Let be a commutative ring. A popular thing to do on this blog is to think about the Morita 2-category of algebras, bimodules, and bimodule homomorphisms over , 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 of -modules, equipped with the usual tensor product over . By the Eilenberg-Watts theorem, the Morita 2-category is equivalently the 2-category whose
- objects are the categories , where is a -algebra,
- morphisms are cocontinuous -linear functors , and
- 2-morphisms are natural transformations.
An equivalent way to describe the morphisms is that they are “-linear” in that they respect the natural action of on given by
This action comes from taking the adjoint of the enrichment of over , which gives a tensoring of over . 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 instead of over . 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.
Let be a symmetric monoidal category (which in this post will be , for a commutative ring), and consider categories enriched over , or -categories for short. If and are two -categories, their naive tensor product is the -category whose objects are pairs of objects in and objects in , and whose homs are given by the tensor products
of the homs in and , with the obvious composition (which requires the ability to switch tensor factors to define). When and have one object and , this reduces to the usual tensor product of -algebras.
Thinking of -categories as many-object generalizations of -algebras (one might call them “-algebroids,” but I won’t), it’s natural to define notions of modules over them. We’ll say that a left module over is a -functor (-enriched functor) , while a right module over is a -functor . If are two -categories, a -bimodule is a -functor .
All of this terminology has its usual meaning when and have one object (so correspond to algebras) and .
The basic analogy
The basic analogy, one piece of structure at a time, goes like this.
- Sets are analogous to categories.
- Abelian groups are analogous to cocomplete categories. (There are several other things we could have said here, but this is the one that’s relevant to thinking about Morita theory. The idea is that taking colimits categorifies addition.)
- Rings are analogous to monoidal cocomplete categories (this includes the condition that the monoidal structure distributes over colimits).
- Commutative rings are analogous to symmetric monoidal cocomplete categories.
- Modules over commutative rings are analogous to cocomplete module categories over symmetric monoidal cocomplete categories.
We won’t get into the generalities of thinking about symmetric monoidal categories or modules over them because, in this post, the only symmetric monoidal categories we care about are those of the form for a commutative ring , and the only module categories over them we care about are the ones that are “free on a category of generators” in the sense that they are categories of -linear presheaves / right modules
on essentially small -linear categories (thinking of taking presheaves as the free cocompletion). By the universal property of the free cocompletion, cocontinuous -linear functors correspond to -linear functors , or equivalently (by an adjunction) to -bimodules; this generalizes, and in particular proves, the Eilenberg-Watts theorem. Composition is given by tensor product of bimodules, which is computed using coends.
In the special case that the categories involved have one object, they correspond to -algebras, and the words “module,” “bimodule,” and “tensor product” all have their usual meaning. More generally, if has finitely many isomorphism classes of objects, then we can replace it with the endomorphism ring of the direct sum of one object from each isomorphism class (because is Morita equivalent to the one-object -linear category with this endomorphism ring), so we get a genuinely bigger Morita 2-category if we allow to have infinitely many objects.
From now on we’ll work in this bigger Morita 2-category, which is now the 2-category deserving the name . It has
- objects essentially small -linear categories ,
- morphisms -bimodules over , and
- 2-morphisms homomorphisms of bimodules.
Equivalently, it has
- objects the cocomplete -linear categories (where is as above),
- morphisms cocontinuous -linear functors , and
- 2-morphisms natural transformations.
From now on, when we say that a -linear category “is” a -algebra, possibly with some further properties, what we mean is that it’s equivalent to a category with one object and endomorphism ring a -algebra, possibly with some further properties.
In what ways do the objects of the Morita 2-category behave like modules?
Proposition: The Morita 2-category has biproducts.
In other words, the product is also the coproduct. This is because on the one hand a functor (cocontinuous and -linear) into is precisely a pair of functors into and , and on the other hand
If and are algebras rather than categories it would be tempting to write rather than , but in the case of algebras these are Morita equivalent as -linear categories. Actually, the above argument, with disjoint unions of -linear categories, applies just as well to infinitely many -linear categories: this means that the bigger Morita 2-category has infinite biproducts. Note that this has nothing to do with itself having biproducts: the same is true for the bigger Morita 2-category over .
Proposition: There is a tensor-hom adjunction
Here the internal hom is the category of cocontinuous -linear functors, which (as we’ve seen above, since it’s a category of bimodules) is itself a cocomplete -linear category, and the tensor product is
(this is a computation, not a definition: the definition is a universal property in terms of functors out of which are cocontinuous and -linear in each variable). Here is the “naive” tensor product over .
Both sides of the equivalence above are equivalent to , which might look familiar as a categorification of a corresponding statement about finite-dimensional vector spaces. This reflects the fact that is always dualizable with respect to the above tensor product, with dual .
With respect to this tensor product, is the unit object. It should be thought of as the “tensor product over ,” exactly analogous to the tensor product on modules over a commutative ring.
Aside: big categories vs. little categories
The biggest secret about category theory that I don’t think is in common circulation is that there are approximately two kinds of categories, and they should be thought of very differently (because the relevant notion of morphism between them is very different):
- Big categories are “categories of mathematical objects.” Typical examples are categories of modules and sheaves. They tend to be cocomplete, and people like to consider cocontinuous functors (really, left adjoints) between them.
- Little categories are “categories as mathematical objects.” Typical examples include categories with one object, or finitely many objects. They tend to be Cauchy complete at best, and people like to consider arbitrary functors, or more generally bimodules, between them.
The Morita 2-categories we discussed above have both little descriptions and big descriptions (via -linear categories and modules over these respectively), and both are important. We can pass from little to big by taking modules / presheaves, and we can pass from big to little by taking tiny objects. (This can at best recover the Cauchy completion of the original little -linear category, but this is fine since we’re only hoping to doing things up to Morita equivalence anyway.)
I think one thing that confuses people when they first start to learn category theory is that the first examples of categories (e.g. groups, rings, modules) tend to be big, even though little categories figure prominently in the theory (e.g. as shapes for diagrams to take limits or colimits over, and/or as things to take presheaves or sheaves on) and feel very different. It’s little categories that can reasonably be thought of as algebraic objects generalizing more familiar objects like monoids and posets (or, in our enriched setting, -algebras), whereas big categories, I think, genuinely require a new set of intuitions.
On the other hand, the ability to pass between big and little categories is also important. Eilenberg-Watts, as we have seen, gives one version of this: another version is Gabriel-Ulmer duality.
The big vs. little nomenclature suggests, but is not equivalent to, the rigorous distinction between large and small categories, and is related to the distinction between big and little toposes (or, depending on your preferences, between gros and petit topoi) in sheaf theory. The basic point of this distinction is that there are two somewhat different sorts of things people mean by sheaf: on the one hand one might mean a functor on a little category like the category of open subsets of a topological space, and on the other hand one might mean a functor on a big category like the category of commutative rings. It would be nice if people emphasized this distinction more.
Bases, coordinates, and matrices
In terms of higher linear algebra, big categories of modules are “higher vector spaces,” while the little categories that can be used to present the big categories as categories of modules are “bases” for them. As we learned previously, a cocomplete abelian category has a “basis” in this sense iff it has a family of tiny (compact projective) generators, for various notions of generator.
Given a “basis” for a “higher vector space” (really, a higher module), any object is described by a module / presheaf ; the components of this presheaf can be thought of as the “coordinates” of in the “basis” . In the same way that a vector is a sum over elements of a basis weighted by its coordinates in that basis, a presheaf is a weighted colimit / coend / functor tensor product
weighted by its “coordinates” of the “basis” of representable presheaves. This is an enriched version of the familiar statement that a presheaf of sets over a category is canonically a colimit of representable presheaves. It is sometimes called the co-Yoneda lemma.
Similarly, the statement that cocontinuous -linear functors are equivalent to functors can be interpreted as saying that such functors can be written as “matrices” indexed by and . Composition, as well as evaluation, are given by the familiar formulas if we consistently reinterpret the relevant products as tensor products and the relevant sums as coends.
Suppose in particular that . Then we get that endomorphisms of correspond to -bimodules, or equivalently to functors . These are precisely the sorts of things we can take coends of, getting an object
which deserves to be called the trace of the endomorphism . This is a generalization of (zeroth) Hochschild homology with coefficients in a bimodule, which it reduces to when is an algebra. (There is also a second trace generalizing Hochschild cohomology and computed using ends.)
The identity functor turns out to be represented by , so taking its trace we get the Hochschild homology or trace of (or, depending on your point of view, of ) itself:
More explicitly, this coend is the result of coequalizing the left and right actions of on (by postcomposition and precomposition respectively), so can be written as
where the two arrows send a pair of morphisms and to the two composites and . In other words, is the quotient of the direct sum of the endomorphism rings of every object in by the subspace of “commutators” of the form , where need not themselves be endomorphisms.
By construction, this means that is the recipient of a “universal trace” map: any endomorphism in has an image satisfying
for all as above, and is universal with respect to this property.
Example. Suppose has one object, so corresponds to an algebra . Then is just the quotient of by the subspace of commutators, hence is the ordinary zeroth Hochschild cohomology of (with coefficients in itself).
The above construction of the trace implies that it is Morita invariant, and is Morita equivalent to the -linear category of finitely generated projective (right) -modules (which is its Cauchy completion). It follows that the trace of this category is also , and this means that for any finitely generated projective -module and any endomorphism there is a trace
This trace is called the Hattori-Stallings trace. For free modules it is computed by taking the image of the sum of the diagonal elements of in . It is a shadow of various more general and more interesting maps from versions of K-theory to versions of Hochschild homology, including a version of the Chern character and the Dennis trace.
In particular, if is commutative, we recover the usual notion of trace of an endomorphism of a finitely generated projective -module without using a monoidal structure (previously we recovered this notion using dualizability with respect to the usual tensor product of -modules).
The simplest interesting case of the above discussion occurs when is a field and the algebras we consider are finite direct sums of matrix algebras ; equivalently, the -module categories we consider are finite direct sums of copies of . These are known as Kapranov-Voevodsky 2-vector spaces. Morphisms from to correspond to matrices of vector spaces, just as for free modules, and the trace of an endomorphism is the direct sum of its diagonal vector spaces.
Higher representation theory
One reason you might want a higher version of linear algebra is to study “higher representation theory,” where groups (or higher versions of groups, such as 2-groups) act on higher vector spaces. Previously we saw such actions occur naturally in Galois descent: namely, if is a Galois extension with Galois group , then naturally acts on the category of -vector spaces, and we used this action to describe Galois descent for vector spaces.
More generally, if is a group acting on a scheme (or some more or less general object, depending on taste), then naturally acts on the category of quasicoherent sheaves on . If is an -scheme for some base , this is naturally a module category over , and if acts on by -scheme automorphisms, the induced action on is -linear. One reason you might want to understand higher representation theory is to understand these sorts of actions. In particular, a natural question is what the homotopy fixed points of this action are, and the answer is that
that is, the homotopy fixed point category (which here is typically called “-equivariant quasicoherent sheaves on “) is the category of quasicoherent sheaves on the quotient regarded as a stack. (It certainly does not suffice to take the ordinary quotient of schemes: for example, if is a point, then is , and is the category of -linear representations of .)
If the stacky quotient happens to be an ordinary scheme (so is a -torsor in schemes), this is a generalization of Galois descent, to which it reduces in the case when is a finite extension, , and is the Galois group.