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## Higher linear algebra

Let $k$ be a commutative ring. A popular thing to do on this blog is to think about the Morita 2-category $\text{Mor}(k)$ of algebras, bimodules, and bimodule homomorphisms over $k$, but it might be unclear exactly what we’re doing when we do this. What are we studying when we study the Morita 2-category?

The answer is that we can think of the Morita 2-category as a 2-category of module categories over the symmetric monoidal category $\text{Mod}(k)$ of $k$-modules, equipped with the usual tensor product $\otimes_k$ over $k$. By the Eilenberg-Watts theorem, the Morita 2-category is equivalently the 2-category whose

• objects are the categories $\text{Mod}(A)$, where $A$ is a $k$-algebra,
• morphisms are cocontinuous $k$-linear functors $\text{Mod}(A) \to \text{Mod}(B)$, and
• 2-morphisms are natural transformations.

An equivalent way to describe the morphisms is that they are “$\text{Mod}(k)$-linear” in that they respect the natural action of $\text{Mod}(k)$ on $\text{Mod}(A)$ given by

$\displaystyle \text{Mod}(k) \times \text{Mod}(A) \ni (V, M) \mapsto V \otimes_k M \in \text{Mod}(A)$.

This action comes from taking the adjoint of the enrichment of $\text{Mod}(A)$ over $\text{Mod}(k)$, which gives a tensoring of $\text{Mod}(A)$ over $\text{Mod}(k)$. Since the two are related by an adjunction in this way, a functor respects one iff it respects the other.

So Morita theory can be thought of as a categorified version of module theory, where we study modules over $\text{Mod}(k)$ instead of over $k$. In the simplest cases, we can think of Morita theory as a categorified version of linear algebra, and in this post we’ll flesh out this analogy further.

## Coalgebraic geometry

Previously we suggested that if we think of commutative algebras as secretly being functions on some sort of spaces, we should correspondingly think of cocommutative coalgebras as secretly being distributions on some sort of spaces. In this post we’ll describe what these spaces are in the language of algebraic geometry.

Let $D$ be a cocommutative coalgebra over a commutative ring $k$. If we want to make sense of $D$ as defining an algebro-geometric object, it needs to have a functor of points on commutative $k$-algebras. Here it is:

$\displaystyle D(-) : \text{CAlg}(k) \ni R \mapsto |D \otimes_k R| \in \text{Set}$.

In words, the functor of points of a cocommutative coalgebra $D$ sends a commutative $k$-algebra $R$ to the set $|D \otimes_k R|$ of setlike elements of $D \otimes_k R$. In the rest of this post we’ll work through some examples.

## Drawing subgroups of the modular group

Previously we learned how to count the finite index subgroups of the modular group $\Gamma = PSL_2(\mathbb{Z})$. The worst thing about that post was that it didn’t include any pictures of these subgroups. Today we’ll fix that.

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.

## Connected components in a distributive category

Previously we claimed that if you want to check whether a category $C$ “behaves like a category of spaces,” you can try checking whether it’s distributive. The goal of today’s post is to justify the assertion that objects in distributive categories behave like spaces by showing that they have a notion of “connected components.”

For starters, let $C$ be a distributive category with terminal object $1$, and let $2 = 1 + 1$ be the coproduct of two copies of $1$. For an object $X \in C$, what does $\text{Hom}(X, 2)$ look like? If $C = \text{Top}$ and $X$ is a sufficiently well-behaved topological space, morphisms $X \to 2$ correspond to subsets of the connected components of $X$, and $\text{Hom}(X, 2)$ naturally has have the structure of a Boolean algebra or Boolean ring whose elements can be interpreted as subsets of the connected components of $X$.

It turns out that $\text{Hom}(X, 2)$ 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 $C = \text{Aff}$ the object this functor outputs is known as the Pierce spectrum.

This construction can be thought of as trying to do for $\pi_0$ what the étale fundamental group does for $\pi_1$.

## Stating Galois descent

After a relaxing and enjoyable break, we’re finally in a position to state what it means for structures to satisfy Galois descent.

Fix a field $k$. The gadgets we want to study assign to each separable extension $k \to L$ a category $C(L)$ of “objects over $L$,” to each morphism $f : L_1 \to L_2$ of extensions an “extension of scalars” functor $f_{\ast} : C(L_1) \to C(L_2)$, and to each composable pair $L_1 \xrightarrow{f} L_2 \xrightarrow{g} L_3$ of morphisms of extensions a natural isomorphism

$\displaystyle \eta(f, g) : f_{\ast} g_{\ast} \cong (fg)_{\ast}$

of functors $C(L_1) \to C(L_3)$ (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms $f_{\ast} g_{\ast} h_{\ast} \cong (fgh)_{\ast}$ we can write down from this data agree. We’ll also want unit isomorphisms $\varepsilon : \text{id}_{C(L)} \cong (\text{id}_L)_{\ast}$ satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of $k$ to the 2-category $\text{Cat}$, and in particular each $C(L)$ naturally acquires an action of $\text{Aut}(L)$ (where we mean automorphisms of extensions of $k$, hence if $L$ 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 $k$) for short. The basic example is the Galois prestack of vector spaces $\text{Mod}(-)$, which sends an extension $L$ to the category $\text{Mod}(L)$ of $L$-vector spaces and sends a morphism $f : L_1 \to L_2$ to the extension of scalars functor

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

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 $f : k \to L$ is an extension, then the functor $f_{\ast} : C(k) \to C(L)$ naturally factors through the category $C(L)^G$ of homotopy fixed points for the action of $G = \text{Aut}(L)$ on $C(L)$. 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 $k \to L$ the natural functor $C(k) \to C(L)^G$ (where $G = \text{Aut}(L) = \text{Gal}(L/k)$) is an equivalence of categories.

In words, this condition says that the category of objects over $k$ is equivalent to the category of objects over $L$ 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.

## The puzzle of Galois descent

Suppose we have a system $f_1, f_2, \dots f_n \in k[x_1, x_2, \dots x_m]$ of polynomial equations over a perfect (to keep things simple) field $k$, and we’d like to consider solutions of it over various field extensions $L$ of $k$. Write $V(L)$ for the set of all solutions to this system over $L$.

As it happens, knowing $V(L)$ for any algebraic extension $L$ of $k$ is equivalent to knowing $V(\bar{k})$, where $\bar{k}$ denotes the algebraic closure of $k$, together with the action of the absolute Galois group $G = \text{Gal}(\bar{k}/k)$. After picking an embedding of $L$ into $\bar{k}$, the infinite Galois correspondence says that $L$ is precisely the set of fixed points of the closed subgroup $H$ of $G$ which stabilizes $L$, and it’s not hard to see that this extends to $V(L)$; that is, $G$ naturally acts on $V(\bar{k})$, and we have a natural identification

$\displaystyle V(L) \cong V(\bar{k})^H$.

Now let’s categorify this situation. Before we considered, for each algebraic extension $L$ of $k$, a set $V(L)$. There are many situations in mathematics in which it’s natural to consider instead a category $F(L)$, such that a morphism $L_1 \to L_2$ induces a functor $F(L_1) \to F(L_2)$, and so forth. The basic example is the case that $F(L) = \text{Mod}(L)$ is the category of $L$-vector spaces, and for $f : L_1 \to L_2$ a morphism the corresponding functor is given by extension of scalars

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

This leads to many other examples coming from equipping vector spaces with extra structure: for example, $F(L)$ might be

• the category of representations of some finite group $G$ over $L$,
• the category of commutative (or associative, or Lie) algebras over $L$, or
• the category of schemes over $L$.

It would be great if understanding all of these categories was in some sense as simple as understanding the category $F(\bar{k})$, which generally tends to be simpler, and the action of the absolute Galois group $G$ on it, whatever that means. For example, the representation theory of finite groups over algebraically closed fields of characteristic zero is well understood, as is, say, the classification of semisimple Lie algebras. The general problem of trying to extract an understanding of $F(L)$ from an understanding of $F(\bar{k})$ is the problem of Galois descent.

We might very optimistically hope that the story here is directly analogous to the story above. This suggests the following puzzle.

Puzzle: In what sense could the statement $F(L) \cong F(\bar{k})^H$ be true for the examples given above?

## Topological Diophantine equations

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let $R$ be a commutative ring, which in the most classical case might be a number field like $\mathbb{Q}$ or the ring of integers in a number field like $\mathbb{Z}$. Suppose we want to find solutions, over $R$, of a system of polynomial equations

$\displaystyle f_1 = \dots = f_m = 0, f_i \in R[x_1, \dots x_n]$.

Then it’s not hard to see that this problem is equivalent to the problem of finding $R$-algebra homomorphisms from $S = R[x_1, \dots x_n]/(f_1, \dots f_m)$ to $R$. This is equivalent to the problem of finding left inverses to the morphism

$\displaystyle R \to S$

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

$\displaystyle \text{Spec } S \to \text{Spec } R$

of affine schemes. Allowing $\text{Spec } S$ to be a more general scheme over $\text{Spec } R$ can also capture more general Diophantine problems.

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

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