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Fixed points of group actions on categories

Previously we described what it means for a group $G$ to act on a category $C$ (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action are.

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 $F(g) F(h) = F(gh)$ with isomorphisms $\eta(g, h) : F(g) F(h) \cong F(gh)$, 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 $F(g) F(h) = F(gh)$ with the structure of a family of isomorphisms between $F(g) F(h)$ and $F(gh)$. 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 $F : G \to \text{Aut}(C)$ is a group action as in the previous post, and $c \in C$ is an object. The structure of a fixed point, or more precisely a homotopy fixed point, is the data of a family of isomorphisms

$\displaystyle \alpha(g) : c \cong F(g) c$

which satisfy the compatibility condition that the two composites

$\displaystyle c \xrightarrow{\alpha(g)} F(g) c \xrightarrow{F(g)(\alpha(h))} F(g) F(h) c \xrightarrow{\eta(g, h)(c)} F(gh) c$

and

$\displaystyle c \xrightarrow{\alpha(gh)} F(gh) c$

are equal, as well as the unit condition that

$\displaystyle \alpha(e) = \varepsilon(c) : c \to F(e) c$

where $\varepsilon$ is the unit isomorphism $\text{id}_C \cong F(e)$. 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 $F$ is trivial (meaning both that $F(g) = \text{id}_C$ and that $\eta(g, h) = e \in Z(C)^{\times}$), this reduces to the definition of a group action of $G$ on $c \in C$ in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on $c \in C$ where the twist is provided by the group action on $C$.

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 $f : c \to d$ and $g : d \to e$ are two morphisms, then we write $fg : c \to e$ for the composite $c \xrightarrow{f} d \xrightarrow{g} e$.

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 $k \to L$ be a Galois extension with Galois group $G = \text{Gal}(L/k)$, and let $V \in \text{Mod}(k)$ be a $k$-vector space.

Claim: $\text{Mod}(L)$ has a natural action of the Galois group $G$. The extension of scalars $V_L = V \otimes_k L$ has a natural homotopy fixed point structure with respect to this Galois action.

All of this structure is natural in $V$ in a strong sense and so also respects various kinds of additional structure $V$ 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 $k$-vector spaces.

The Galois action, at least, should be clear: it comes from precomposing the module structure map $\varphi : L \to \text{End}(W)^{op}$ (for $W \in \text{Mod}(L)$; we want right modules here or else this precomposition won’t be an action but an anti-action) with some automorphism $g : L \to L$ in the Galois group. In fact it’s possible to show that $G$ is precisely the group of $K$-linear automorphisms of $\text{Mod}(L)$, up to natural equivalence.

This action may be hard to appreciate because it is trivial on $\pi_0$; that is, it sends every $L$-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 $L$ componentwise.

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

$\displaystyle \sigma(vc) = \sigma(v) \sigma(c)$

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

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

Suppose that the action of $G$ on $C$ can be strictified in the sense that we can arrange for equalities $F(g) F(h) = F(gh)$ on the nose and take the $\eta(g, h)$ to be identities (and $F(1) = \text{id}_C$ on the nose as well). Suppose furthermore that $C$ has one object $c$, so all of the objects $F(g) c$ are identical, and so the only interesting information in the action is an action of $G$ on $\text{Aut}(c)$. Then homotopy fixed point data consists of a collection $\alpha(g) \in \text{Aut}(c)$ of elements subject to the condition that $\alpha(e) = e$ and

$\displaystyle \alpha(gh) = \alpha(g) F(g)(\alpha(h)) \in \text{Aut}(c)$

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

Note that if each $F(g)$ is also the identity then this is just the condition that $\alpha$ is a group homomorphism $G \to \text{Aut}(c)$, and so as we noted above, homotopy fixed point data is just the data of an action of $G$ on $c$.

To get the usual equivalence relation on 1-cocycles we should talk about equivalences between two homotopy fixed point structures $\alpha_1, \alpha_2$. (Here we are temporarily switching back to the fully general case.) These are isomorphisms $f : c \cong c$ such that the composites

$\displaystyle c \xrightarrow{f} c \xrightarrow{\alpha_1(g)} F(g) c$

and

$\displaystyle c \xrightarrow{\alpha_2(g)} F(g) c \xrightarrow{F(g)(f)} F(g) c$

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

$\displaystyle f \alpha_1(g) = \alpha_2(g) F(g)(f) \in \text{Aut}(c)$

which is precisely the condition that $\alpha_1, \alpha_2$ 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 $c$ is precisely the cohomology set $H^1(BG, \text{Aut}(c))$ with nonabelian local coefficients.

In the setting of Galois descent, isomorphism classes of homotopy fixed point data on an object $c$ living “over $L$” (where $k \to L$ is a Galois extension) will turn out, in nice cases, to correspond to isomorphism classes of objects living “over $k$” which “extend by scalars” to $c$. 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 $G$, the category of $G$-sets is equivalent to the category of locally constant sheaves, or local systems, of sets on the classifying space $BG$. Given a $G$-set $X$, its subset $X^G$ of fixed points can be recovered as the global sections of this local system.

Similarly, the category of $G$-modules (in abelian groups, say) is equivalent to the category of local systems of abelian groups on $BG$. 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 $H^0$ computes $G$-invariants, but there are higher derived functors $H^i$ computing “derived $G$-invariants,” and we recover group cohomology with nontrivial coefficients.

The analogous statement in this setting is that the 2-category of $G$-categories (categories equipped with an action of $G$) is equivalent to the 2-category of locally constant stacks or local systems of categories on $BG$. 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 $\infty$-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 $H^2$ reflects the fact that categories only live in a 2-category and hence only have automorphism 2-groups. We can get to $H^n$ by generalizing this story to group actions on objects living in an n-category, which have automorphism n-groups.