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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.

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Distributive categories

Among all of the standard algebraic structures that a student typically encounters in an introduction to abstract algebra (groups, rings, fields, modules), commutative rings are somehow special: the opposite category \text{CRing}^{op} behaves like a category of spaces, so much so that an entire field of mathematics is dedicated to doing geometry based on it.

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 C with finite products \times and finite coproducts + such that finite products distribute over finite coproducts; more explicitly, the natural maps

X \times Y+ X \times Z \to X \times (Y + Z)

should be isomorphisms, and also the natural maps 0 \to X \times 0 should be isomorphisms, where 0 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:

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

  • \text{Grp}
  • 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 \text{Gpd} is distributive!

Let 2 be a set with two elements. The category of Boolean functions is the category whose objects are the finite powers 2^k, k \in \mathbb{Z}_{\ge 0} of 2 and whose morphisms are all functions between these sets. For a computer scientist, the morphisms of this category have the interpretation of functions which input and output finite sequences of bits.

Since this category has finite products and is freely generated under finite products by a single object, namely 2, it is a Lawvere theory.

Question: What are models of this Lawvere theory?

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Previously we learned how to count the number of finite index subgroups of a finitely generated group G. But for various purposes we might instead want to count conjugacy classes of finite index subgroups, e.g. if we wanted to count isomorphism classes of connected covers of a connected space with fundamental group Gi.

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 X is a groupoid, then LX = [S^1, LX], the free loop space or inertia groupoid of X, is the groupoid of maps S^1 \to X, where S^1 is the groupoid B\mathbb{Z} with one object and automorphism group \mathbb{Z}. Explicitly, this groupoid has

  • objects given by automorphisms f : x \to x of the objects x \in X, and
  • morphisms (f_1 : x_1 \to x_1) \to (f_2 : x_2 \to x_2) given by morphisms g : x_1 \to x_2 in X such that

x_1 \xrightarrow{f_1} x_1 \xrightarrow{g} x_2 = x_1 \xrightarrow{g} x_2 \xrightarrow{f_2} x_2.

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

In particular, when G is finite, this quotient always has groupoid cardinality 1. Hence:

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

I promise this is relevant to counting subgroups!

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My MaBloWriMo 2015 run met an untimely end on the 18th, when LaTeX stopped working on WordPress for me; I could no longer see any of the LaTex I was writing. It’s still not working for me in Chrome, but it’s now working for me in another browser, so hopefully I’ll get some posts up soon.

Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence

\displaystyle C(k) \cong C(L)^G

where f : k \to L is a Galois extension, G = \text{Aut}(L) is the Galois group of L (thinking of L as an object of the category of field extensions of k at all times), C(k) is a category of “objects over k,” and C(l) is a category of “objects over L.”

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

Today we’ll recover from this abstract description the somewhat more concrete punchline that k-forms c_k \in C(k) of an object c_L \in C(L) can be classified by Galois cohomology H^1(BG, \text{Aut}(c_L)), and we’ll give some examples.

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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.

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