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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.

This identity reflects, in a way we made precise in the previous post, the decomposition of a finite -set (the terms on the LHS) into a disjoint union of transitive -sets (the terms on the RHS).

Noam Zeilberger commented on the previous post that he had seen results like this for more specific groups in the literature; in particular, Samuel Vidal describes a version of this analysis for , the modular group. In this post we’ll use the above formula to compute the number of subgroups of index in using a computer algebra system that can manipulate power series. We’ll also say something about how to visualize these subgroups.

**The count**

The key is that , as an abstract group, is very easy to describe: it’s isomorphic to the free product of a cyclic group of order and a cyclic group of order . The corresponding elements of of order and order , regarded as fractional linear transformations , are

.

We’ll name them and in what follows, so that

.

This free product decomposition means that it’s very easy to describe homomorphisms ; such a homomorphism is the same thing as a pair of permutations such that . That makes these homomorphisms easy to count. If denotes the number of permutations such that , then we know (e.g. by applying the general formula above to ) that

and hence that

and that

.

Finally, we have

and so we can compute the number of subgroups of index in the modular group by extracting the coefficients of the logarithm

which is easy to do with a computer algebra system such as WolframAlpha. For example, suppose we wanted to compute the first six terms . We have

and

which tells us the first six terms of . This gives

and taking the logarithm of this produces

.

That is, the first six terms of the sequence are . Plugging this into the OEIS gives A005133, an entry contributed by Samuel Vidal.

**Visuals**

Of course we can do better than just count these subgroups: we can also visualize them in various ways. Subgroups of of index are naturally in bijection with pointed transitive -sets, and we can visualize these using a generalized Cayley graph construction. To describe the action of on a finite set , we can draw a graph whose vertices are the elements of . The action of the generator of order pairs together some disjoint pairs of vertices, and we can draw undirected edges between these. The action of the generator of order groups together some disjoint triples of elements of ; we can draw directed cycles of length between these.

The resulting graph is connected iff the action of is transitive, and so subgroups of index correspond to pointed connected graphs of this form with vertices. Moreover, by changing the basepoint we conjugate the subgroup, so by grouping the pointed connected graphs together according to the isomorphism class of the underlying unpointed graph, we can describe the orbits of acting on subgroups by conjugation. The normal subgroups are the orbits of size , and these correspond to graphs whose automorphism groups act transitively on vertices. For a normal subgroup the corresponding graph can be identified with the Cayley graph of the quotient, with generators .

There should be a better visualization coming from Bass-Serre theory that will even describe the isomorphism classes of the resulting groups, analogous to how drawing covering spaces of graphs shows that subgroups of free groups are free, but I’m not sure how to describe it rigorously off the top of my head. The analogous statement here, I think, is that subgroups of are all free products of copies of , and , but don’t quote me on that. Topologically, the idea is to think of as a 1-dimensional orbifold consisting of an orbifold point with stabilizer joined to an orbifold point with stabilizer by an edge (reflecting the wedge sum decomposition ), and then to consider coverings of this orbifold, which are also 1-dimensional orbifolds.

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Today we’ll go in the other direction. Given an action of on explicitly described by a 2-cocycle , we’ll recover the category of -projective representations, or equivalently the category of modules over the twisted group algebra , by taking the homotopy fixed points of this action. We’ll end with another puzzle.

**Spelling it out**

It’s straightforward to verify this. Given a 2-cocycle , which we can normalize to be unital, the corresponding action of on is given by taking each to be the identity and taking multiplication by to be the natural isomorphism . Then a homotopy fixed point for this action is a -vector space together with isomorphisms

such that the composites

and

agree (and ), and up to matching our conventions (we again need to interpret compositions in diagrammatic order) this is precisely the condition that

so we get that homotopy fixed point data for is precisely the data of an -projective representation . Moreover, the definition we gave of a morphism between -projective representations generalizes to a definition of morphism between homotopy fixed points, so we really get an identification between the category of homotopy fixed points of the -action on and the category of -projective representations.

The claim that if are cohomologous then the categories of -projective and -projective representations are equivalent is then subsumed in the claim that taking homotopy fixed points is functorial in the appropriate sense (there is really a 2-category of actions of on categories which we haven’t described, and taking homotopy fixed points is a 2-functor from this 2-category to ).

The observation that the Schur class is multiplicative with respect to tensor products of projective representations also has an interpretation here. It reflects the fact that is an object of a symmetric monoidal 2-category, namely the Morita 2-category , and that taking homotopy fixed points of group actions is lax monoidal: it gives rise to functors

where denotes taking homotopy fixed points. This is a natural categorification of the corresponding fact for ordinary linear representations of finite groups, where taking fixed points in the usual sense is also lax monoidal in the sense that we get natural maps , provided you believe that the categories are natural categorifications of vector spaces (sometimes called 2-vector spaces).

**What’s the topological picture?**

As usual, by passing through the homotopy hypothesis, we can get a topological picture of what’s going on here. The 2-groupoids fit into a fiber sequence

(where denotes the universal Schur class) which expresses a generalization of the interpretation of the Schur class in terms of an obstruction to lifting: this fiber sequence says that is the homotopy fiber of the Schur class, which means that if

is any map of spaces (where, again, by “spaces” I mean weak homotopy types, or equivalently -groupoids, or equivalently spaces with the homotopy type of a CW complex), which can be interpreted as classifying a “projective bundle” on , then the space of lifts of this projective bundle to a vector bundle is the space of nullhomotopies of the composite map

.

In particular, any projective bundle on has a Schur class in which vanishes iff it lifts to a vector bundle.

How can we think about the Schur class in this generality? A map can also be thought of as classifying a bundle of matrix algebras over (while a lift of this map to a map exhibits a bundle of Morita equivalences between these matrix algebras and , or equivalently a -vector bundle and an isomorphism between this bundle of matrix algebras and ). We might call such a thing an Azumaya algebra over .

When we fiberwise take module categories over these matrix algebras, we get a “2-line bundle,” namely a bundle of categories equivalent to , thought of as a “free module of rank ” over . Taking global sections of this 2-line bundle in an appropriate sense is a generalization of taking homotopy fixed points, which is the special case where . This interpretation exhibits as a topological analogue of the Brauer group.

We can also use this topological picture to describe in exactly what sense an -projective representation has more structure than a projective representation with Schur class . Thinking of as a map (which can be represented by a 2-cocycle on for ), the space of -dimensional “-projective bundles” on (which reduces to -projective representations of for ) can be identified with the homotopy pullback of the diagram

.

In other words, an -projective bundle is a map together with a choice of homotopy between the Schur class of this map and another fixed map . It’s this structure of a choice of homotopy, rather than the property that such a homotopy exists, that is the extra structure on an -projective representation. This extra structure is hard to see if we only think about as a group rather than, according to taste, either the set of connected components of a space (namely the space of maps ) or the set of isomorphism classes of objects in a 2-groupoid (namely the 2-groupoid of 2-line bundles on ).

**The puzzle**

Nothing we’ve said above has explicitly mentioned the twisted group algebra .

**Puzzle:** How can we fit the twisted group algebra into this story?

I like this puzzle because I know of two good answers to it.

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(There is a second half that goes in the other direction.)

For the remainder of this post we’ll restrict our attention to finite-dimensional projective representations .

**Preparatory background**

The first observation is that naturally acts on the matrix algebra . More precisely, the conjugation action of on naturally factors through . In fact, is precisely the automorphism group of . This is guaranteed by the following theorem.

**Theorem (Noether-Skolem):** Let be a finite-dimensional simple -algebra, let be a central simple -algebra (meaning is finite-dimensional, simple, and ), and let be -algebra homomorphisms. Then there is an invertible element such that for all .

**Corollary:** Every automorphism of a central simple algebra is inner, and hence the automorphism group of a central simple algebra is .

*Proof.* are finite-dimensional simple algebras, and hence have unique simple modules . The restriction of to an -module along either or produces an -module which (by simplicity of ) is a direct sum of copies of ; in particular, the isomorphism type of this module is determined by its dimension. Hence these two restrictions are isomorphic. An isomorphism between them is some -linear automorphism of , so by the Jacobson density theorem (and centrality of ) it is given by multiplication by some invertible . This is the desired .

Hence an equivalent description of (finite-dimensional) projective representations of over is that they are actions of on matrix algebras .

Next, it will be useful to describe something about the functoriality of the construction associating to a -algebra the category of (right) -modules. This is both a covariant and a contravariant functor, although we’ll only be interested in the covariant version, which sends a morphism of -algebras to the extension of scalars functor

where acquires the structure of an -bimodule over as follows: the left action of is given by left multiplication by , while the right action is given by right multiplication as usual.

This construction is in fact a 2-functor from the category of -algebras to the Morita 2-category of -algebras, -bimodules, and bimodule homomorphisms, or equivalently of module categories , cocontinuous -linear functors, and natural transformations.

**Lemma:** Let be two morphisms of -algebras. Natural transformations of functors can be identified with elements acting by left multiplication on such that

.

*Proof.* By Eilenberg-Watts, natural transformations of functors can be identified with morphisms of bimodules, where the first bimodule has its left -module structure coming from and the second coming from . By the Yoneda lemma, morphisms of right -modules can be identified with elements acting by left multiplication, and then the additional compatibility with the left -module structures is precisely the above condition.

In other words, the 2-functor factors through a 2-category whose

- objects are -algebras ,
- morphisms are -algebra morphisms ,
- 2-morphisms are elements such that ,

and the resulting 2-functor is fully faithful on hom categories.

**The functor**

Now, recall that is Morita equivalent to . Hence an action of on gives rise not only to an action of on (by applying the above 2-functor), but also to an action on (since one payoff of writing down the 2-categorical notion of action is that it transports across equivalences of categories). In fact, we can be more precise about which action we get.

**Theorem:** Let be a finite-dimensional projective representation of over . It induces a -linear action of on , and the isomorphism class of this action is the Schur class .

*Proof.* First, we observe that a -linear Morita equivalence induces a -linear isomorphism on centers

.

We’ll be talking about 2-cocycles with values in , but because of the Morita equivalence above this is equivalent to talking about corresponding 2-cocycles with values in , which will allow us to match up the 2-cocycles that are about to appear with the 2-cocycles that appeared when we classified -linear actions of on . Apart from this observation we will no longer need to explicitly talk about the Morita equivalence.

Now, given , consider the corresponding automorphism . Either because of the Morita equivalence to or by Noether-Skolem, we know that is equivalent to the identity. Explicitly, we know that admits a lift to some , and that , thought of as acting on by left multiplication, furnishes a natural isomorphism between and the identity.

Hence we can describe natural isomorphisms between various composites of the by first trivializing them using composites of the lifts , then describing natural automorphisms of the identity. The identity is as an -bimodule, and hence its natural automorphisms can naturally be identified with invertible elements in the center acting by left multiplication.

There are natural isomorphisms relating and which, after trivializing both using the lift and the product of lifts , can be described as multiplication by scalars such that

and so, up to making sure our conventions are consistent, we find that writing down the 2-cocycle representing the Schur class of corresponds precisely to writing down the 2-cocycle representing the corresponding action of on , as desired.

**The Schur class as a characteristic class**

The Schur class can be thought of as a characteristic class of projective representations, and in the same way that characteristic classes of vector bundles in classical algebraic topology come from universal cohomology classes of classifying spaces, the Schur class comes from a universal characteristic class

which classifies the universal projective representation . In other words, the universal Schur class is a map

of 2-groupoids, and the content of the above discussion is that it admits a functorial description in terms of the 2-functor sending a matrix algebra to its category of modules.

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.

This is the same group that appears in the classification of projective representations of over , and we asked whether this was a coincidence.

Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.

**The Schur class of a projective representation**

**Definition:** A **projective representation** of a group over a field is a homomorphism , where is the projective linear group of a -vector space .

A natural question to ask about a projective representation is when it can be lifted to a linear representation . We can certainly individually lift each to some ; what we can’t necessarily do is guarantee that . Since , we instead have

for some scalars (uniquely determined by our choices of lifts ). These scalars can’t be arbitrary, though, since can be decomposed in two different ways as a product, which gives

on the one hand and

on the other hand. Hence must satisfy

which is just the 2-cocycle condition. If we also decided that we might as well lift to , then this is even a unital 2-cocycle.

We made some arbitrary choices when choosing the lifts , so now let’s see what happens if we choose different lifts . These must differ from our original lifts by some scalars (that is, ), and after some computation the relationship between our old 2-cocycle and our new 2-cocycle is that

which is just the condition that differ by the 2-coboundary .

Altogether we’ve proven the following.

**Theorem:** Associated to any projective representation is a cohomology class , the **Schur class**. It can be constructed using any choice of lifts as above and does not depend on the choice. lifts to a linear representation iff the Schur class vanishes.

This is perhaps the simplest interesting example of a cohomology class being the obstruction to a lifting problem.

Although there is no natural way to take the direct sum of two projective representations, it is possible to make sense of the tensor product of two projective representations, and Schur classes are multiplicative with respect to tensor product: that is, if is the Schur class of a projective representation , then

.

**Categories of projective representations**

There’s a natural notion of isomorphism of projective representations: namely, a **projective isomorphism** between two projective representations is an element such that

for every . However, it’s less clear how to generalize this definition to describe a notion of not-necessarily-invertible morphism between projective representations.

Rather than do this directly, we’ll observe that the Schur class is a projective isomorphism invariant of projective representations, so the study of projective representations naturally breaks up into the study of projective representations of a fixed Schur class , for each . This suggests the following refined definition.

**Definition:** Fix and fix a 2-cocycle representing . An **-projective representation** of is a map such that

.

A morphism of -projective representations is a -linear map such that

for every .

Note that an -projective representation has more structure than a projective representation with Schur class , and that it really is necessary to pick a 2-cocycle in order to state this definition. There is a functor from the groupoid of -projective representations to projective representations, but it is not faithful: the notion of morphism above does not involve quotienting by scalar multiplication.

Unlike ordinary projective representations, -projective representations admit a direct sum, and in fact the category of -projective representations is about as well-behaved as possible in the following sense.

**Theorem:** The category of -projective representations is equivalent to the category of modules over the **twisted group algebra** . This algebra is as a -vector space, but with the modified multiplication

.

If are cohomologous, then are isomorphic, and hence the categories of -projective and -projective representations are equivalent.

This theorem guarantees that -projective representations exist, such as the regular representation of the twisted group algebra on itself. Note that there is no analogue of the trivial representation here, so this isn’t obvious. In other words:

**Corollary:** The Schur class map from projective representations to is surjective.

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John Baez likes to describe (vertical) categorification as *replacing equalities with isomorphisms*, which we saw on full display in the previous post: we replaced the equality with isomorphisms , and as a result we found 2-cocycles lurking in this story.

I prefer to describe categorification as *replacing properties with structures*, in the nLab sense. That is, the real import of what we just did is to replace the property (of a function between groups, say) that with the structure of a family of isomorphisms between and . The use of the term “structure” emphasizes, as we also saw in the previous post, that unlike properties, structures need not be unique.

Accordingly, it’s not surprising that being a fixed point of a group action on a category is also a structure and not a property. Suppose is a group action as in the previous post, and is an object. The structure of a fixed point, or more precisely a **homotopy fixed point**, is the data of a family of isomorphisms

which satisfy the compatibility condition that the two composites

and

are equal, as well as the unit condition that

where is the unit isomorphism . This is, in a sense we’ll make precise below, a 1-cocycle condition, but this time with nontrivial (local) coefficients.

Curiously, when the action is trivial (meaning both that and that ), this reduces to the definition of a group action of on in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on where the twist is provided by the group action on .

**Warning**

Previously we didn’t have to worry about this because we ended up only working in abelian groups, but today we need to worry about what convention we’re using for composition, and for the rest of this post the convention is that compositions are in diagrammatic order: that is, if and are two morphisms, then we write for the composite .

This is necessary to keep things consistent with the other conventions we’ve been using while writing down as few inverses as possible, which I didn’t realize before picking them, but on the other hand I will also continue to write functions as acting on the left. Oops.

**Okay, but does this really have anything to do with Galois descent?**

Let be a Galois extension with Galois group , and let be a -vector space.

**Claim:** has a natural action of the Galois group . The extension of scalars has a natural homotopy fixed point structure with respect to this Galois action.

All of this structure is natural in in a strong sense and so also respects various kinds of additional structure may be equipped with, such as the structure of an associative, commutative, or Lie algebra, or more generally the structure of an algebra over an operad in -vector spaces.

The Galois action, at least, should be clear: it comes from precomposing the module structure map (for ; we want right modules here or else this precomposition won’t be an action but an anti-action) with some automorphism in the Galois group. In fact it’s possible to show that is precisely the group of -linear automorphisms of , up to natural equivalence.

This action may be hard to appreciate because it is trivial on ; that is, it sends every -vector space to another vector space of the same dimension, and hence acts trivially on isomorphism classes of objects. However, it acts nontrivially on morphisms. It can be thought of very explicitly as acting on a matrix with entries in componentwise.

On to the homotopy fixed point structure. It’s clear that is acted on, as an abelian group (and in fact as a -vector space), by the Galois group in the second coordinate. This action is *not* -linear, so it’s not correct to describe this structure by saying that has a -action in the naive sense as an object in . Its relationship to the -module structure is that if is an element of the Galois group, is a scalar, and is a vector, then

Such a transformation is said to be **-semilinear**. From our perspective, what this really says is that is an -linear morphism from to , namely with the -module structure modified by . It’s not hard to check that the necessary compatibilities are satisfied, so this gives a natural homotopy fixed point structure as desired.

Note that given the homotopy fixed point structure on , we can recover as the fixed points (in the usual sense) of the Galois action.

**What about 1-cocycles?**

Suppose that the action of on can be strictified in the sense that we can arrange for equalities on the nose and take the to be identities (and on the nose as well). Suppose furthermore that has one object , so all of the objects are identical, and so the only interesting information in the action is an action of on . Then homotopy fixed point data consists of a collection of elements subject to the condition that and

and this is precisely the usual (nonabelian) 1-cocycle condition (with nontrivial coefficients), plus an additional unit condition as before.

Note that if each is also the identity then this is just the condition that is a group homomorphism , and so as we noted above, homotopy fixed point data is just the data of an action of on .

To get the usual equivalence relation on 1-cocycles we should talk about equivalences between two homotopy fixed point structures . (Here we are temporarily switching back to the fully general case.) These are isomorphisms such that the composites

and

are equal. Again, in the special case where the action is strict and has a single object, this condition reads

which is precisely the condition that are cohomologous in the usual sense. Hence in this special case we conclude the following.

**Theorem:** With the above hypotheses, the set of isomorphism classes of homotopy fixed point data on is precisely the cohomology set with nonabelian local coefficients.

In the setting of Galois descent, isomorphism classes of homotopy fixed point data on an object living “over ” (where is a Galois extension) will turn out, in nice cases, to correspond to isomorphism classes of objects living “over ” which “extend by scalars” to . So we’re getting closer to understanding Galois descent in this language.

**Where’s all this cohomology coming from, anyway?**

Recall that for a group , the category of -sets is equivalent to the category of locally constant sheaves, or local systems, of sets on the classifying space . Given a -set , its subset of fixed points can be recovered as the global sections of this local system.

Similarly, the category of -modules (in abelian groups, say) is equivalent to the category of local systems of abelian groups on . Now instead of taking global sections, we can apply the machinery of homological algebra and take derived global sections, or sheaf cohomology. When we do this, the zeroth derived functor computes -invariants, but there are higher derived functors computing “derived -invariants,” and we recover group cohomology with nontrivial coefficients.

The analogous statement in this setting is that the 2-category of -categories (categories equipped with an action of ) is equivalent to the 2-category of locally constant stacks or local systems of categories on . Taking homotopy fixed points corresponds to taking global sections, but in a refined 2-categorical sense. Since taking derived global sections corresponds to taking global sections in a refined -categorical sense, it’s not surprising to find some cohomology (with nontrivial coefficients) showing up.

The fact that we’ve only made it up to reflects the fact that categories only live in a 2-category and hence only have automorphism 2-groups. We can get to by generalizing this story to group actions on objects living in an n-category, which have automorphism n-groups.

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The condition we need on units is first that we should have a unit isomorphism

and second that this unit isomorphism should be compatible with the isomorphisms in the sense that the composites

and

should both be the identity. If we use the unit isomorphism to replace with (which changes the ), this is just the condition that and should both be the identity.

Similarly, in our definition of an equivalence between two group actions , needs to respect these unit isomorphisms in the sense that

Again, if we use the unit isomorphisms to replace with on the nose, this is just the condition that should be the identity.

Fortunately, in the special case we considered in the previous post, where vanishes (and perhaps in general), this produces the same classification of group actions as before, so nothing has gone too badly wrong. Details below the fold.

**Concrete details**

Recall that in the special case of group actions where each is the identity , the only information left is in the natural isomorphisms . Yesterday we only imposed the usual 2-cocycle condition on these, and saw that equivalences of group actions corresponded to 2-coboundaries. Today we want the additional unit conditions above. This produces a variant of 2-cocycles and 2-coboundaries which we’ll call **unital**. Explicitly, for 2-cocycles this means , and for 2-coboundaries it means we look at 2-coboundaries of the form where .

We want to show that these give rise to the same cohomology group , and we’ll do this very explicitly, by showing that any 2-cocycle is cohomologous to a unital 2-cocycle, and that two unital 2-cocycles are 2-cohomologous iff they are unitally 2-cohomologous. First, using the 2-cocycle condition

and substituting first and then , we get

and

from which it follows that and are both constant, and hence must both be equal to . Now we just need to modify this 2-cocycle by the 2-coboundary where and for ; we get a new 2-cocycle satisfying

and we see that as desired. Hence every 2-cocycle is cohomologous to a unital 2-cocycle.

Next, suppose are two unital 2-cocycles which are cohomologous via a 2-coboundary , so that

.

Then setting gives

and hence must be a unital 2-coboundary.

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