Let be a ring. Previously we characterized the finitely presented projective (right) -modules as the tiny objects in : the objects such that
preserves colimits. We also highlighted the key role that these modules play in Morita theory.
If is a commutative ring, then 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 -modules.
Dualizability implies that we can treat finitely presented projective -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.
All compositions will be written in “diagrammatic order”: If is a morphism and is another morphism, then will denote the composite . This means that it’s necessary to be careful when trying to match up the definitions below to other definitions.
“2-category” means bicategory and not strict 2-category, although for simplicity we’ll be pretending that all of the 2-categories we write down are strict, and in particular we’ll ignore associators, etc.
Definition and examples
Let be a 2-category. For lack of a better notation, below denotes horizontal composition of 2-morphisms, while vertical composition of 2-morphisms will be denoted by concatenation.
(the unit or coevaluation map) and
(the counit or evaluation map) satisfying the zigzag identities
Switching the order of composition of morphisms, or equivalently switching the role of and , gives the definition of left dualizable and left duals / adjoints. We will write to mean that is right dualizable with right dual / adjoint , or equivalently that is left dualizable with left dual / adjoint . If we don’t want to single out a particular interpretation, we will say that and (in that order) are a dual / adjoint pair.
Suppose in particular that is the delooping of a monoidal category , or equivalently that it has one object, so that the endomorphisms of that object form a monoidal category . Then right dualizability is a condition on an object , and we will denote its right dual by . Similarly, if we need it, the left dual will be denoted by . If we need to distinguish this notion of dual from another one, we will use the term monoidal dual.
In this special case right dualizability takes the following form. If denote the identity object of , the unit and counit maps are
and the zigzag identities are
If where is at least braided monoidal, then the braiding turns left duals into right duals, so the term “dualizable” is unambiguous in this case.
An important observation about this definition is that it is purely equational: it asks only that some objects, morphisms, and 2-morphisms satisfy some equations between compositions, and checking that this is the case is “local” in the sense that it doesn’t involve any other objects, morphisms, and 2-morphisms in the ambient 2-category. It also follows that adjoints are preserved by arbitrary 2-functors and that duals are preserved by arbitrary monoidal functors.
Example. Let be the symmetric monoidal category of vector spaces over a field equipped with the usual tensor product. Then a vector space is dualizable iff it is finite-dimensional, in which case is the usual vector space dual , and the unit and counit map are the familiar coevaluation and evaluation maps respectively. This is the basic source of the intuition that dualizability is a “finiteness” condition on an object.
Example. Let be the 2-category of categories, functors, and natural transformations. Then the above is precisely the definition of a unit-counit adjunction (provided that we interpret composition of functors in diagrammatic order, to get the lefts and rights to match up).
Example. Let be a category with finite limits and let be the symmetric monoidal category whose objects are the objects of but whose morphisms are (isomorphism classes of) spans in , with monoidal structure given by the product in . Then every object is self-dual. If is an object, then the unit map can be taken to be the span
(where is the diagonal), and similarly the counit map can be taken to be the span
As defined, fails to be locally small if isn’t small. One way to fix this is to notice that is naturally a symmetric monoidal 2-category whose morphisms are spans and whose 2-morphisms are morphisms of spans, and as a 2-category it is locally small (if is) in the sense that between morphisms there is a set of 2-morphisms.
Example. Let be a commutative ring and let be the Morita 2-category, whose
- objects are -algebras ,
- morphisms are -bimodules over (meaning that is equipped with a left action of and a commuting right action of such that the two resulting actions of agree), with composition given by tensor product, and
- 2-morphisms are bimodule homomorphisms.
This example strictly generalizes the example of the monoidal category of vector spaces, which we get when is a field and we look at the endomorphism category of in . By the Eilenberg-Watts theorem, an equivalent description of this 2-category is that its
- objects are the -linear categories ,
- morphisms are the cocontinuous -linear functors, which are precisely the functors of the form for an -bimodule over , and
- 2-morphisms are natural transformations.
As we’ll see, a bimodule is right dualizable iff it is finitely presented projective as a -module, in which case its right dual is .
is itself symmetric monoidal with monoidal structure the tensor product , and so we can also ask when an object is dualizable. (In a monoidal 2-category this means we only ask that the zigzag identities hold up to 2-isomorphism.)
It turns out that every object is dualizable, with dual . The unit morphism is regarded as a -bimodule and the counit morphism is regarded as a -bimodule.
Traces and dimensions
Suppose is a dualizable object in a symmetric monoidal category and that is an endomorphism.
Definition: The trace of is the composite
where denotes the braiding in . The dimension of is the trace .
We won’t use traces in the rest of the post, but it’s good to know that they’re there. The fact that they can be defined and manipulated using string diagrams is an aspect of -dimensional topological field theory.
Example. Both of these names come from the fact that in the special case that is the symmetric monoidal category of vector spaces over a field , we get back the usual notion of the trace of an endomorphism of a finite-dimensional vector space and the dimension of respectively (as an element of , which loses some information if has positive characteristic).
Example. Let as above, and let be an endomorphism of an object in the usual sense, regarded as the span . The trace turns out to be the pullback of the diagram ; in other words, it’s the intersection of the diagonal and the graph of inside , which can in turn be identified with the object of fixed points of (e.g. if we get back the set of fixed points in the usual sense), regarded as a span .
Example. Let as above. If is a -algebra, then an endomorphism of in this 2-category is an -bimodule , and its trace turns out to be
also known as the zeroth Hochschild homology . It can be thought of as the universal quotient of on which the left and right -actions agree. In particular, the dimension of turns out to be the zeroth Hochschild homology
where denotes the subspace (not ideal!) of commutators.
Let be a commutative ring as above. We will now classify the left and right dualizable 1-morphisms in ; that is, we will classify left and right dual pairs of bimodules . Throughout this section it may be useful to keep in mind the special case that and that is a field, in which case we are just saying some familiar facts about duality for finite-dimensional vector spaces.
Recall that dualizability means that we have a unit 2-morphism
(here a morphism of -bimodules) and a counit 2-morphism
(here a morphism of -bimodules) satisfying the zigzag identities. Explicitly, is completely determined by a particular element
of satisfying , and is a bilinear map
satisfying and , as well as . The first zigzag identity is
and the second is
At this point we need the following lemma.
Dual basis lemma: A right -module is finitely presented projective iff there exist and (which is a left -module) such that, for any , we have
Under these hypotheses, we furthermore have as right -modules, and for any , we have
In particular, is also finitely presented projective.
Proof. The first statement just a restatement of the condition that is a retract of a finite rank free module . The “basis” define a morphism , the “dual basis” define a morphism , and the above condition is precisely the condition that (keeping mind that we’re still writing compositions in diagrammatic order).
Applying to the previous paragraph, we obtain dual morphisms satisfying , hence is finitely presented projective and the , together with the images under the canonical map , form a dual basis of .
It remains to show that the canonical map is an isomorphism. Here is a cheap trick for doing so: it is clearly true if is finite free, and because splitting idempotents is an absolute colimit, it is preserved by all functors, including . Hence it’s true for retracts of finite free modules.
Theorem: A bimodule has a right dual iff it is finitely presented projective as a right -module, in which case . Dually, a bimodule has a left dual iff it is finitely presented projective as a left -module, in which case .
Proof. With notation as above, the identity from above implies that and form a dual basis, so by the dual basis lemma, if has a right dual then it is finitely presented projective as a right -module. Similarly, the identity implies that and form a dual basis, so by the dual basis lemma, if has a left dual then it is finitely presented projective as a left -module. Now, the counit defines morphisms
of bimodules which we want to show are isomorphisms. We’ll do this by using the unit to exhibit morphisms of bimodules
which we want to show are their inverses. This comes down to checking four identities, two of which are the zigzag identities, and two of which are their -linear duals.
Conversely, suppose is finitely presented projective as a right -module, and set . We want to show that is the right dual of . The counit of the adjunction will just be the dual pairing
so the interesting question is how to find the unit. Explicitly, we do this by finding a basis and a dual basis , which we assemble into a proposed unit
It may not be immediately clear why this is equal to . To see this, apply both sides to an element ; we get either way. Then we need to use the fact that , which follows from the stronger statement that
This statement follows in turn from the fact that is finitely presented projective, hence is cocontinuous, and then from the Eilenberg-Watts theorem. This lets us interpret the unit more abstractly as the map
describing the left action of on .
It remains to verify the zigzag equations. The first one reads
which is the first half of the dual basis lemma. The second one reads
which is the second half of the dual basis lemma.
Corollary: Let be a commutative ring. A -module is dualizable iff it is finitely presented projective as a -module, in which case its dual (both left and right) is its linear dual .
Proof. Apply the above theorem to the case that .
Example. If is a morphism of -algebras, we can consider the corresponding bimodule , where the left -module structure is provided by . This defines a functor from the ordinary category of -algebras and -algebra homomorphisms to the Morita 2-category . This bimodule is always finitely presented projective as a right -module, so it always has a right dual, namely , where now the right -module structure is provided by . This reflects the fact that the extension of scalars functor
always has a cocontinuous right adjoint, namely the restriction of scalars functor
However, the condition that this bimodule has a left dual is quite strong: this is the condition that is finitely presented projective as a left -module. For example, if and is a PID, this means that is finite free as a -module.