Archive for the ‘math.CT’ Category

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.


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Separable algebras

Let k be a commutative ring and let A be a k-algebra. In this post we’ll investigate a condition on A which generalizes the condition that A is a finite separable field extension (in the case that k is a field). It can be stated in many equivalent ways, as follows. Below, “bimodule” always means “bimodule over k.”

Definition-Theorem: The following conditions on A are all equivalent, and all define what it means for A to be a separable k-algebra:

  1. A is projective as an (A, A)-bimodule (equivalently, as a left A \otimes_k A^{op}-module).
  2. The multiplication map A \otimes_k A^{op} \ni (a, b) \xrightarrow{m} ab \in A has a section as an (A, A)-bimodule map.
  3. A admits a separability idempotent: an element p \in A \otimes_k A^{op} such that m(p) = 1 and ap = pa for all a \in A (which implies that p^2 = p).

(Edit, 3/27/16: Previously this definition included a condition involving Hochschild cohomology, but it’s debatable whether what I had in mind is the correct definition of Hochschild cohomology unless k is a field or A is projective over k. It’s been removed since it plays no role in the post anyway.)

When k is a field, this condition turns out to be a natural strengthening of the condition that A is semisimple. In general, loosely speaking, a separable k-algebra is like a “bundle of semisimple algebras” over \text{Spec } k.


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It’s common to think of monads as generalized algebraic theories; the most familiar examples, such as the monads on \text{Set} encoding groups, rings, and so forth, have this flavor. However, this intuition is really only appropriate for certain monads (e.g. finitary monads on \text{Set}, which are the same thing as Lawvere theories).

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.


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Lie algebras are groups

Once upon a time I imagine people were very happy to think of Lie algebras as “infinitesimal groups,” but presumably when infinitesimals fell out of favor this interpretation did too. In this post I’ll record an observation that can justify thinking of Lie algebras as groups in a strong sense: they are group objects in a certain category which can be interpreted as a category of “infinitesimal spaces.”

Below we work throughout over a field of characteristic zero.

For starters, the universal enveloping algebra functor \mathfrak{g} \mapsto U(\mathfrak{g}), 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 H, its primitive elements are those elements x \in H satisfying

\Delta x = x \otimes 1 + 1 \otimes x

where \Delta 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 \mathfrak{g} \to U(\mathfrak{g}) is fully faithful; that is, there is a natural bijection between Lie algebra homomorphisms \mathfrak{g} \to \mathfrak{h} and Hopf algebra homomorphisms U(\mathfrak{g}) \to U(\mathfrak{h}). This is more or less equivalent to the claim that the natural inclusion \mathfrak{g} \to U(\mathfrak{g}) induces an isomorphism from \mathfrak{g} to the Lie algebra of primitive elements of U(\mathfrak{g}), 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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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.


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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!

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