An important structure possessed by the 2-category is the duality involution . Here we consider what the important properties of this involution are that should be generalized to other 2-categories.
The most obvious property of is that it is coherently self-inverse. To state this formally, let be the walking isomorphism , considered as a 3-category with only identity 2-cells and 3-cells. Thus, a (pseudo) functor from to any 3-category can be considered an “internal adjoint (bi)equivalence in .” If is a 3-category, let denote the 3-cell dual of ; note that and that is a functor . Finally, let be the evident involution that switches and .
A 2-contravariant involution (hereafter merely an involution) on a 2-category is functor such that and the square
commutes (on the nose).
We write (or equivalently ) for the functor ; the rest of the data of simply says that in a coherent way.
Any (2,1)-category admits a canonical involution that is the identity on objects and morphisms and sends each 2-cell to its inverse.
Let be a 2-category equipped with an involution and let . Then has a canonical induced involution defined by
Conversely, if is Cauchy-complete, then can be identified with the full subcategory of indecomposable projectives in , and thus an involution on induces an involution on .
For any 2-category , the 2-category of involutions on , if nonempty, is a torsor for the 3-group of (covariant) automorphisms of .
In particular, any automorphism of a (2,1)-category induces an involution on .
As observed by WeberYS2T, experience suggests that the most important additional property of the involution on is that fibrations over can be identified with opfibrations over , since both are equivalent to functors . So it is reasonable to consider, together with an involution on a 2-category , a family of equivalences .
The next question is what additional properties should be required of such a family. The following definition is provisional, but I believe whatever eventual definition we settle on should allow the construction of all the data given below.
A duality involution on a 2-category is an involution together with the following data.
which are natural in (that is, is a natural equivalence between functors ),
in which all the displayed maps are fibrations, an invertible modification between the two composites
which commutes with the previous two equivalences.
Perhaps there should also be some coherence data and axioms relating these data to each other.
The first two data should be fairly self-explanatory, but the third datum may seem quite complicated. In fact, it requires some thought to see that the two displayed composites are even well-defined. Note first that since , by naturality of we have , and thus . Also, , so again by naturality . Thus , and the final two equivalences in each displayed composite are instances of the theorem on iterated fibrations. One way to think about this datum is as sort of an “inclusion-exclusion” property for reversal of arrows.
We call the final datum commutation of opposites. To explain it, observe that because is a 2-contravariant involution, we automatically have equivalences (these are the vertical maps in the displayed square). In , this equivalence corresponds to composing a functor with as well as reinterpreting it as an opfibration. Commutation of opposites then says that this operation commutes with .
One immediate application of commutation of opposites is to deduce a version of the third datum for a pullback square of opfibrations, by passage along the equivalences . We will see others below.
If is a (2,1)-category, then its canonical involution extends to a canonical duality involution, since .
If is a (2,1)-category, then the canonical involution on extends in a natural way to a duality involution. This does not seem to be the case if is merely a 2-category equipped with an involution.
The 3-group also acts on the 2-category of duality involutions on ; the action is free but (seemingly) no longer necessarily transitive. However, I do not know an example of two inequivalent duality involutions having the same underlying involution (as would be necessary for non-transitivity).
Of course, since any equivalence of 2-categories preserves discrete objects, our definition implies an equivalence . More surprisingly, it turns out that the seemingly stronger two-sided version is a consequence of our definition.
If is equipped with a duality involution, then we have natural equivalences .
We first construct an equivalence . Note that since is the identity, naturality implies that . Also, since is an equivalence, and products in are the same as products in , we have . Now we have the composite equivalence (using the theorem on iterated fibrations):
Note that there is also another equivalence
but the second and third data in the definition of a duality involution supply a canonical equivalence between these composites, so it doesn’t matter which we use. Finally, for the general case, we simply apply this twice:
This gives our desired equivalence.
Note, though, that this proof crucially requires that the duality involution come with an equivalence , and not merely .
Commutation of opposites, which we have not yet used, is necessary for the following important, and perhaps also surprising, result: duality involutions are preserved by (fibrational) slicing.
Any duality involution on induces a duality involution on each fibrational slice and .
We define to be the composite
To show that is an involution, we verify
Here we use commutation of opposites in to commute past . We now define
to be the composite
It is easy to see that , and to construct the pullback-commutation equivalence. Finally, we construct commutation of opposites in as the composite
The case of is dual.
We saw above that in general, a 2-category admitting an involution or duality involution will admit many. One way to get rid of some of these spurious involutions is to require that fix groupoidal objects, as is clearly the case for the involution of and the “canonical” involutions of (2,1)-truncated 2-presheaf 2-toposes.
Note that any involution takes groupoidal objects to groupoidal objects.
An involution on fixes groupoids it it restricts to the canonical involution on , i.e. if coherently whenever is groupoidal.
Suppose that is an involution on that fixes groupoids. Then for any and any groupoidal , we have
Now given as well, with an eso where is groupoidal, any morphism is determined by a map with an action by . But is also groupoidal, so morphisms are completely determined by morphisms to out of groupoidal objects. Thus, by the Yoneda lemma, any two values of must be canonically isomorphic.
Now suppose is exact, and let be an eso where is groupoidal. Then its kernel is also groupoidal, and so is a homwise-discrete category in . Thus we can consider its opposite, which is also a homwise-discrete category in . In general, the opposite of a 2-congruence in a 2-category will not be a 2-congruence, since the condition of being a two-sided fibration is not preserved. But in a (2,1)-category such as , this extra condition is automatic, so this opposite of is also a 2-congruence in (and, in fact, an -congruence when is an -category). Thus, since is -exact, this opposite has a quotient, which we call . It is straightforward to verify that this defines an involution on that fixes groupoids.
Of course, the only interesting values of in the above lemma are 2 and (1,2), since any groupoid-fixing involution on a (2,1)-category is equivalent to the canonical one. Of particular note is that any (2,1)-truncated Grothendieck 2-topos admits a unique groupoid-fixing involution.
In fact, this groupoid-fixing involution should also extend to a duality involution. For when is -exact, the functor ought to be a 3-sheaf? (a 2-stack). This means that if is an eso with groupoidal, then can be reconstructed from and with descent data. Similarly, should also be a 3-sheaf, and so can be recovered from and with “op-descent data.” But since is groupoidal, , and likewise for . Therefore, descent data for over , which determines an object of , is the same as op-descent data for the opposite of over , which determines an object of . This gives the equivalence .
Alternately, there is also a stronger theorem that if is 2-exact with enough groupoids, then it is equivalent to a particular 2-category of internal categories and anafunctors in the (2,1)-category (more precisely, it is the 2-exact completion of ). And it is fairly easy to see, by its construction, that this 2-category has a duality involution, just as the 2-category of internal categories in any 1-category does.
Modulo the action of automorphisms, these are the only examples of duality involutions that I know. I do not know an example of a duality involution on a 2-category that does not have enough groupoids.
Now, since a duality involution on extends to a duality involution on and , it is natural to strengthen the notion of groupoid-fixing so that it carries over to fibrational slices. We write and similarly .
A duality involution on fixes groupoids locally if we have a equivalence modification between and the composite .
The latter equivalence is the canonical involution on the (2,1)-category . Taking , together with the second condition in the definition of a duality involution, this implies that fixes groupoids in the previous sense. Moreover, we have:
If is a duality involution that fixes groupoids locally, then the induced duality involutions on and also fix groupoids locally.
For a fibration , on we have and , so the equivalence between and in induces one between and in .
Now, suppose that is a duality involution that fixes groupoids locally. Then for any and , the pullback-commutation datum shows that the composite equivalence
derived from Theorem 1 is equivalent to . But if we restrict to groupoidal fibrations, then
is equivalent to . Together with commutation of opposites, this implies that the canonical equivalence
is, up to equivalence, simply given by . In particular, since preserves powers, the canonical equivalence
takes to . In the 2-internal logic, this corresponds to the statement that “,” which is certainly an expected part of the behavior of opposite categories. Of course, for this it suffices that fix discretes locally, which has the evident definition.
Another natural requirement is that when is groupoidal, should be equivalent to the composite
Note that this includes the second datum in the definition of a duality involution as a special case, since is groupoidal. There seems no reason for this condition to be stable under slicing, but it could be stabilized.