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## Tiny objects

The starting observation of Morita theory is that the abelian category $\text{Mod}(R)$ of (right) modules over a (not necessarily commutative) ring $R$ does not uniquely determine $R$, since for example we always have Morita equivalences of the form

$\displaystyle \text{Mod}(R) \cong \text{Mod}(M_n(R))$.

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

The next best thing we can try to do is to classify all of the rings $S$ such that $\text{Mod}(R) \cong \text{Mod}(S)$ by isolating the corresponding modules $S \in \text{Mod}(R)$ by some categorical property. The crucial property turns out to be that the hom functor

$\displaystyle \text{Hom}(S, -) : \text{Mod}(R) \cong \text{Mod}(S) \to \text{Ab}$

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” ($\text{Set}$-enriched) or “linear” ($\text{Ab}$-enriched). We will be proving facts for both kinds of categories simultaneously because the proofs are almost identical.

If $D$ is an ordinary category then $\widehat{D}$ denotes the category $[D^{op}, \text{Set}]$ of presheaves of sets on $D$, while if $D$ is a linear category then $\widehat{D}$ denotes the category $[D^{op}, \text{Ab}]$ of presheaves of abelian groups on $D$. Edit, 2/1/16: As Ingo Blechschmidt points out in the comments, this is a little ambiguous, so to be clear, by a presheaf of abelian groups on a linear category $D$ I mean a linear ($\text{Ab}$-enriched) functor $D^{op} \to \text{Ab}$.

“Module” means “right module.” In particular, if $D$ is a ring $R$ regarded as a linear category with one object, then $\widehat{D}$ is the linear category of $R$-modules.

Definition and examples

Let $C$ be a cocomplete category (possibly linear).

Definition: A tiny object is an object $c \in C$ such that $\text{Hom}(c, -)$ 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 $\text{Ab} \to \text{Set}$ preserves filtered colimits, and so in the definition of compactness we did not need to specify whether $\text{Hom}(c, -)$ 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 $C = \text{Set}$, then the one-object set $1$ is the only tiny object. To see this, if $X$ is a set, then in order for $\text{Hom}(X, -)$ to preserve colimits it must in particular preserve finite coproducts, so the map

$\displaystyle \text{Hom}(X, 1) \sqcup \text{Hom}(X, 1) \to \text{Hom}(X, 1 \sqcup 1)$

must be an isomorphism. The RHS can be identified with the set of subsets of $X$, while the inclusion of the LHS can be identified with the inclusion of the empty subset and the subset consisting of all of $X$. Hence if $X$ preserves finite coproducts, then $X$ must have exactly two subsets, the empty subset and $X$, and this condition is satisfied precisely when $X = 1$.

Example. The motivating class of examples is that if $D$ is a category (possibly linear), then in its presheaf category $\widehat{D}$, the representable presheaves $d \in D$ are tiny by the Yoneda lemma, since colimits of presheaves are computed pointwise. These are the “free modules of rank $1$” on each $d \in D$. In particular, if $D$ is a one-object linear category with endomorphisms $R$, we conclude that the free $R$-module of rank $1$ is tiny in $\hat{D} = \text{Mod}(R)$.

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

Subexample. Suppose $C$ has a zero object $0$. Then $F(0)$ is empty. Every object $c \in C$ admits a map $c \to 0$, but in $\text{Set}$ the only set which admits a map to the empty set is the empty set, so $F(c), c \in C$ is also empty. (In particular, abelian categories have no tiny objects when regarded as enriched over $\text{Set}$ as opposed to enriched over $\text{Ab}$.)

Subexample. Let $C = \text{CRing}$ be the category of commutative rings. Then $F(\mathbb{Z})$ is empty. The zero ring $0$ (which is not a zero object) is the coequalizer of the two projections $\mathbb{Z} \times \mathbb{Z} \rightrightarrows \mathbb{Z}$. $F$ preserves coequalizers, so $F(0)$ is also empty. But every ring admits a morphism $R \to 0$, so $F(R)$ admits a morphism to the empty set, hence is also empty. (While $\text{CRing}$ 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 $\text{Set}$ and $\text{Ab}$), 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:

1. Let $c$ be an object and let $d$ be a retract of it, so that there are morphisms $f : c \to d$ and $g : d \to c$ such that $fg = \text{id}_d$. Let $m = gf$ be the corresponding split idempotent. Then the retract $d$, equipped with the map $f$, is the coequalizer $\text{coeq}(\text{id}_c, m)$, and this coequalizer is preserved by any functor whatsoever. Dually, $d$, together with the map $g$, is the equalizer $\text{eq}(\text{id}_c, m)$.
2. Conversely, suppose $m : c \to c$ is an idempotent in a category such that either the equalizer or the coequalizer of $m$ and $\text{id}_c$ exists. Then they both exist and are canonically isomorphic. The resulting object $d$ and the corresponding maps $f : c \to d, g : d \to c$ exhibit $d$ as a retract of $c$, so $fg = \text{id}_d$ and $gf = m$, and $m$ is a split idempotent. In particular, the equalizer and coequalizer of $m$ and $\text{id}_c$ 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 $d$, together with the map $f$, satisfies the universal property of the above coequalizer. The universal property says that a map $h_d$ out of $d$ is the same thing as a map $h_c$ out of $c$ such that $h_c m = h_c gf = h_c$. So let’s verify this: given a map $h_d$ out of $d$, the pullback $h_c = h_d f$ is a map out of $c$ such that

$\displaystyle h_c gf = h_d fgf = h_d f = h_c$.

In the other direction, given a map $h_c$ out of $c$ such that $h_c m = h_c gf = h_c$, the pullback $h_d = h_c g$ is a map out of $d$, and

$\displaystyle h_d f = h_c gf = h_c$.

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 $f$ and $g$, but it is still true that in the opposite category, $d$ is a retract of $c$.)

Conversely, suppose $m : c \to c$ is an idempotent and that the coequalizer of $m$ and $\text{id}_c$ exists; call it $d$, and write $f : c \to d$ for the corresponding map out of $c$. By construction, a map $d \to c$ corresponds, via pulling back along $f$, to a map $c \to c$ coequalizing $\text{id}_c, m$. Since $m$ is idempotent, it is itself such a map; write $g : d \to c$ for the corresponding map out of $d$, which by construction satisfies $m = gf$. In particular,

$\displaystyle m^2 = gfgf = m = gf$.

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

$\displaystyle gfg = mg = g$.

So $g$ equalizes $\text{id}_c, m$. In fact $g$ exhibits $d$ as the equalizer of $\text{id}_c, m$: if $h$ is a map into $c$ equalizing $\text{id}_c, m$, so $mh = gfh = h$, then $h$ factors through $g$ since $h = g(fh)$.

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

$\displaystyle fg = \text{id}_c$

and hence the maps $f, g$ exhibit $d$ as a retract of $c$ as desired. $\Box$

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 $c$ is an absolute colimit $\text{colim}_{i \in I} c_i$ of tiny objects $c_i$ and $d$ is an arbitrary colimit $\text{colim}_{j \in J} d_j$. Then $\text{Hom}(c, d) \cong \text{lim}_{i \in I} \text{Hom}(c_i, d)$ by absoluteness, while $\text{Hom}(c_i, d) \cong \text{colim}_{j \in J} \text{Hom}(c_i, d_j)$ by tininess, and hence

$\displaystyle \text{Hom}(c, d) \cong \text{lim}_{i \in I} \text{colim}_{j \in J} \text{Hom}(c_i, d_j)$.

By absoluteness, $\text{lim}_{i \in I} \text{Hom}(c_i, d_j)$ is an absolute limit in the diagram category $[J, \text{Set}]$ (since limits are computed pointwise), and hence is preserved by the colimit functor $[J, \text{Set}] \to \text{Set}$ (or, for linear categories, $[J, \text{Ab}] \to \text{Ab}$), so we can exchange the order of the limit and the colimit to get

$\displaystyle \text{lim}_{i \in I} \text{colim}_{j \in J} \cong \text{colim}_{j \in J} \text{lim}_{i \in I} \text{Hom}(c_i, d_j)$.

But $\text{lim}_{i \in I} \text{Hom}(c_i, d_j) \cong \text{Hom}(c, d_j)$ by a final application of absoluteness, and the conclusion follows. $\Box$

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

Example. In $\text{Mod}(R)$, it follows that the retracts of finite free modules are tiny. These are precisely the finitely presented projective modules. More generally, if $D$ 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 $\widehat{D}$.

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 $c$ is tiny, then because $\text{Hom}(c, -)$ preserves colimits, $c$ satisfies any other property which can be expressed in terms of $\text{Hom}(c, -)$ preserving some restricted class of colimits. We’ll highlight three in particular:

1. $c$ is connected: $\text{Hom}(c, -)$ preserves finite coproducts. (This differs slightly from the nLab’s first definition but is equivalent to it in nice cases.)
2. $c$ is compact: $\text{Hom}(c, -)$ preserves filtered colimits.
3. $c$ is projective: $\text{Hom}(c, -)$ preserves epimorphisms (since it preserves pushouts).

The terminology of connectedness stems from the fact that in topological spaces, graphs, $G$-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 $1$ is the only tiny object in $\text{Set}$, since we showed above that it’s even the only connected object in $\text{Set}$.

What can we say about the converse?

Theorem: An object $c$ in a cocomplete category $C$ is tiny iff it is connected, compact, and $\text{Hom}(c, -)$ preserves coequalizers. If $C$ is linear, then $c$ is tiny iff it is compact and $\text{Hom}(c, -)$ preserves coequalizers, or equivalently cokernels. If $C$ is abelian, then $c$ 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, $\text{Hom}(c, -)$ preserves coequalizers iff $c$ is projective (as we showed earlier). $\Box$

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 $\text{Mod}(R)$, 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 $R, S$ are Morita equivalent rings, then $S$ is the endomorphism ring of a finitely presented projective $R$-module.

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

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 $\text{Mod}(R)$ 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 $C$ to be an object such that $\text{Hom}(c, -)$ 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 $c$ is in addition assumed to be projective, so that $\text{Hom}(c, -)$ preserves coequalizers, then $\text{Hom}(c, -)$ preserves coproducts iff it preserves colimits iff $c$ 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 $C$ was to write every object $c \in C$ as a filtered colimit of some readily identifiable class of compact objects, from which it would follow that if $c$ is projective then the identity $\text{id}_c : c \to c$ factors through a map to some member of this class, hence $c$ 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 $C$ is a cocomplete category (possibly linear) and $S$ is an essentially small full subcategory of tiny objects of $C$ such that every object of $C$ is a colimit of objects in $S$.

Example. Let $D$ be an essentially small category (resp. linear category) and let $C = \widehat{D}$ be its category of presheaves of sets (resp. abelian groups). If $D$ is an ordinary category, then by the Yoneda lemma every presheaf is a colimit of representables, and hence we can take $S = D$ to be the representables. If $D$ 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 $S$ to be the finite biproducts of representables. This reduces to a familiar fact when $D$ has one object, namely that every $R$-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 $C$ is ordinary then the tiny objects of $C$ are precisely the retracts of the objects in $S$, and if $C$ is linear then the tiny objects of $C$ are precisely the retracts of finite biproducts of the objects in $S$.

Proof. Write a tiny object $c \in C$ as a colimit $\text{colim}_{i \in I} c_i$ where $c_i \in S$. By hypothesis, $\text{Hom}(c, -)$ preserves colimits, so

$\displaystyle \text{Hom}(c, c) \cong \text{colim}_{i \in I} \text{Hom}(c, c_i)$.

If $C$ is an ordinary category, so that the above colimit is a colimit of sets, it follows that the identity $\text{id}_c : c \to c$ factors through some $c_i$, hence that $c$ is a retract of some $c_i$.

If $C$ is a linear category, so that the above colimit is a colimit of abelian groups, it follows that the identity $\text{id}_c : c \to c$ is a finite sum of morphisms factoring through some $c_i$s, hence that it factors through some finite biproduct of the $c_i$s, hence that $c$ is a retract of a finite biproduct of some $c_i$s. $\Box$

Corollary: If $D$ is an essentially small ordinary category, the tiny objects in $\widehat{D}$ are precisely the retracts of representable presheaves. If $D$ 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 $\text{Mod}(R)$ are precisely the retracts of finite free modules, or equivalently the finitely presented projective modules.

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

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

Morita equivalences and Cauchy completion

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

Definition: Two essentially small categories $D_1, D_2$ (possibly linear) are Morita equivalent if $\widehat{D}_1 \cong \widehat{D}_2$.

When $D_1, D_2$ 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 $\mathbb{Z}[\Delta]$ of the simplex category $\Delta$ (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 $[i], i \in \mathbb{Z}_{\ge 0}$ and morphisms the free abelian groups on generators $d_i : [i-1] \to [i], i \ge 1$ satisfying $d_{i+1} d_i = 0$. This is, by construction, the linear category such that presheaves on it are connective chain complexes.

If $F : \widehat{D}_1 \to \widehat{D}_2$ is a (Morita) equivalence, then it induces an equivalence on tiny presheaves. This motivates the following definition.

Definition: If $D$ is an essentially small category (possibly linear), then its Cauchy completion $\overline{D}$ is the full subcategory of $\widehat{D}$ on the tiny presheaves.

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

1. The objects of $\overline{D}$ are pairs $(d, m)$ of an object $d \in D$ and an idempotent $m : d \to d$.
2. The morphisms $(d_1, m_1) \to (d_2, m_2)$ are the morphisms $f : d_1 \to d_2$ such that $f m_1 = m_2 f = f$. Equivalently, they are the morphisms of the form $m_2 f m_1, f : d_1 \to d_2$. Composition is as in $D$.

The description of morphisms follows from the fact that the formal splitting of an idempotent $m : d \to d$ has two universal properties: it is both the equalizer and the coequalizer of $m$ and $\text{id}_d : d \to d$. Note that this construction does not depend on $D$ 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 $\overline{D}$ is obtained from $D$ 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:

1. The objects are tuples $(d_1, d_2, \dots d_n)$ of objects in $D$.
2. The morphisms $(d_1, d_2, \dots d_n) \to (e_1, e_2, \dots e_m)$ are matrices $f_{ij} : d_i \to e_j$ of morphisms in $D$. 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 $\text{Hom}(c, -)$.

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 $C$ (possibly linear) is Cauchy complete if it has all absolute colimits.

Note that we do not require that $C$ is essentially small, and in fact we will need to know that $\text{Set}$ and $\text{Ab}$ are Cauchy complete (because they are cocomplete; for $\text{Ab}$ 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 $\overline{D}$ as defined above is Cauchy complete. Let’s show that the Cauchy completion really deserves its name.

Theorem: Let $D$ be an essentially small (possibly linear) category and let $C$ be Cauchy complete (linear if $D$ is). Then the restriction functor $[\overline{D}, C] \to [D, C]$ from the category of (linear if $D$ is) functors $\overline{D} \to C$ to the category of functors $D \to C$ is an equivalence. In particular, any functor $D \to C$ extends essentially uniquely to a functor $\overline{D} \to C$.

With the right construction of the Cauchy completion we can remove the hypothesis that $D$ 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 $\overline{D}$ above, we’ll explicitly exhibit an inverse. If $F : D \to C$ is a functor, its extension $\overline{F} : \overline{D} \to C$ is uniquely determined by the fact that functors preserve absolute colimits: $\overline{F}$ splits idempotents and, in the linear case, preserves finite biproducts. Similarly, if $F, G : D \to C$ are functors and $\eta : F \to G$ is a natural transformation, then its extension $\overline{\eta} : \overline{F} \to \overline{G}$ 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.) $\Box$

Corollary: With the above hypotheses, $D$ is Cauchy complete iff $D \to \overline{D}$ is an equivalence iff the tiny objects in $\widehat{D}$ are precisely the representables iff all idempotents in $D$ split and, in the linear case, $D$ has finite biproducts.

Corollary: With the above hypotheses, the restriction functor $\widehat{\overline{D}} \to \widehat{D}$ is an equivalence; that is, $D$ is Morita equivalent to its Cauchy completion $\overline{D}$.

Corollary: Two essentially small categories $D_1, D_2$ (possibly linear) are Morita equivalent iff their Cauchy completions $\overline{D}_1, \overline{D}_2$ are equivalent.

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

Corollary: Let $M_1, M_2$ be two monoids with no nontrivial idempotents (e.g. groups). Then $M_1, M_2$ are Morita equivalent (meaning that their categories of modules in $\text{Set}$ are equivalent) iff they are isomorphic.

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

Presheaf categories

The hypotheses on a cocomplete category $C$ 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 $C$ be a cocomplete (possibly linear) category and $S$ be an essentially small full subcategory of tiny objects in $C$ such that every object of $C$ is a colimit of objects in $S$. Then the natural functor

$\displaystyle C \ni c \mapsto \text{Hom}(-, c) \in \widehat{S}$

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 $C \to \widehat{S}$ is fully faithful and essentially surjective. Let $c, d$ be two objects in $C$, and write them as colimits $\text{colim}_{i \in I} c_i, \text{colim}_{j \in J} d_j$ of objects in $S$. Since the $c_i$ are tiny,

$\text{Hom}(c, d) \cong \text{colim}_{j \in J} \text{lim}_{i \in I} \text{Hom}(c_i, d_j)$.

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

Now let $F \in \widehat{S}$ be a presheaf. By the Yoneda lemma, $F$ is a colimit of (in the linear case, finite biproducts of) representables. We can take the same colimit in $C$, and because $C \to \widehat{S}$ is cocontinuous, it gets preserved. Hence $C \to \widehat{S}$ is essentially surjective, and the conclusion follows. $\Box$

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

$\displaystyle C \ni c \mapsto \text{Hom}(s, c) \in \text{Mod}(\text{End}(s))$

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 $S = \{ 0, s, s \sqcup s, \dots \}$ (rather than to $S = \{ s \}$), then use the fact that $\widehat{S}$ is equivalent to the category of $\text{End}(s)$-modules because finite biproducts are absolute. $\Box$

The hypothesis that $C$ is abelian is somewhat misleading: if “projective” is interpreted to mean “$\text{Hom}(s, -)$ preserves coequalizers” or “compact projective” is interpreted to mean tiny, then we only need to know that $C$ is a cocomplete linear category, and in particular we do not need to assume anything about limits in $C$.

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 $R$ is a ring, the rings Morita equivalent to $R$ are precisely the endomorphism rings of finitely presented projective modules which generate $\text{Mod}(R)$ in the above sense.

### 7 Responses

1. […] from little to big by taking modules / presheaves, and we can pass from big to little by taking tiny objects. (This can at best recover the Cauchy completion of the original little -linear category, but this […]

2. on February 12, 2016 at 3:14 pm | Reply Ingo Blechschmidt

Thank you very much for this very enlightening and useful blog post; your work is very much appreciated.

Just to be absolutely clear, if $D$ is a linear category, then by $\hat D$ you mean the category of $\mathrm{Ab}$-enriched functors $D^{\mathrm{op}} \to \mathrm{Ab}$. That is, not the full category of $\mathrm{Ab}$-valued presheaves, but only the $\mathrm{Ab}$-enriched ones.

• You’re welcome! And yes, that’s right.

3. […] can be thought of as a categorification of the usual method of producing an idempotent from a section-retraction pair, namely a pair of morphisms and such […]

4. […] be a ring. Previously we characterized the finitely presented projective (right) -modules as the tiny objects in : the […]

5. […] « Tiny objects […]