It’s also common to think of monads as generalized monoids; previously we discussed why this was a reasonable thing to do.

Today we’ll discuss a different intuition: monads are (loosely) categorifications of idempotents.

**Conventions**

As previously, in this post compositions will be done in diagrammatic order, so if and are two morphisms, their composite will be denoted , or sometimes (which is independent of, but strongly suggests, the diagrammatic order).

This will end up switching the role of “left” and “right” in a few statements relative to the usual order of composition. For example, it will switch which adjoint goes first when constructing a monad out of an adjunction, so be careful when matching up statements in this post to statements elsewhere. But as we’ll see this convention has nice properties.

It will also force us to use the following curious-looking notation: if is a functor and is an object, we’ll denote the value of on by (thinking of as a morphism ). Read this in your head the same way you would read “c squared.”

Several different kinds of composition throughout this post will be denoted by concatenation, and it should hopefully be clear from the types of the objects involved what kind of composition is meant. For example, if is a 2-morphism and is a 1-morphism such that the compositions are defined, then denotes the vertical composition of with the identity .

**Monads in a 2-category**

Like the definition of adjoints, the definition of monads is purely 2-categorically equational, so makes sense in any 2-category and is preserved by any 2-functor, and we’ll introduce it at this level of generality.

In short, a monad on an object in a 2-category is a monoid in the monoidal category of endomorphisms of . More explicitly, a **monad** is a 1-morphism from an object to itself together with a **unit** 2-morphism and a **multiplication** 2-morphism satisfying the following compatibilities. The first compatibility (**associativity**) says that the two ways of using to write down a 2-morphism agree; explicitly,

.

The second compatibility (**unit**) says that

.

Dually, a **comonad** is a monad in the 1-opposite 2-category (reversing the order of composition of 1-morphisms): it still starts out as a 1-morphism but has a counit and a comultiplication .

Our favorite way of producing monads and comonads will be via adjunctions, as follows. Let be an adjunction, so is the left adjoint and is the right. Let

denote the unit of the adjunction, and let

denote the counit. (This is a place where diagrammatic order matters.)

**Proposition:** is a monad on , with unit the unit of the adjunction and multiplication the map

Dually, is a comonad on , with counit the counit of the adjunction and comultiplication derived from the unit.

This 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 that . (This is another place where diagrammatic order matters. is the retraction and is the section.) This condition means that satisfies , which categorifies to the above.

*Example.* Let be the one-object 2-category corresponding to a monoidal category . Then a monad in is just a monoid in , and a comonad in is a comonoid in . (Working backwards, if monads and comonads in a 2-category categorify idempotent morphisms in a category, then monoids and comonoids in a monoidal category categorify idempotent elements of a monoid.)

An adjoint pair in is a dual pair of objects of (here indicates the right dual of ). So the construction above specializes to the observation that the tensor product has a natural monoid structure, which should be familiar from the case of vector spaces, where it is a matrix algebra. (What might be less familiar is the dual statement that has a natural comonoid structure.)

More generally, if is a closed symmetric monoidal category with internal hom , then

so by the Yoneda lemma we conclude that for a dualizable object , the tensor product is canonically isomorphic to the internal endomorphism object (and a little more work shows that this is even an isomorphism of monoid objects).

In general, we get that is a monoid in which naturally acts on from the left, and on from the right. Furthermore,

where denotes the monoidal unit, so the “points” of (the result of applying ) are endomorphisms of .

*Example.* Let be the 2-category of posets. Then a monad in is a closure operator on a poset .

Explicitly, must first of all be a morphism of posets, so implies . Next, the unit becomes the condition , and finally, multiplication becomes the condition . Since implies , we have . (So all monads on posets are genuinely idempotents.)

The construction of monads from adjunctions specializes here to the construction of closure operators from adjunctions between posets, also known as Galois connections. This reproduces various familiar closure operators in mathematics, such as Zariski closure.

*Example.* Let be the Morita 2-category of algebras, bimodules, and bimodule homomorphisms over a commutative ring . Then a monad in is an algebra object in -bimodules over , where is some -algebra; we’ll call this an -algebra for short, although note that even if is commutative it doesn’t reduce to the usual notion of algebra over a commutative ring.

An adjoint pair in is, as we saw previously, a pair consisting of an -bimodule which is f.g. projective as a right -module (the left adjoint) and its -linear dual regarded as a -bimodule (the right adjoint). The notation is meant to again evoke the special case of vector spaces, which we recover when is a field. The corresponding monad is the -algebra , which can be thought of as the algebra of -linear endomorphisms of (acting on the left) or of (acting on the right).

**Left and right modules**

Just like monoids, monads have modules over them. A **right module** over a monad is an object , a 1-morphism , and an action 2-morphism

satisfying the associativity condition

and the unit condition

.

Dually, a **left module** is an object , a 1-morphism , and an action 2-morphism satisfying the obvious duals of the above conditions.

Our favorite way of producing modules is again via adjunctions. If is an adjunction giving rise to , then the left adjoint is naturally a left module over , and dually the right adjoint is naturally a right module over . (If we had stuck to compositions in the usual rather than diagrammatic order this would be reversed.) In fact the two together form a kind of bimodule over .

Classically, in , what is usually called a **module** or **algebra** for a monad is an object together with an action map satisfying the same axioms as above. This is a special kind of right module where is the terminal category. More generally, in a right module can be interpreted as a family of -algebras parameterized by . It’s less clear what a left module is.

Thinking of monads as idempotents , right modules categorify morphisms such that , or equivalently that equalize and , and dually left modules categorify morphisms such that , or equivalently that coequalize and . These are used to state the universal property of the equalizer and coequalizer of and respectively, which can be thought of as the object of invariants (fixed points) or the object of coinvariants (orbits), respectively, under the action of , and (because is an idempotent) which are canonically isomorphic.

This categorifies as follows. The categorification of invariants is the **Eilenberg-Moore object** of a monad . This is, if it exists, the universal right module over : that is, it is equipped with a 1-morphism and an action 2-morphism making it a right module, and any other right module uniquely factors through it. Said another way, right module structures on an object are equivalent to 1-morphisms .

Dually, the categorification of coinvariants is the **Kleisli object** . This is, if it exists, the universal left module over : that is, it is equipped with a 1-morphism and an action 2-map making it a left module, and any other left module uniquely factors through it. Said another way, left module structures on an object are equivalent to 1-morphisms .

*Example.* In , the Eilenberg-Moore category of a monad on a category turns out to be the category of -algebras (categorifying how the invariants of an idempotent endomorphism of a set is the set of its fixed points). The right module structure on has 1-morphism the forgetful functor given by forgetting the -algebra structure and action 2-morphism the natural transformation whose components are given by the action maps

of the -algebras .

The universal property of says that a right module structure on a category is a functor , or in other words a family of -algebras parameterized by , which at least makes sense at the level of objects.

The Kleisli category turns out to have the same objects as , but where a morphism from to is a **Kleisli morphism**, namely a morphism . Composition is as follows: if and are two Kleisli morphisms, then their composite is

.

The left module structure on has 1-morphism the functor which is the identity on objects and which sends an ordinary morphism to the Kleisli morphism

.

Its action 2-morphism is the natural transformation whose components are the identity , regarded as a Kleisli morphism from to .

The universal property of says that a left module structure on a category is a functor . It’s less clear to me what this means.

**Adjunctions from monads**

A **splitting** of an idempotent is a pair of morphisms (a section-retraction pair) such that and ; we say that **splits** (or, in the terminology we used earlier, is a **split idempotent**) if it admits a splitting. Under very mild hypotheses (e.g. the existence of either equalizers or coequalizers), every idempotent admits a unique (up to unique isomorphism) splitting, where is simultaneously both the object of invariants and the object of coinvariants of . To exhibit this isomorphism we start by writing down a map from coinvariants to invariants (when they both exist), and this map exists because an idempotent both equalizes and coequalizes itself.

This categorifies as follows. (All of the terminology I’m about to introduce is nonstandard.) A **splitting** of a monad is a pair of adjoint 1-morphisms together with an isomorphism of monads; we say that **splits** (or is a **split monad**) if it admits a splitting.

*Example.* As above, let be the one-object 2-category corresponding to a monoidal category . A monad in is a monoid in , and a monoid splits iff there is a right dualizable object such that as monoids.

Specializing to the case that is the symmetric monoidal category of modules over a commutative ring , a monad in is a -algebra , and an algebra splits iff there is a f.g. projective -module such that as -algebras, or equivalently iff

.

Such a need not either exist or be unique. Lack of existence is clear; for example, when is a field the above condition says that is a matrix algebra, and there are plenty of non-matrix algebras. To see lack of uniqueness we can observe that if then we can take to be any invertible -module, so need not be unique if the Picard group is nontrivial.

(There’s something interesting to say here even when is a field. It’s possible for a -algebra not to split over but to split over a finite extension of ; such algebras correspond to nontrivial classes in the Brauer group . Incidentally, in this area there’s an existing definition of “split,” and it’s a happy accident as far as I can tell that the two uses coincide in this special case.)

We’ve learned that in general, splittings of monads neither exist nor are unique. However, it’s true that the Eilenberg-Moore object and Kleisli object (when they exist) both give rise to splittings, as follows. In particular, all monads in split (so arise from adjunctions).

A monad is naturally both a left and a right module over itself. Because is a right module over itself, admits a factorization

through the universal right module . The pair of 1-morphisms are in fact adjoint: the unit of the adjunction comes from the unit of , while the counit comes from the action 2-morphism , as follows. Part of the definition of a right module implies that the action 2-morphism is in fact a morphism of right -modules, and so by the universal property of the Eilenberg-Moore object it factors through , giving a 2-morphism which is our counit.

The verification of the zigzag identities isn’t hard but is annoying without good notation, so we’ll omit it. One of them follows from the unit condition for the right -module structure on , and the other one follows from the unit condition for the monad structure on , together with the same factoring-through- argument as above.

Hence whenever the Eilenberg-Moore object exists, it exhibits a splitting of . Dually (by reversing 1-morphisms), whenever the Kleisli object exists, it also exhibits a splitting of . (Hence Eilenberg-Moore and Kleisli objects can’t always exist in 2-categories with monads that aren’t split.)

But we can say more than this. The left adjoint now admits a natural left -module structure (since it’s the left adjoint of an adjunction that splits ), so by the universal property of the Kleisli object (if it exists), we get a further factorization

of . If our situation were exactly analogous to the case of idempotents, the middle morphism would be an equivalence. This isn’t true, but it’s very close, and it’s true if in addition is an **idempotent monad** (meaning that the multiplication 2-morphism is an isomorphism). In these arise from reflective subcategories, for example the adjunction between and .

*Example.* In the case of , let be a monad on a category . Then, as we saw above, the Eilenberg-Moore category is the category of -algebras, so objects equipped with action maps satisfying unit and associativity conditions. There’s an obvious forgetful functor given by forgetting the action map, and its left adjoint acts on objects as

.

This exhibits as the **free -algebra** on (categorifying how, if is an idempotent endomorphism of a set, then for every , the element is a fixed point of ). The -algebra structure on comes from the right -module structure on itself; explicitly, the action map is

and unit and associativity for it reduce to unit and associativity for . The claim that is the free -algebra says concretely that if is an -algebra (with action map ), then we have a natural bijection

.

This bijection is, explicitly, the following. The natural map from the LHS to the RHS sends an -algebra morphism to the composite

The natural map from the RHS to the LHS sends a morphism to the composite

.

The verification that these two maps are inverses to each other is purely equational. In one direction (starting from an -algebra morphism ), we need the identity

which we can prove as follows. being an -algebra morphism means, by definition, that

.

The identity we need then becomes

which follows from the unit condition on .

In the other direction (starting from a morphism ), we need the identity

The naturality of means precisely that

so the identity we need becomes

which follows from the unit condition on the -algebra structure on .

According to our abstract argument above, the left adjoint of the forgetful functor from the Eilenberg-Moore category should naturally factor through the Kleisli category . This factorization can be described as follows. There is a natural functor

sending an object to the free -algebra , and sending a Kleisli morphism to the composite

.

The free -algebra functor factors through this functor in the obvious way. In fact more is true: because we know that is the free -algebra on , we have

from which it follows that in fact the functor from the Kleisli category to the Eilenberg-Moore category is fully faithful: it exhibits the Kleisli category as the full subcategory of free -algebras among all -algebras.

*Subexample.* If is a poset, so that is a closure operator, then both the Eilenberg-Moore and Kleisli posets can be identified with the poset of -closed elements of (those elements such that ), and the natural map is an equivalence. This reflects the fact that closure operators are idempotent, so every closed element is a “free” closed element .

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Below we work throughout over a field of characteristic zero.

For starters, the universal enveloping algebra functor , which a priori takes values in algebras (it’s left adjoint to the forgetful functor from algebras to Lie algebras), in fact takes values in Hopf algebras. This upgraded functor continues to be a left adjoint, although the forgetful functor is less obvious. Given a Hopf algebra , its primitive elements are those elements satisfying

where is the comultiplication. The primitive elements of a Hopf algebra form a Lie algebra, and this gives a forgetful functor from Hopf algebras to Lie algebras whose left adjoint is the upgraded universal enveloping algebra functor.

The key observation is that this upgraded functor is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms and Hopf algebra homomorphisms . This is more or less equivalent to the claim that the natural inclusion induces an isomorphism from to the Lie algebra of primitive elements of , which can be proven using the PBW theorem.

Hence Lie algebras embed as a full subcategory of Hopf algebras; that is, they can be thought of as Hopf algebras satisfying certain properties, rather than having extra structure (in the nLab sense). What are these properties? For starters, they are all cocommutative. This is important because cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (this is *not* true with “cocommutative” dropped!), which in turn can be interpreted as a category of infinitesimal spaces. (For example, this category is cartesian closed, and in particular distributive.)

Hence Lie algebras are group objects in cocommutative coalgebras satisfying some property (for example, “conilpotence”; see Theorem 3.8.1 here).

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The pictures in this post can be interpreted in at least two ways. On the one hand, they are graphs of groups in the sense of Bass-Serre theory, and on the other hand, they are also dessin d’enfants (for the rest of this post abbreviated to “dessins”) in the sense of Grothendieck. But you don’t need to know that to draw and appreciate them.

**How to draw a group**

Using the fact that is a free product, we can think of the classifying space as the wedge sum of the classifying spaces and . These spaces are the homotopy quotients of points (which we give two different names to make the diagrams we’re about to draw easier to parse) by the trivial actions of and . Intuitively, they can be thought of as “half a point” and “a third of a point,” respectively. We’re going to draw as their wedge sum using the following dessin (using QuickLaTeX, because WordPress doesn’t support xymatrix):

The reason we want to draw is that we’re going to think of finite index subgroups as finite covers , and we’ll be drawing covers of this picture in a way closely analogous to how we can draw finite index subgroups of free groups by drawing covers of graphs.

What do covering spaces of this thing look like? The preimage of the edge will be a disjoint union of edges, so we expect to see something like a graph. The interesting question is what happens locally around the “orbifold points” . The covering space theory of these spaces is very simple: they both admit a single nontrivial connected cover, namely points (a double and a triple cover, respectively), and every covering space is a disjoint union of some copies of these nontrivial covers and the trivial cover. (This reflects the corresponding decomposition of -sets resp. -sets into transitive components.)

What’s more interesting is how the edge and the orbifold points interact. When gets covered by in some cover, the portion of the edge adjacent to it is doubled; similarly, when gets covered by in some cover, the portion of the edge adjacent to it is tripled and acquires a cyclic ordering (which describes how acts on the fiber over a basepoint in ).

For example, here is the unique connected double cover of .

The orbifold point has “unfolded” itself into an ordinary point , in the process doubling the edge, but the orbifold point has no interesting double covers, so it can’t “unfold” yet.

We can extract a lot of information from this dessin. Recall that in order to get a subgroup of of a space from a cover of it, we need to pick both a basepoint and a lift of that basepoint to the cover. Here it’s easiest to pick a basepoint in the middle of the edge, where there’s no funny orbifold business going on, and so a lift of the basepoint corresponds to an edge in the cover. The conjugates of the subgroup we get come from picking different edges, and two edges give the same subgroup iff they are related by an automorphism of the cover. (Moreover, “automorphism of the cover” means more or less the obvious thing in terms of dessins; we’ll be more precise about this later.) In particular, we can see visually when a subgroup is normal: it’s iff the automorphism group of the corresponding dessin acts transitively on edges, which is the case here. More generally, the number of subgroups conjugate to a given subgroup is given by the number of orbits of the action of the automorphism group on the edges of its dessin.

After picking an edge, the corresponding subgroup of is given by the stabilizer of this edge with respect to the action of on edges given as follows: if we write as a free product

then sends an edge which is connected to an “unfolded” white point to the other edge connected to (and otherwise fixes the edge), and sends an edge which is connected to an “unfolded” black point to the next edge in the cyclic order (and otherwise fixes the edge). We can find a set of generators for the stabilizer by finding a spanning tree of the dessin (which is unnecessary here) and looking at the corresponding loops we get in the same way as in the case of graphs, while remembering that there are extra loops coming from and .

More topologically, this procedure computes the (“orbifold”) fundamental group of the dessin, regarded as a picture of a space (“1-dimensional orbifold”) obtained by gluing copies of and to some of the vertices of a graph, via iterated application of Seifert-van Kampen.

(This argument, cleaned up and made fully rigorous, shows that every subgroup of , not necessarily finite index, is a free product of copies of , and . For a generalization of this result to arbitrary free products, which you should be able to guess from here, see the Kurosh subgroup theorem.)

From this description we see that the unique subgroup of index in (which can be described as the kernel of the natural map killing the second factor) is the free product

.

Now let’s move on to index . Here there is an obvious triple cover, namely

given by “unfolding” but covering trivially. (We need to pick a cyclic ordering of these three edges, but it doesn’t matter too much because the resulting dessins are isomorphic.) The corresponding subgroup is normal (as we can see because the automorphism group of this dessin acts transitively on edges); in fact it is the kernel of the natural map killing the first factor. This dessin shows that it is the free product

.

But we know that there are subgroups of index . The other are conjugate and come from the same dessin, namely

where has now also been “unfolded” once. The cyclic ordering on the edges around (which we haven’t drawn) matters now: without it this dessin would have a nontrivial automorphism exchanging the two edges around .

This is the first example we’ve looked at where the underlying graph is not a tree, and accordingly we get our first factor in the free product decomposition: this subgroup (we’ll pick our basepoint to be the odd edge out) is the free product

.

We get the free generator from the loop in the dessin as follows: takes us from our basepoint to (depending on the cyclic ordering) one of the other two edges around . Going around the loop involves passing through , which corresponds to acting on one of the edges around using to get the other one. Finaly, going back to our basepoint involves passing through , which corresponds to using again.

As we’ll see later, this subgroup is in fact (conjugate to) the smallest congruence subgroup .

Let’s move faster now. The subgroups of index form conjugacy classes of size coming from the following dessins, which have trivial automorphism group:

The dessins tell us that these subgroups are free products and respectively. The second one is (conjugate to) the second smallest congruence subgroup .

The subgroups of index form a single conjugacy class coming from the following dessin, which again has trivial automorphism group:

This dessin tells us that these subgroups are free products .

Index is when things start getting exciting because we can now have two trivalent vertices : among other things, we see our first appearance of subgroups which are torsion-free (and hence free). The dessin

has both and fully “unfolded.” Abstractly the corresponding subgroups are free products (so free groups on two generators).

There are actually two dessins here depending on whether or not the cyclic orderings around the two copies of match, and they are not the same, although they both have transitive automorphism group and so correspond to normal subgroups. One of them corresponds to the kernel of the natural map

while the other corresponds to the kernel of the natural map

and hence to the congruence subgroup .

But there are also torsion-free subgroups of index that are not normal, coming from the dessin

which does not have transitive automorphism group. There are orbits of edges under the action of the automorphism group (which is , coming from the obvious reflection), so we get conjugate subgroups, one of which them is the congruence subgroup .

The congruence subgroup also has index , and its dessin is

so in particular it is not free: instead, it is the free product . This dessin has trivial automorphism group, so we get conjugates.

There are dessins left for index :

The first dessin here is actually dessins depending on whether the cyclic orderings match again. Altogether we find that there are conjugacy classes of subgroups of of index , with sizes (in order based on the order in which we drew the dessins)

in agreement with our previous count. This means that the sequence of conjugacy classes of subgroups of index in begins

which we can look up on the OEIS, getting the sequence A121350.

**Dessins as graphs**

The graphs appearing here as dessins are bipartite graphs ( vertices can only be connected to vertices) where vertices have degree or and vertices have degree or . In addition, the vertices of degree are equipped with a cyclic ordering on their edges.

These dessins can be thought of as built up of copies of a single “puzzle piece,” namely an edge with “half a point” on one side and a “third of a point” on the other (perhaps a half and a third of a sphere, with appropriate connecting bits). The name of the game is that you take of these pieces and combine them by combining two half-points to make a whole point and three third-points to make a whole point . You can even encode the cyclic orders around the points by arranging the connecting bits so that three pieces can only connect in one cyclic order, and one of the connecting bits has an arrow pointing to the other. I wonder if someone would be interested in manufacturing these…

**Euler characteristic**

It’s not hard to see that torsion-free subgroups can only exist when the index is divisible by . This is because, thinking in terms of the corresponding transitive -sets , torsion-freeness is equivalent to both the generator of order and the generator of order acting with no fixed points. The first condition means is divisible by , while the second means is divisible by .

A more geometric way to see this is using the notion of virtual or **orbifold Euler characteristic**, which in some sense generalizes both the usual Euler characteristic and groupoid cardinality. We will only compute this number for (classifying spaces of) finitely generated groups which are virtually free, meaning that they contain a free subgroup of finite index (necessarily also finitely generated), which is the case for all finite index subgroups of . In this case the Euler characteristic is determined by the following two axioms:

- , and
- If is a subgroup of of index , then .

Note that is the Euler characteristic in the usual sense of , which can be described as a wedge of circles. The second axiom is motivated by the fact that the map is an -fold cover.

Since has a free subgroup on two generators of index , its Euler characteristic is

.

But the Euler characteristic axioms also imply that if is a finite group, then (since is a subgroup of of index , and , being the free group on generators, has Euler characteristic ), and we can arrive at this formula in another way assuming that inclusion-exclusion holds in a suitable form for this notion of Euler characteristic: namely, is obtained by connecting and by an edge, so if the usual inclusion-exclusion formula holds, then we should have

.

Assuming that Euler characteristic is well-defined, it now follows that a subgroup of of index has Euler characteristic , which is an integer iff is divisible by . Hence we learn not only that the free subgroups must have index but that they must be free on generators.

You can now go back through the examples above and verify that their Euler characteristics computed using either their index or inclusion-exclusion agree: for example, the unique subgroup of index has Euler characteristic

.

**Drawing congruence subgroups**

The dessins we’re drawing describe sets equipped with a transitive action of , namely the edges of the dessins, in the manner described above. For the congruence subgroups , the corresponding -set is given by the natural action of on itself, while for the congruence subgroups , the corresponding -set is given by the natural action of on the projective line .

may be a little unfamiliar if is not prime; for an arbitrary positive integer , it can be described as the quotient of the set of coprime pairs by the equivalence relation of multiplication by a unit in . Loosely speaking, these pairs can be thought of as fractions modulo , although if isn’t prime they have some funny and somewhat unexpected behavior. The size of , and hence the index of , is

and so we can compute that have indices respectively, and these are the only ones of index at most . The index of is the size of ; the only one of these groups of size at most is .

Here’s how to draw the dessin corresponding to . The generator of order can be taken to be the fractional linear transformation

acting on the elements of thought of as fractions as above, while the generator of order can be taken to be the fractional linear transformation

.

Since the action of is transitive, we can reach every element of by repeatedly applying each of these transformations to any particular point, say (by which we mean the ordered pair ). We can build up the dessin from here by repeatedly applying and and adding vertices and edges as appropriate.

For example, here is the dessin for drawn in this way, now with the edges labeled by elements of :

The implied cyclic order around the vertices, corresponding to the action of , is clockwise.

None of what we’ve said requires the subgroups to have finite index, so here is a small slice of the dessin for the subgroup generated by translation , which corresponds to the action of on the projective line (since translation generates the subgroup fixing ). The dessins are all quotients of this one.

**Higher-dimensional pictures**

Dessins give a “1-dimensional” picture of the finite index subgroups of the modular group in terms of certain 1-dimensional orbifolds modeling the classifying spaces . It is also possible to give 2-dimensional, 3-dimensional, and 4-dimensional pictures.

- In 2 dimensions, we can look at the quotient . These quotients are generalizations of modular curves, and their points describe elliptic curves equipped with various extra data. Geometrically they are 2-dimensional orbifolds which become ordinary surfaces if is torsion-free. Remarkably, Belyi’s theorem implies that in this case, the corresponding Riemann surfaces arise from algebraic curves defined over , and that all algebraic curves over arise in this way.
- In 3 dimensions, we can look at the quotient . These quotients are the unit tangent bundles of the modular quotients above, and are always ordinary 3-manifolds with PSL geometry. itself is the complement of the trefoil knot in , so the quotients are finite covers of this complement. See this blog post by Bruce Bartlett for more on this.
- In 4 dimensions, we can look at the quotient , where denotes the preimage of in . (I won’t write out this action because I have very little to say about this construction.) These quotients are elliptic surfaces with base and fiber over a point the elliptic curve corresponding to that point. Elliptic surfaces arising in this way are called elliptic modular surfaces.

We can recover dessins as drawn above from the modular curves as follows (although these are not quite the dessins described in the Wikipedia article, which I believe come from finite index subgroups of rather than ). These curves come with a natural map , where is the usual modular curve parameterizing elliptic curves with no extra structure. There is a map

sending an elliptic curve (where ) to its j-invariant which is almost an isomorphism. However, the action of on is not free, so we shouldn’t just take the ordinary quotient: the moduli space of elliptic curves is really a stack, and we can think about this stacky structure in terms of two special “orbifold points” in the quotient .

One of these special points corresponds to the elliptic curve given by quotienting by the square lattice, which has an extra automorphism of order given by multiplication by , while the other corresponds to the elliptic curve ( a primitive sixth root of unity) given by quotienting by the hexagonal lattice, which has an extra automorphism of order given by multiplication by . These extra automorphisms are reflected in the fact that these points have nontrivial stabilizers under the action of . Namely, the generator of order stabilizes , and the generator of order stabilizes .

So we can draw the dessin for itself as a path in (say a geodesic with respect to the hyperbolic metric) connecting the two special orbifold points. The preimage of this path in for a subgroup (not necessarily finite index) of is then the dessin associated to . When the corresponding dessin is a tree sitting inside (I think it is half of the 1-skeleton of the usual tesselation of the hyperbolic plane on which acts by symmetries), and this tree can in fact be used to prove that .

It follows that if is torsion-free, then the dessin of can be drawn on a surface whose genus it is possible to calculate. In the simplest case this surface has genus zero and is a punctured sphere; the corresponding dessins are then planar. This is the case, for example, for the modular groups , where the corresponding dessins encode Cayley graphs for the groups

.

These dessins turn out to be essentially 1-skeleta of the tetrahedron, cube, and dodecahedron respectively. More precisely, the dessins can be obtained by coloring the vertices of the 1-skeleta , then adding a to the middle of each edge (and then cyclically ordering appropriately).

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For starters, let be a distributive category with terminal object , and let be the coproduct of two copies of . For an object , what does look like? If and is a sufficiently well-behaved topological space, morphisms correspond to subsets of the connected components of , and naturally has have the structure of a Boolean algebra or Boolean ring whose elements can be interpreted as subsets of the connected components of .

It turns out that naturally has the structure of a Boolean algebra or Boolean ring (more invariantly, the structure of a model of the Lawvere theory of Boolean functions) in any distributive category. Hence any distributive category naturally admits a contravariant functor into Boolean rings, or, via Stone duality, a covariant functor into profinite sets / Stone spaces. This is our “connected components” functor. When the object this functor outputs is known as the Pierce spectrum.

This construction can be thought of as trying to do for what the étale fundamental group does for .

**The proof**

In general, in a category with finite products, the set naturally acquires the structure of a model of the Lawvere theory generated by , namely the full subcategory on the finite products . (In our previous post on Lawvere theories we made use of the analogous construction for finite coproducts of .)

How can we calculate this Lawvere theory when in a distributive category? Using distributivity! By induction, it’s not hard to see that is literally the coproduct of copies of in any distributive category. It follows that a morphism is a -tuple of morphisms . There are two distinguished such morphisms, namely the two coproduct inclusions into , and using these two morphisms we can write down for any Boolean function a map , in a way that agrees with products and composition.

(**Edit: **This section previously contained a mistake which was pointed out by Zhen Lin in the comments.)

More abstractly, if is any distributive category then there is a natural functor given by sending a finite set to the coproduct , and by distributivity this functor preserves finite products in addition to finite coproducts. Above we’re looking at the image of the sets and morphisms between them under this functor, which are the objects and some distinguished morphisms between them.

(Note that distributivity can even be regarded as the crucial ingredient in the proof we outlined previously that Boolean functions can all be generated by constants and the if-then-else ternary operator; the key observation in that proof is that we can distribute , which is what let us express -ary Boolean functions in terms of pairs of -ary Boolean functions using if-then-else.)

It follows that in any distributive category, naturally acquires the structure of a model of the Lawvere theory of Boolean functions (so, according to taste, either a Boolean algebra or a Boolean ring).

**The intuition**

Intuitively, a morphism is a way to disconnect into two pieces, namely the pullbacks where the morphism is either of the two coproduct inclusions, although is probably not the coproduct of these two pieces unless we assume the stronger condition that the ambient category is extensive. These can in turn be thought of as subsets of the connected components of , and the Boolean algebra / ring structure then comes from the usual logical operations on such subsets, e.g. intersection and union.

**The example of affine schemes**

The example of affine schemes is worth working through in detail. First, the terminal object in affine schemes is , and so is . It’s enlightening to rewrite this using the isomorphism

which reveals that is the free commutative ring on an idempotent, and hence that morphisms , for a commutative ring, are naturally in bijection with idempotents in . Idempotents are in turn in bijection with direct product decompositions

and so morphisms of affine schemes really do correspond to ways to disconnect into pieces . Abstractly, this reflects the fact that is extensive, and not only distributive.

The abstract discussion above implies that the set of idempotents in canonically acquires the structure of a Boolean ring, which we’ll denote . The multiplication in is just the usual multiplication on idempotents, but the addition is the following modified addition : if are idempotents, then

.

Note that the third term disappears if has characteristic . Geometrically we can think of as indicator functions of unions of connected components of ; then the RHS describes the operation that must be performed on indicator functions, regarded as taking values in , to get XOR.

Altogether, we find that taking idempotents gives a functor from commutative rings to Boolean rings. (Curiously, it is not the right adjoint to the inclusion of Boolean rings into commutative rings, although it does preserve limits.) Taking opposite categories, we get a functor from affine schemes to profinite sets called the **Pierce spectrum** functor, which we’ll denote

.

consists off a point iff is connected, meaning it has exactly two idempotents, and (which are not equal, so the zero ring is not connected). This condition is equivalent to the Zariski spectrum being connected as a topological space, and holds, for example, for any integral domain.

The Pierce spectrum organizes the Zariski spectrum into “connected components” as follows. If is a prime ideal of , then the quotient map induces a map

on Pierce spectra. Since the Pierce spectrum of is a point, we can associate to a unique point in , which intuitively is the connected component to which the point belongs. This construction organizes into a natural map

where on the LHS denotes the prime spectrum as a topological space. (Curiously, this is a map on underlying topological spaces between two ringed spaces which cannot be promoted to a map of ringed spaces, basically because the natural inclusion of the Boolean ring object into the affine line is not a morphism of ring objects.)

The fibers of this map can be given natural affine scheme structures, as follows. An element of the Pierce spectrum can be thought of as a homomorphism of Boolean rings. These can be regarded as generalizations of ultrafilters, which they reduce to in the special case that is a product of copies of (so that the Pierce spectrum is the Stone-Čech compactification ). This occurs whenever is a product of connected rings (e.g. integral domains).

Accordingly, there is a generalization of an ultraproduct construction we can perform here: given , we can quotient by the ideal generated by the elements of (which, recall, are idempotents in ). The result, which we’ll call in deference to ultraproduct notation, is a connected ring, since every idempotent in belongs to either or and hence gets identified with either or in this quotient, and in fact any morphism from to a connected ring must factor through one of these morphisms . Geometrically this says that any morphism from a connected affine scheme to factors through some , so these quotients really do deserve the name “connected components.”

*Example.* Let . On math.SE, Martin Brandenburg recently asked what one can say about the Zariski spectrum of , and Eric Wofsey gave an excellent answer in terms of looking at the fibers of over its Pierce spectrum exactly as above (which in fact motivated the writing of this post).

In this case is , so the Pierce spectrum is , which can be identified with the space of ultrafilters on , and the fibers are given by ultraproducts. These ultraproducts admit many interesting prime ideals: for example, if is any sequence of primes, then we get a quotient map

to an ultraproduct of fields, which is again a field. So there is a prime ideal for every “nonstandard prime.”

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In general, suppose you find yourself in some category. What sort of behavior could you look for that might qualify as “behaving like a category of spaces”?

One thing to look for is **distributivity**. Recall that a distributive category is a category with finite products and finite coproducts such that finite products distribute over finite coproducts; more explicitly, the natural maps

should be isomorphisms, and also the natural maps should be isomorphisms, where denotes the initial object. (Curiously, distributive categories are themselves like categorified versions of commutative rings.)

This is a pretty good test. The following familiar categories are distributive:

- More generally, any bicartesian closed category, and in particular any topos

These are all reasonable candidates for categories of “spaces.” On the other hand, the following familiar categories are not distributive:

- More generally, any nontrivial category with a zero object, and in particular any abelian category

You might object that there is also an entire field of mathematics dedicated to treating groups as geometric objects. I contend that the geometric object a group describes is actually a groupoid, and is distributive!

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Since this category has finite products and is freely generated under finite products by a single object, namely , it is a Lawvere theory.

**Question: **What are models of this Lawvere theory?

**The answer**

The answer is that they are Boolean algebras, and also that they are Boolean rings. Equivalently, the standard axiomatizations of Boolean algebras resp. Boolean rings describe two different presentations of the Lawvere theory of Boolean functions. In the Boolean algebra presentation, the generators are

- two constants (true and false, or and ),
- a unary operator (NOT), and
- two binary operators (AND and OR),

while in the Boolean ring presentation, the generators are

- two constants ( and ),
- two binary operators (AND and XOR).

The relations are given by the various axioms that these satisfy.

It’s an interesting observation that either of these small sets of Boolean functions already suffice to build arbitrary ones. This means that logic circuits built out of these basic components can compute any Boolean function, which is maybe the zeroth interesting theorem in computer science. (The first interesting theorem, which is much more interesting, is the existence of universal Turing machines.)

Here’s a proof that either of these sets of Boolean functions generate all Boolean functions, although I won’t prove anything about relations. We only need to worry about Boolean functions of the form . By conditioning on whether or not the first bit of the input is or , the data of such a Boolean function is equivalent to the data of two other Boolean functions , namely the Boolean function of the other bits of the input you get if the first bit of the input is , and the function you get if the first bit of the input is . In terms of these two functions, itself can be described as “if the first bit is , then return ; else, return .”

Inducting on , this argument shows that all Boolean functions can be generated by

- two constants ( and ), and
- a ternary operator “if-then-else.”

This ternary operator is known in computer science as the conditional operator or ternary operator and in logic as conditional disjunction.

(I once spent some time asking myself what a programming language is from the point of view of category theory. My tentative answer is that a programming language is a tool for naming morphisms in categories. We can think about the generators above as a toy programming language whose only data type is booleans and whose only control structure is if-then-else, and the way in which I happened upon it was as a tool for naming morphisms in the Lawvere theory of Boolean functions. More complicated programming languages can be thought of as tools for naming morphisms in more complicated categories, such as cartesian closed categories.)

It follows that in order to show that some collection of Boolean functions generates all Boolean functions (a condition known in logic as functional completeness), it suffices to show that it can generate , and if-then-else.

In terms of logical implication , and as a function of three bits , if-then-else can be written

.

Logical implication can itself be written as , so if-then-else can be written in terms of (NOT, OR, AND). This shows that the Boolean algebra generators generate all Boolean functions. To show that the Boolean ring generators also generate all Boolean functions, we need to write NOT, OR, and AND in terms of XOR () and AND (). We can get using , and we already have AND, which means we can get OR using

.

Hence the Boolean ring generators also generate all Boolean functions.

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There is also a generating function we can write down that addresses this question, although it gives the answer less directly. It can be derived starting from the following construction. If is a groupoid, then , the **free loop space** or **inertia groupoid **of , is the groupoid of maps , where is the groupoid with one object and automorphism group . Explicitly, this groupoid has

- objects given by automorphisms of the objects , and
- morphisms given by morphisms in such that

.

It’s not hard to see that , so to understand this construction for arbitrary groupoids it’s enough to understand it for connected groupoids, or (up to equivalence) for groupoids with a single object and automorphism group . In this case, is the groupoid with objects the elements of and morphisms given by conjugation by elements of ; equivalently, it is the homotopy quotient or action groupoid of the action of on itself by conjugation.

In particular, when is finite, this quotient always has groupoid cardinality . Hence:

**Observation: **If is an essentially finite groupoid (equivalent to a groupoid with finitely many objects and morphisms), then the groupoid cardinality of is the number of isomorphism classes of objects in .

I promise this is relevant to counting subgroups!

**Where are those subgroups?**

Now let be the groupoid of actions of a finitely generated group on -element sets. The number of isomorphism classes of objects in this groupoid is the number of isomorphism classes of -sets with elements, and so this number can also be identified with the groupoid cardinality of the free loop space . But this is just

which can in turn be identified with the groupoid of -sets with elements. In other words, the number of isomorphism classes of -sets with elements is

.

Now, the collection of all isomorphism classes of finite -sets is a free commutative monoid (under disjoint union) on the isomorphism classes of finite transitive -sets. In other words, such an isomorphism class is described by describing the multiplicity with which each finite transitive -set occurs within it. This gives us the following count.

**Theorem:** Let denote the number of conjugacy classes of subgroups of index in . Then

.

Incidentally, this result and the previous result about subgroups of index are both exercises in Stanley’s *Enumerative Combinatorics: Volume II* (more precisely, Exercise 5.13a and c).

It would be nice to write this in a form that lets us more clearly extract the coefficients from the LHS. If denotes the number of subgroups of of index , then taking logarithms gives

.

Extracting the coefficient of from both sides gives

and hence Möbius inversion gives the following.

**Theorem:** With and as above, we have

.

*Example. *Let . Then for all . This recovers

as expected.

*Example.* Now let . For abelian groups counting conjugacy classes of subgroups is the same as counting subgroups, so and turns out to be

.

In the same way that has Dirichlet series , this function has Dirichlet series . By induction, we find that the number of subgroups of of index is

which has Dirichlet series . We leave it as an entertaining exercise for the reader to give a direct proof of this.

**A slower discussion**

The generating function given above can be interpreted as the weighted groupoid cardinality of the groupoid of -sets, and the Möbius inversion formula gives some sort of relationship between transitive -sets and transitive -sets. What exactly is this relationship, and can we use it to give a more direct proof of the Möbius inversion formula?

For starters, a -set is the same thing (in the sense that we have an equivalence of categories) as a pair consisting of a -set and an automorphism of . (We already used this fact when we passed from the free loop space description to talking about above.) The -set is transitive if the combination of the action of and the automorphism is transitive. And we can identify subgroups of with pointed transitive -sets. So what do these look like?

If is a finite transitive -set, then its decomposition as a -set consist of a number of copies of the same finite transitive -set of size , which are cyclically permuted by the automorphism . If is pointed, then one of these copies has a basepoint in it, so can canonically be identified with where is the stabilizer of . The automorphism can be used to identify all of the other copies of with the copy containing the basepoint, so the only remaining data in this automorphism is the induced automorphism of .

The automorphism group of is , where denotes the normalizer

of in , and acts by left multiplication.

Altogether we’ve described a bijection between

- subgroups of of index and
- triples of a divisor of , a subgroup of of index , and an element .

This is close to, but not quite, the count we wanted, which was in terms of conjugacy classes of subgroups of of index . To get this count we need to know how many conjugates a given subgroup of index has. Every conjugate appears as the stabilizer of a point in , but two points that are related by the action of the automorphism group will have the same stabilizer, and conversely two points with the same stabilizer are related by the action of the automorphism group. The automorphism group itself acts freely, so every orbit has size . Altogether we find that has

conjugates, so after grouping all of the conjugates of together in the above bijection we find that the contribution of conjugacy classes of subgroups of of index to the count of subgroups of of index is as desired.

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where is a Galois extension, is the Galois group of (thinking of as an object of the category of field extensions of at all times), is a category of “objects over ,” and is a category of “objects over .”

In fact this description is probably only correct if is a finite Galois extension; if is infinite it should probably be modified by requiring that every function of that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on . To avoid this difficulty we’ll stick to the case that is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that -forms of an object can be classified by Galois cohomology , and we’ll give some examples.

**Basepoints**

Way back when we defined a homotopy fixed point structure, we saw that it could be regarded as a generalization of a 1-cocycle, which it reduces to in the special case that the action of on a category can be strictified so that it becomes an action of on the automorphism group of a single object . This is possible in concrete examples; for example, if is the standard -dimensional vector space over , then , and the action of the Galois group on this is just componentwise. When is it possible abstractly?

Topologically, we’re starting from an action of a group on a space (in our examples, a groupoid), and we want to know when such an action gives rise to an action on the fundamental group . Naively, this is only possible if is fixed by the action of . But this is not a homotopy-theoretic condition, and the homotopy-theoretic refinement is that should be a homotopy fixed point of ; then the action of on the unpointed space can be upgraded to an action of on a pointed space , and taking the fundamental group is a functor on pointed spaces. In fact this defines an equivalence from pointed connected groupoids to groups, and so an equivalence from actions of on pointed connected groupoids to actions of on groups.

For our purposes, what this means is that in order to strictify the Galois action on to an action on for a particular object , we need to pick a homotopy fixed point structure on to act as a basepoint; equivalently, we need to pick a -form . The corresponding homotopy fixed point structure is encoded by maps

as usual, and we can now use these maps to coherently identify each with . The corresponding strictified action of on is given by sending an automorphism to the automorphism

which we’ll write as for simplicity.

Fixing this homotopy fixed point structure allows us to describe other homotopy fixed point structures by describing their difference, which we’ll write as

.

After some simplification, we compute that satisfies the compatibility condition that

(where as usual we’ll need to interpret compositions in diagrammatic order for consistency), which is the usual definition of a 1-cocycle on with coefficients in (with respect to the action defined above). Similarly we get that isomorphisms of homotopy fixed point data correspond to 1-cocycles being cohomologous. Hence:

**Theorem:** Suppose that has at least one -form . Using this -form as a basepoint, isomorphism classes of -forms on can be identified with elements of the Galois cohomology set with respect to the strictified action above.

Note that this set is naturally pointed by the trivial 1-cocycle, whereas the set of isomorphism classes of homotopy fixed points does not have a natural “trivial” object in it. This reinforces the need to pick a basepoint / -form.

Note also that we haven’t yet provided a prescription for actually writing down a -form given homotopy fixed point data on some .

**The real and complex numbers**

Galois cohomology becomes particularly easy to describe in the special case that (or more generally any quadratic extension), which is already useful for many applications. Here is generated by a single nontrivial element , namely complex conjugation. The action of on will generally also have the interpretation of complex conjugation (e.g. on matrices), and so the data of a -cocycle amounts to (after trivializing ) the data of a single element such that

.

In other words, is an automorphism of whose inverse is its complex conjugate. Two such automorphisms are cohomologous as 1-cocycles iff there is some such that

.

**Some examples**

In all of the examples below we are claiming without proof that some Galois prestack is in fact a Galois stack.

*Example.* Let be the Galois stack of vector spaces, and let , with distinguished -form . The strictified Galois action on is componentwise, and this will tell us what the Galois action is in many other examples involving vector spaces with extra structure. Since every -form of must be an -dimensional vector space over and hence must be isomorphic to , we conclude that

.

This is a generalization of Hilbert’s Theorem 90, which it reduces to in the special case that .

When we learn the following: a 1-cocycle is a matrix such that , and the fact that every such 1-cocycle is cohomologous to zero means that every such matrix can be written in the form for some . This recently came up on MathOverflow.

When , and we learn the following: a 1-cocycle is an element such that

(that is, an element of norm ), and the fact that every such 1-cocycle is cohomologous to zero means that every such element can be written in the form

for some . This gives a parameterization

of the set of rational solutions to the Diophantine equation , and in particular when we recover the usual parameterization of Pythagorean triples. (I learned this from Noam Elkies.)

*Example.* Let be the Galois stack of commutative algebras, and consider . Its automorphism group as an -algebra is with trivial Galois action, so -forms of are classified by

.

Because the Galois action is trivial, this is the set of conjugacy classes of homomorphisms , or equivalently isomorphism classes of actions of on -element sets. Such an isomorphism class is a disjoint union of transitive actions of on -element sets, which by the Galois correspondence can be identified with finite separable extensions of of degree , and in fact it turns out that -forms of are precisely -algebras of the form

where each is a subextension of and . So in this case we more or less get ordinary Galois theory back.

*Example.* Let be the Galois stack of algebras, not necessarily commutative, and consider . Its automorphism group as an -algebra is with Galois action inherited from , so -forms of are classified by

.

This classification is related to the Brauer group of , namely the part involving those central simple -algebras which become isomorphic to after extension by scalars to . It is also related to Severi-Brauer varieties, which are -forms of projective space.

To connect this to a previous computation, the short exact sequence gives rise to a longish exact sequence part of which goes

.

Since vanishes by Hilbert’s theorem 90, by exactness we conclude that if the relative Brauer group vanishes, then so does ; equivalently, in this case the only -form of is .

**Good properties**

One last comment. Galois descent gives us a reason to single out certain properties P that some objects satisfying Galois descent (such as algebras or commutative algebras) can have as particularly good: namely, those properties which also satisfy Galois descent. This means that

- The extension of scalars of a P-object is P.
- The -forms of any P-object are P.

For example, for algebras, being semisimple is not a good property in this sense: the extension of scalars of a semisimple -algebra can fail to be semisimple (e.g. if is an extension of which is not separable). The good version of this property is being separable, which is equivalent to being “geometrically semisimple” in the sense that is semisimple for all field extensions .

Similarly, for commutative algebras, being isomorphic to a finite product of copies of the ground field is not a good property in this sense: although it is preserved under extension of scalars, since , it is not preserved under taking -forms, as we saw above.

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Fix a field . The gadgets we want to study assign to each separable extension a category of “objects over ,” to each morphism of extensions an “extension of scalars” functor , and to each composable pair of morphisms of extensions a natural isomorphism

of functors (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms we can write down from this data agree. We’ll also want unit isomorphisms satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of to the 2-category , and in particular each naturally acquires an action of (where we mean automorphisms of extensions of , hence if is Galois this is the Galois group) in precisely the sense we described earlier.

We’ll call such an object a **Galois prestack** (of categories, over ) for short. The basic example is the Galois prestack of vector spaces , which sends an extension to the category of -vector spaces and sends a morphism to the extension of scalars functor

.

Every example we consider will in some sense be an elaboration on this example in that it will ultimately be built out of vector spaces with extra structure, e.g. the Galois prestacks of commutative algebras, associative algebras, Lie algebras, and even schemes. In these examples, fields are not really the natural level of generality, and to make contact with algebraic geometry we should replace them with commutative rings, but for now we’ll ignore this.

In order to state the definition, we need to know that if is an extension, then the functor naturally factors through the category of homotopy fixed points for the action of on . We’ll elaborate on why this is in a moment.

**Definition:** A Galois prestack **satisfies Galois descent**, or is a **Galois stack**, if for every Galois extension the natural functor (where ) is an equivalence of categories.

In words, this condition says that the category of objects over is equivalent to the category of objects over equipped with homotopy fixed point structure for the action of the Galois group (or **Galois descent data**).

(**Edit, 11/18/15:**) This definition is slightly incorrect in the case of infinite Galois extensions; see the next post and its comments for some discussion.

**Intuitions**

The usage of the term stack above is meant to activate two intuitions. On the one hand, the way we stated Galois descent is meant to look like a sheaf condition. Since for a Galois extension, the Galois descent condition just says that the 2-functor sends certain limits to certain homotopy limits (or 2-limits, depending on taste).

On the other hand, stacks are supposed to be generalizations of schemes, and we can also think of the assignment as the functor of points of some sort of moduli space of objects such that maps correspond to objects over (that is, objects in ).

In addition, as mentioned in the first post in this series on Galois descent, this condition should also be regarded as a categorification of the observation that if denotes the set of -points of a variety over , then the natural map is an isomorphism.

**The natural factorization**

In order to state the above condition we needed to know that naturally factors through the category of homotopy fixed points. Let’s think about the analogous question one category level down: suppose is a map of sets, and is a group acting on . When does factor through the set of fixed points? The answer is precisely when

for all . This reflects the universal property of taking fixed points: it’s a certain limit, and so it’s preserved by functors of the form in the sense that we have a natural isomorphism

.

It shouldn’t be surprising that taking homotopy fixed points has an analogous universal property. In fact, it’s not hard to check that if denotes the functor category between two categories and acts on , then we have a natural equivalence of categories

where refers to taking homotopy fixed points on both the LHS and the RHS. In words, this equivalence says that the category of functors is equivalent to the category of functors equipped with homotopy fixed point structure for the induced action of .

Now we just need to observe that the extension of scalars functor is always equipped with homotopy fixed point structure for the action of . This follows from functoriality: because is, by definition, fixed by the action of , we have

for all , and applying gives natural isomorphisms

satisfying the appropriate compatibilities to give homotopy fixed point data. This is the abstract version of the concrete argument we gave previously that the extension of scalars functor naturally factors through homotopy fixed points .

**Forms**

We can extract somewhat more concrete information from this abstract version of Galois descent as follows.

**Definition:** Fix an object . An object is a **-form** of if there is an isomorphism .

If we understand the objects in well, we can hope to understand the objects in by understanding the -forms of objects in .

*Example.* Let , and let be the Galois prestack of semisimple Lie algebras. This turns out to be a Galois stack; that is, semisimple Lie algebras satisfy descent. The classification of complex semisimple Lie algebras reduces the classification of real semisimple Lie algebras to the classification of real forms of complex semisimple Lie algebras; explicitly, a real form of a complex Lie algebra is a real Lie algebra such that

.

These can be quite varied. For example, the complex special orthogonal Lie algebra has real forms the indefinite special orthogonal Lie algebras for any nonnegative integers such that .

If Galois descent holds, then the extension of scalars functor can be described simply as the functor that takes an object of equipped with homotopy fixed point structure and forgets that structure. Hence:

**Observation:** If is a Galois stack, then isomorphism classes of -forms of an object can be identified with isomorphism classes of homotopy fixed point structures on .

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