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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.

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.

## Group actions on categories

Yesterday we decided that it might be interesting to describe various categories as “fixed points” of Galois actions on various other categories, whatever that means: for example, perhaps real Lie algebras are the “fixed points” of a Galois action on complex Lie algebras. To formalize this we need a notion of group actions on categories and fixed points of such group actions.

So let $G$ be a group and $C$ be a category. For starters, we should probably ask for a functor $F(g) : C \to C$ for each $g \in G$. Next, we might naively ask for an equality of functors

$\displaystyle F(g) F(h) = F(gh) : C \to C$

but this is too strict: functors themselves live in a category (of functors and natural transformations), and so we should instead ask for natural isomorphisms

$\displaystyle \eta(g, h) : F(g) F(h) \cong F(gh)$.

These natural isomorphisms should further satisfy the following compatibility condition: there are two ways to use them to write down an isomorphism $F(g) F(h) F(k) \cong F(ghk)$, and these should agree. More explicitly, the composite

$\displaystyle F(g) F(h) F(k) \xrightarrow{F(g) \eta(h, k)} F(g) F(hk) \xrightarrow{\eta(g, hk)} F(ghk)$

should be equal to the composite

$\displaystyle F(g) F(h) F(k) \xrightarrow{\eta(g, h) F(k)} F(gh) F(k) \xrightarrow{\eta(gh, k)} F(ghk)$.

(There’s also some stuff going on with units which I believe we can ignore here. I think we can just require that $F(e) = \text{id}_C$ on the nose and nothing will go too horribly wrong.)

These natural isomorphisms $\eta(g, h)$ can be regarded as a natural generalization of 2-cocycles, and the condition above as a natural generalization of a cocycle condition. Below the fold we’ll describe this and other aspects of this definition in more detail, and we’ll end with two puzzles about the relationship between this story and group cohomology.

## Dualizable objects (and morphisms)

Let $R$ be a ring. Previously we characterized the finitely presented projective (right) $R$-modules as the tiny objects in $\text{Mod}(R)$: the objects $P$ such that

$\displaystyle \text{Hom}(P, -) : \text{Mod}(R) \to \text{Ab}$

preserves colimits. We also highlighted the key role that these modules play in Morita theory.

If $k$ is a commutative ring, then $\text{Mod}(k)$ has a natural symmetric monoidal structure which allows us to describe another finiteness condition called dualizability. Unlike tininess, dualizability makes no reference to colimits; instead, it is a purely equational definition involving the monoidal structure. The dualizable modules are again the finitely presented projective $k$-modules.

Dualizability implies that we can treat finitely presented projective $k$-modules like finite-dimensional vector spaces in many ways: for example, dualizability allows us to define the trace of an endomorphism. Moreover, since dualizability is defined using only a monoidal structure, it makes sense in very general settings, and we’ll look at some more exotic examples of dualizable objects as well.

Duals are also a special case of a 2-categorical notion of adjunction which, in the 2-category of categories, functors, and natural transformations, reproduces the usual notion of adjunction. In a suitable 2-category it will also reproduce another characterization of finitely presented projective modules, this time over noncommutative rings.

This post should, but will not, include diagrams, so pretend that I’ve inserted some string diagrams or globular diagrams where appropriate.

## The homotopy groups are only groups

Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

• Given a Lie group $G$, its tangent space $T_e(G)$ at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{Z})$ is a priori a graded abelian group, but it ends up having the structure of a graded ring.
• Given a space $X$, its cohomology $H^{\bullet}(X, \mathbb{F}_p)$ over $\mathbb{F}_p$ is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-$p$ Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups $\pi_n(X)$ of a space: that is, we will show that, in a suitable sense, each individual homotopy group $\pi_n(X)$ is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)

## Groupoids

My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.

In the previous post we learned that it is possible to recover the center $Z(R)$ of a ring $R$ from its category $R\text{-Mod}$ of left modules (as an $\text{Ab}$-enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).
For noncommutative rings, the situation is more interesting. Two rings $R, S$ are said to be Morita equivalent if the categories $R\text{-Mod}, S\text{-Mod}$ are equivalent as $\text{Ab}$-enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.