more generally of compact oriented manifolds in dimension . We’ll focus our attention on three such invariants.

- The
**rank**of a lattice is its rank as an abelian group; in the case of 4-manifolds this is just the second Betti number . - The
**signature**of a lattice is the signature of its bilinear form on . More explicitly, by Sylvester’s law of inertia any nondegenerate bilinear form on a real vector space can be diagonalized so that the corresponding quadratic form isfor two integers , which can equivalently be described as the number of positive resp. negative eigenvalues of a matrix describing the bilinear form. The signature is then ; note that the rank is , so the signature and the rank together determine the ordered pair , which is also sometimes called the signature. This gives an invariant of compact oriented manifolds in dimension also called the signature and denoted .

- The
**parity**of a lattice is defined as follows: if is always divisible by , then the lattice is**even**, and otherwise the lattice is**odd**. In other words, where the signature comes from looking at , the parity comes from looking at .

*Remark.* In general this is very far from being a complete set of invariants of lattices. In the case that the signature is equal to the rank (so that the lattice is positive definite), the Smith-Minkowski-Siegel mass formula implies that the number of isomorphism classes of lattices grows very rapidly with the rank.

*Remark.* The signature is a particularly interesting invariant: its definition can be extended to manifolds in dimension not divisible by by declaring the corresponding signatures to be , and then the signature is a genus, although we won’t use this fact.

The intersection form turns out to be a surprisingly strong invariant. Milnor and Whitehead showed that compact, oriented, simply connected 4-manifolds are determined up to oriented homotopy by their intersection forms as lattices. Freedman showed that every unimodular lattice arises in this way and that the only additional data required to determine such a 4-manifold up to homeomorphism is a class in called the Kirby-Siebenmann invariant; moreover,

- if the lattice is even, then there is a unique corresponding 4-manifold up to homeomorphism with Kirby-Siebenmann invariant , and
- if the lattice is odd, then there are exactly two corresponding 4-manifolds, one with each possible value of the Kirby-Siebenmann invariant.

The Kirby-Siebenmann invariant vanishes whenever a manifold has a smooth structure, and so in the odd case at least one of the two 4-manifolds does not have a smooth structure.

There are also other obstructions to having a smooth structure involving the intersection form. For example, by the above the lattice occurs as the intersection form of a unique homeomorphism class of compact, orientable, simply connected 4-manifold, the manifold. The lattice is positive definite but not diagonalizable, so by Donaldson’s theorem the manifold does not have a smooth structure.

The computations we’ve done so far don’t tell us what the intersection form is. Fortunately, we’ll be able to compute the intersection form, and hence the cup product structure on cohomology, as follows. First, we can compute the signature using the Hirzebruch signature theorem in terms of Pontryagin classes. Second, if the signature is not equal to plus or minus the rank (so the lattice is indefinite) then the possible lattices have been completely classified. There are only two possibilities if the rank and signature are fixed, depending only on the parity:

- if the lattice is odd, it must be the lattice of vectors with integer entries in , the real vector space equipped with the symmetric bilinear form of signature ;
- if the lattice is even, the signature must be divisible by , and the lattice must be the lattice of vectors in whose entries are either all integers or all integers plus and which sum to an even number.

In other words, for indefinite unimodular lattices the rank, signature, and parity form a complete set of invariants. Hence if we compute that the signature is not equal to the rank , the only additional information we need to determine the lattice is its parity. It will turn out that this is determined by whether the second Stiefel-Whitney class vanishes, or equivalently by whether admits a spin structure.

**The signature via Pontryagin classes**

Recall that if is a real vector bundle over a space then it admits a complexification which is a complex vector bundle, and that the Pontryagin classes of are characteristic classes defined in terms of the Chern classes of the complexification via

.

For a compact smooth oriented 4-manifold , the Hirzebruch signature theorem asserts that the signature is given by

where is the first Pontryagin class

of (the tangent bundle of) and is the fundamental class as usual. In particular, it implies that the first Pontryagin number is divisible by .

Hence to compute the signature of a hypersurface we need to compute the second Chern class of the complexification of its tangent bundle, regarded as a real vector bundle (whereas above we computed the Chern classes of the tangent bundle, which already had a complex structure). In general we can compute the Chern classes of the complexification of a complex vector bundle in terms of the Chern classes of the original bundle as follows.

**Theorem:** Let be a complex vector bundle. Then the complexification of the underlying real vector bundle of is isomorphic, as a complex vector bundle, to , where is the conjugate vector bundle.

**Corollary:** The Pontryagin classes of the underlying real vector bundle of a complex vector bundle can be computed in terms of its Chern classes via the Whitney sum formula as

.

In particular,

.

*Proof.* Write

.

This tells us that to understand the endofunctor on complex vector bundles it suffices to understand as a -bimodule; the left -module structure tells us how to take the tensor product and the right -module structure tells us what the complex structure on the tensor product is. The theorem is then equivalent to the claim that, as a bimodule,

where

- denotes the identity bimodule, with acting on the left and right by left and right multiplication, so that tensoring with this bimodule is the identity endofunctor , and
- denotes the bimodule where left and right multiplication by disagree by a sign of (more explicitly, we can take the left module structure to be the usual one and the right module structure to be multiplication by the conjugate), so that tensoring with this bimodule is the endofunctor .

To see this, we will first think of as a right -module with basis , and then we will diagonalize left multiplication by . When we do this we find that on

left and right multiplication by agree, whereas on

left and right multiplication differ by a sign. The left, or equivalently right, -submodules generated by these vectors gives the desired decomposition.

Now let be a hypersurface of degree in . Above we computed the total Chern class to be

so we compute that

and hence that

and, using again the fact that , we compute the signature of a smooth projective hypersurface of degree in to be

.

Above the numerator has been written in a form that makes it clear that it is divisible by .

We conclude that for the signature is not equal to plus or minus the rank, and so the intersection form is indefinite in this case, which tells us that to uniquely identify the intersection form we only need to know the parity as we hoped.

*Example.* When the signature is . This reflects the fact that the intersection form on is positive definite, since it is just given by .

*Example.* When the signature is . This reflects the fact that the intersection form on is indefinite, since by the Kunneth formula is generated by two elements (where denotes a generator of ) which square to zero but whose cup product is a generator of . An explicit diagonalization of the intersection form over is given by the basis .

*Example.* When the signature is . In particular it is not divisible by , so the intersection form is odd and hence must be the lattice .

*Example.* When the signature is . We’ll see later that in this case the intersection form is even, and hence must be the lattice .

In general, when is odd the signature is odd, so the intersection form is odd and hence is uniquely determined. When is even the signature is divisible by , and in particular is divisible by , so the intersection form could be even or odd.

**The parity via Stiefel-Whitney classes**

To summarize, the story so far is the following:

- If is a smooth projective hypersurface in of degree , then in particular it is a smooth, compact, oriented, and simply connected 4-manifold.
- For such a manifold, is a free abelian group of finite rank, and is determined up to homeomorphism by the intersection form on , which gives the structure of a unimodular lattice.
- The rank and the signature of are given by
and in particular, for , is indefinite.

- By the classification of indefinite unimodular lattices, the only remaining bit of information we need about to completely determine it is its parity. More specifically, if the parity is odd then must be and if the parity is even then must be , where
.

In this section we’ll compute the parity. It will turn out to depend only on , which via Freedman’s work gives an independent confirmation that when the homeomorphism type of a smooth projective hypersurface of degree only depends on (since the Kirby-Siebenmann invariant vanishes when a 4-manifold has a smooth structure).

Let be a smooth, compact, oriented, simply connected 4-manifold. Since vanishes, we have , and so the parity of is determined by whether or not the map

is identically zero. Over this map is linear; in fact it can be identified with the Steenrod square . By PoincarÃ© duality (this step only requires that is compact, since every compact manifold is orientable over ) there must therefore be a unique cohomology class such that

.

This class is called the second Wu class, and by definition vanishes identically iff vanishes, so is even iff vanishes.

So it remains to compute . The Wu classes turn out to be closely related to the Stiefel-Whitney classes (of the tangent bundle). More precisely, the total Stiefel-Whitney class is the total Steenrod square of the total Wu class:

.

*Remark.* In particular, the Stiefel-Whitney classes of a compact smooth manifold depend only on its cohomology over as a module over the Steenrod algebra, which is surprising: a priori the Stiefel-Whitney classes also depend on the additional data of the tangent bundle.

This gives

and hence

.

Since we assumed that is oriented, vanishes (although this also follows from the fact that vanishes as well), from which it follows that vanishes iff the second Stiefel-Whitney class vanishes. Hence we have proven the following.

**Theorem:** Let be a compact oriented simply connected 4-manifold. Then is even iff vanishes.

*Remark.* Even if is not equipped with a smooth structure, hence is not equipped with a tangent bundle, as long as is compact we can still define its Stiefel-Whitney classes in terms of its Wu classes, and these will agree with the Stiefel-Whitney classes computed from any smooth structure on . If is equipped with a smooth structure and is oriented, then the vanishing of is equivalent to also admitting a spin structure.

*Remark.* The lattice is even; in fact it is the unique even positive definite unimodular lattice of rank . It follows that if the manifold had a smooth structure, it would also admit a spin structure, and then Rokhlin’s theorem would imply that its signature is divisible by . But its signature is ; contradiction. This gives a second proof that the manifold has no smooth structure.

*Remark.* If is not simply connected, or more precisely if has -torsion, then it is still true that if vanishes then is even, but the converse need not hold owing to the presence of an additional direct summand in coming from universal coefficients.

It remains to compute the second Stiefel-Whitney class. We can in fact compute all Stiefel-Whitney classes of a hypersurface of degree in as follows.

**Theorem:** Let be a complex vector bundle. Then the Stiefel-Whitney classes of the underlying real vector bundle are determined by the Chern classes as follows: the odd Stiefel-Whitney classes vanish, and the even Stiefel-Whitney classes satisfy

*Proof.* We’ll first prove this in the case when is a line bundle . (This is the only case we need but it’s not much harder to prove the general statement.) In this case we only need to show that vanishes and that .

First, vanishes if and only if has an orientation. But any complex structure induces an orientation, so this is clear.

To compute we can use the fact that the top Stiefel-Whitney class of an oriented vector bundle is the reduction of its Euler class while the top Chern class of a complex vector bundle is its Euler class, which gives since they are both the Euler class . If we want to avoid the Euler class, we can also argue as follows:

The functor from complex line bundles to real plane bundles is induced, at the level of classifying spaces, by the map

induced by the standard embedding . Since as subgroups of , the map above factors as a composite

where the first map is a homotopy equivalence, showing that the classification of complex line bundles is in fact equivalent to the classification of oriented real plane bundles.

From standard results about characteristic classes we know that on the one hand

is a polynomial algebra on the universal second Stiefel-Whitney class , while on the other hand

is a polynomial algebra on the universal first Chern class . In particular, generates while is the unique generator of , so the homotopy equivalence necessarily identifies the latter with the reduction of the former.

We have the desired result for line bundles. To obtain the result for all bundles we appeal to the splitting principle, which tells us in particular that to prove an equality of characteristic classes it suffices to prove it on a direct sum of line bundles.

So let be complex line bundles. We now know that the total Stiefel-Whitney class of the underlying real vector bundle of can be computed, using the Whitney sum formula, as

since we know that vanishes. This implies that all of the odd Stiefel-Whitney classes vanish. Since we also know that , this tells us that the total Stiefel-Whitney class is

and this is the reduction of the total Chern class as desired.

Now again let be a hypersurface of degree in . Above we computed the first Chern class to be

where , as before, denotes the pullback of the generator of to . By the Lefschetz hyperplane theorem, or from the fact that we know , the cohomology class is nonzero, hence the reduction

vanishes if and only if is even. We conclude that the parity of is precisely the parity of . This completes our computation of the cohomology ring of .

*Remark.* When is even we also conclude that the hypersurfaces have a spin structure, and in particular we get an independent confirmation of Rokhlin’s theorem that the signature is divisible by in this case.

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into its presheaf category (where we use to denote the category of functors ). The Yoneda lemma asserts in particular that is full and faithful, which justifies calling it an embedding.

When is in addition assumed to be small, the Yoneda embedding has the following elegant universal property.

**Theorem:** The Yoneda embedding exhibits as the **free cocompletion** of in the sense that for any cocomplete category , the restriction functor

from the category of cocontinuous functors to the category of functors is an equivalence. In particular, any functor extends (uniquely, up to natural isomorphism) to a cocontinuous functor , and all cocontinuous functors arise this way (up to natural isomorphism).

Colimits should be thought of as a general notion of gluing, so the above should be understood as the claim that is the category obtained by “freely gluing together” the objects of in a way dictated by the morphisms. This intuition is important when trying to understand the definition of, among other things, a simplicial set. A simplicial set is by definition a presheaf on a certain category, the simplex category, and the universal property above says that this means simplicial sets are obtained by “freely gluing together” simplices.

In this post we’ll content ourselves with meandering towards a proof of the above result. In a subsequent post we’ll give a sampling of applications.

**A toy version of the above result**

Coproducts in particular are examples of colimits, so if we think of coproducts as being analogous to addition, we can think of a cocomplete category as being analogous to a commutative monoid and a cocontinuous functor as being analogous to a morphism of commutative monoids. The universal property above can then be thought of as analogous to the following. Let be a set and let be the set of functions which vanish except at finitely many points in . There is an inclusion sending a point in to the indicator function which is equal to at that point and elsewhere.

**Theorem:** The natural inclusion exhibits as the the free commutative monoid on in the sense that for any commutative monoid , the restriction map

from the set of monoid homomorphisms to the set of functions is a bijection.

(Of course an intriguing difference between the toy theorem and the real theorem is that being cocomplete is a property of a category, while being a commutative monoid is a structure placed on a set.)

In the setting of commutative monoids, a shorter description of the above theorem is that there’s a forgetful functor from commutative monoids to sets and that describes its left adjoint. Similarly, we’d like to be able to say that there’s a forgetful functor from cocomplete categories to categories and that the Yoneda embedding is its left adjoint. Unfortunately, there are nontrivial size issues that get in the way: is never small, and in fact, the only cocomplete small categories are preorders by a theorem of Freyd.

In any case, before we get to discussing the result in full generality, let’s look at some illustrative examples.

**Sets**

Take to be the terminal category. Then is just the category of sets. This example already says something interesting: the universal property implies that is the free cocomplete category on an object in the sense that if is a cocomplete category, then the category of cocontinuous functors is equivalent to itself. The inverse of this equivalence sends an object to the functor

which, given a set , returns the coproduct of copies of , and conversely every cocontinuous functor has this form. This statement should be thought of as analogous to the statement that is the free commutative monoid on a point.

**Graphs**

Take to be the category with two objects and two parallel morphisms between them. (This category is in fact a truncation of the simplex category.) Think of as a vertex, as an edge, and the two morphisms as the two inclusions of the endpoints of the edge. A presheaf is then precisely a pair of sets together with a pair of functions

.

The two maps have been named because we can think of them as source and target maps: in fact, is precisely a (directed multi)graph with vertex set and edge set . Here the universal property of presheaves can be interpreted as the claim that graphs are obtained by freely gluing together edges along vertices.

The universal property also gives a natural way of describing graphs as topological spaces, as follows: is a cocomplete category, and there is a functor sending to a point, to an interval , and the two arrows to the two inclusions of the endpoints of the interval. By the universal property, this functor extends to a cocontinuous functor sending a graph to its underlying topological space (with directions on the edges ignored). This is a simple version of geometric realization.

But of course the universal property implies that there are many other more exotic notions of geometric realization for graphs. For example, instead of using topological spaces we could use affine schemes: fixing a field , the category of affine schemes over is cocontinuous, and there is a functor sending to a point , to , and the two maps to the inclusions of the two points into (for example). By the universal property we obtain a geometric realization functor which, for example, sends the loop (the graph consisting of a vertex and an edge from that vertex to itself) to the affine scheme with ring of functions

.

This affine scheme is precisely the nodal cubic. To see this, write the loop as the coequalizer of the two maps , thought of as natural transformations between the corresponding presheaves. To compute the ring of functions on the resulting affine scheme means computing the equalizer of the two maps given by evaluation at and respectively.

**Species**

Write for the category (really groupoid) of finite sets and bijections. This is equivalently the core of the category of finite sets and functions. It is equivalent, as a category, to the disjoint union

of the one-object groupoids corresponding to the symmetric groups , hence the name . We will often think of the objects of as the non-negative integers. A presheaf is, depending on who you ask, a **species**, **-module**, or **symmetric sequence** in sets; we’ll use the term species. More concretely, a species is a collection of sets indexed by the non-negative integers such that each set is equipped with a (right) action of the symmetric group .

Species are surprisingly fundamental objects in mathematics. Under the name species, they were introduced by Joyal to study combinatorics, and among other things to categorify the theory of exponential generating functions; see, for example, Bergeron, Labelle, and Leroux. I think the names -module and symmetric sequence are used by authors studying operads, as operads are species with extra structure (see the nLab for details).

The universal property tells us that we can extend any functor from to a cocomplete category to a cocontinuous functor . An important source of functors is given by taking to be a symmetric monoidal category, to be an object, and considering the functor

.

This observation can be codified as the following universal property.

**Theorem:** , equipped with disjoint union, is the free symmetric monoidal category on an object in the sense that for any symmetric monoidal category , the restriction functor from the category of symmetric monoidal functors to the category of functors , which is just , is an equivalence.

If is in addition cocomplete, in such a way that the monoidal operation is cocontinuous in both arguments (**symmetric monoidally cocomplete**), then after choosing an object , we get not only a symmetric monoidal functor but even a functor , which turns out to be symmetric monoidal if is given a monoidal structure via Day convolution. (Day convolution is the monoidal structure categorifying the product of exponential generating functions.) This observation can in turn be codified as a universal property.

**Theorem:** , equipped with Day convolution, is the free symmetric monoidally cocomplete category on an object in the sense that for any symmetric monoidally cocomplete category , the restriction functor from the category of symmetric monoidal cocontinuous functors to the category of functors (thinking of as a representable presheaf), which is just , is an equivalence.

What do these symmetric monoidal cocontinuous functors actually look like? For an object , the corresponding functor is

where is shorthand for taking coinvariants with respect to the diagonal action of , and is shorthand for the coproduct of an -indexed family of s (see copower for some motivation behind this notation). This is an important construction: in the special case that is an operad, so that describes the set of -ary operations in the operad, the above construction describes the free -algebra on . If all of the are finite sets, the above construction can also be thought of as categorifying the exponential generating function

(thinking of taking coinvariants with respect to an -action as categorifying dividing by , in accordance with the general yoga of groupoid cardinality.)

*Example.* Let be the associative operad. Here consists of operations of the form

for each permutation and hence, as a right -set, is isomorphic to . is then naturally isomorphic to , so the free associative algebra (monoid) on an object in a symmetric monoidally cocomplete category is the infinite coproduct

.

Regarded just as a combinatorial species, categorifies the generating function .

*Example.* Let be the commutative operad. Here consists of the single operation

and hence, as a right -set, is trivial. is then the quotient , so the free commutative algebra (commutative monoid) on an object in a symmetric monoidally cocomplete category is the infinite coproduct

.

Regarded just as a combinatorial species, categorifies the generating function .

**Intuitions about the proof**

Recall that we are trying to show that the restriction functor

is an equivalence. By analogy with the corresponding statement about sets, commutative monoids, and free commutative monoids, one way to proceed with this proof is to figure out how to write every presheaf as a colimit of representable presheaves (the image of the Yoneda embedding ), then turn this colimit into a colimit in by applying a given cocontinuous functor . This will show, roughly speaking, that the restriction map is “injective” (although we need to be careful about what this means because we’re dealing with categories, not sets).

To show that the restriction map is “surjective,” we need to extend a functor to a cocontinuous functor . We’d like to do this “by linearity,” by choosing an expression for a presheaf as a colimit of representable presheaves and turning this colimit into a colimit in by applying our functor; however, we need to be able to make this choice functorially, and then we still need to verify that the resulting functor is actually cocontinuous.

**Presheaves as colimits of representable presheaves**

The following result is at least implicit in the use of the terminology “free cocompletion” and is important in getting the above proof to work, as well as being a generally useful thing to know in category theory. It is sometimes called the co-Yoneda lemma for reasons that are a little difficult to explain without more background. Previously it showed up when we discussed operations and pro-objects, but there we rushed through the proof and here we’ll take a more leisurely pace.

**Theorem:** Let be a (locally small) category. Then every presheaf is canonically a colimit of representable presheaves.

*Idea #1.* One relevant intuition here is to think of a presheaf as a recipe for writing down a colimit in by prescribing how many copies of each object and morphism in appear in the diagram, in the same way that one can think of a function from a set to the non-negative integers (with finite support) as a recipe for writing down an element of the free commutative monoid on by prescribing how many copies of each element of to add up. This intuition is hopefully quite clear in the case of graphs, where a presheaf on tells you how many edges and vertices to glue together as well as how to glue them together.

*Idea #2.* For the more categorically minded, a related intuition is the following. Let be a diagram in . The colimit , if it exists, is defined by a universal property describing how maps out of it behave. This determines the covariant functor it represents uniquely, but says very little about the contravariant functor it represents. However, there is in some sense a “minimal” possibility for this contravariant functor. For example, if the colimit in question is the coproduct of two objects, then by definition

but the only thing we know about is that there are natural inclusion maps , hence we know that admits a natural map from , but this is all we know without further information. Now, since colimits in functor categories are computed pointwise, is none other than the coproduct of , but regarded as lying in the presheaf category. In general, the sense in which presheaves are “free colimits” of objects of is that, as contravariant functors, they describe the “minimal” contravariant functors that a colimit of objects in could represent.

Now we turn to the proof itself.

*Proof.* Let by a presheaf. Since we want to describe as a colimit, let’s think about the contravariant functor that represents. By definition, consists of families of maps satisfying the naturality condition that if is a morphism, then the diagram

(drawn using QuickLaTex) commutes. We want to write as a colimit of representable functors, and we know that by the Yoneda lemma, if (which we use to designate the representable functor ) is a representable functor, then . To go from elements of to maps we need copies of .

A clean way to obtain these copies is to write down a diagram whose objects are given by pairs of an object and an element , equipped with the map to given by forgetting . The preimage of is then precisely , and if we don’t specify any morphisms then a cocone over this diagram in is precisely a family of maps satisfying no naturality conditions.

To get the naturality conditions back we need to equip with morphisms. Choosing the morphisms such that sends to enforces precisely the naturality condition desired on the maps , and furthermore the maps canonically exhibit as the colimit of the corresponding diagram in as desired.

(The diagram we constructed above is the opposite of the **category of elements** of , which is a special case of the **Grothendieck construction**. As described in the nLab article, we can think of as the classifying space of -bundles, and then is the classifying map of a -bundle on and is the total space of the bundle. admits other more sophisticated descriptions that won’t concern us at the moment.)

**The actual proof**

Now we return to the proof of the theorem. Let be a small category and be a (locally small) cocomplete category. Recall, again, that we are trying to show that the restriction functor

is an equivalence of categories. If we wanted to show that a map of sets was a bijection, we’d just have to show that it’s injective and surjective, and we sketched some intuition for why this should be the case above. But an equivalence of categories is more subtle, and instead of verifying two conditions we need to verify three: needs to be full, faithful, and essentially surjective.

To show that is fully faithful, let be a natural transformation between two cocontinuous functors . We want to show that knowing the restriction of to representable functors uniquely determines for all presheaves , and moreover that given such a restriction we can always extend it to a natural transformation on all presheaves. But since is cocontinuous and is a colimit of representables, is freely determined by the universal property of colimits: in particular it is determined by its restriction to every representable , which is just composed with the inclusion by naturality, and given such a compatible family of restrictions it exists.

To show that is essentially surjective, let be a functor. We want to extend to a cocontinuous functor , which we will do “by linearity”: if is a presheaf, we’ll write it canonically as a colimit of representable presheaves using the diagram of shape we described above (which is small since is small), then apply to this diagram to obtain a diagram in , then take the colimit in . In symbols,

.

Every step of this process, including the formation of the category of elements, is functorial, so really is a functor. (It is crucial that be small to ensure that is a small diagram; “cocomplete” only means that all small colimits exist, and in fact the theorem of Freyd alluded to above also implies that a category with all colimits is a preorder.)

It remains to verify first that really is cocontinuous and second that it really does restrict to (a functor naturally isomorphic to) . These will both be a corollary of the following.

**Proposition:** is the left adjoint of the functor

.

(A version of this construction, the “left pro-adjoint,” appeared previously on this blog.)

(There is some mild abuse of notation going on here. should really denote the functor given by precomposition with , and should really denote the left adjoint of this functor, also known as **left Kan extension**. The decorations (pronounced “upper star” and “lower shriek” respectively) on these functors are by analogy with some of Grothendieck’s six operations on sheaves.)

*Proof.* We want to show that there is a natural bijection

.

We know that , hence we can write the RHS as

first by the universal property of colimits and second by the Yoneda lemma. On the other hand, by definition is also a colimit over , hence we can write the LHS as

by the universal property of colimits. The conclusion follows.

In particular, since is a left adjoint, it is necessarily cocontinuous, and if above is a representable presheaf then the above adjunction gives

by the Yoneda lemma, so by a second application of the Yoneda lemma. It follows that is essentially surjective, hence an equivalence as desired.

(In fact should really have denoted the functor given by precomposition with , and what we really wrote down above is the left adjoint to this functor, which is a genuine left Kan extension along . We could’ve written the proof so as to show that is not only a left adjoint but in fact an inverse once we restrict to cocontinuous functors.)

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Although I’m sure there are more, I’m only aware of two other students at Berkeley who’ve posted transcripts of their quals, namely Christopher Wong and Eric Peterson. It would be nice if more people did this.

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