What is the category of covariant and contravariant functors?

Question

Let $\bf Cat'$ be the category that has as objects small categories $A, B...$, and as arrows functors $F:A\to B$ that are either covariant or contravariant. The identity on $A\in\bf Cat'$ is the usual identity functor; the composition of a covariant and a contravariant functor is contravariant and the composition of two contravariant functors is covariant. (Here, by a contravariant functor $F:A\to B$, I mean a mapping $F: obj A \to obj B$ and for any arrow $f:x\to y$ an arrow $Ff:Fy\to Fx$ with the usual axioms on identities and composition.)

I wonder which are the abstract properties of $\bf Cat'$. Note for instance that in $\bf Cat'$ any category is isomorphic to its dual; so I suppose that it is not possible to recover the covariant arrows in an abstract way. Maybe the obvious functor $\bf Cat \to \bf Cat'$ has some universal property?

Answer 1

$\mathbf{Cat'}$ can be thought of as a semi-direct product. There is an action $G=(\mathbb{Z}/2\mathbb{Z})$ on $\mathbf{Cat}$ given be the oposite category endofunctor and $\mathbf{Cat'}$ is isomorphic to the semi-direct product $G \ltimes \mathbf{Cat}$.

In general the semi-direct product $G \ltimes C$ of a category $C$ by a group $G$ acting on $C$, is the category whose objects are the $c \in C$ and the morphisms $x \to y$ are pair $(g,f)$ where $g$ is an element of $G$ and $f$ is an arrow $x \to gy$, composition being given by $(g,f) \circ (g',f') = (gg',f \circ gf')$.

In our case, morphisms of the form $(0,f)$ are covariant functor, while the $(1,f)$ are the contravariant functors !

Semi-direct product are also a special case of the Grothendieck construction: the action of $G$ on $C$ can be described as a functor $BG \to \mathbf{Cat}$ where $BG$ is the one object groupoid with $G$ has its unique automorphism group. And the semi-direct product described above is the corresponding Grothendieck construction.

Now, this Grothendieck construction point of view provide us with a universal property for the semi-direct product: The Grothendieck construction is the Lax colimit of a diagram. Here, given that $BG$ only has invertible cell, this is also the pseudo-colimit of the diagram $BG \to \mathbf{Cat}$. That is it is the "pseudo-quotient" of $C$ by the action of $G$.

Comming back to the action of $\mathbb{Z}/2\mathbb{Z}$, this means that $\mathbf{Cat} \to \mathbf{Cat'}$ is universal for making the action of $\mathbb{Z}/2\mathbb{Z}$ trivial, in an appropriate 2-categorical sense.

If you prefer, the functor from the $2$-category of categories to the $2$-category of categories with an action of $\mathbb{Z}/2\mathbb{Z}$ has a left 2-adjoint $Q$, and if $\mathbf{Cat}$ is endowed with its action of $\mathbb{Z}/2\mathbb{Z}$ by the opposite category, then (up to equivalence of categories) $\mathbf{Cat'} \simeq Q(\mathbf{Cat})$.

Answer 2

You may also be interested in my paper Contravariance through enrichment, which shows that $\bf Cat'$ is enriched over a certain non-symmetric monoidal structure on $\bf Cat\times Cat$. This allows distinguishing the covariant and contravariant functors while still keeping them both present in the structure. Thus for instance we can distinguish the "contravariant isomorphism" $A\cong A^{\rm op}$ from a normal covariant isomorphism, and indeed use the former to characterize $A^{\rm op}$ as "the unique (up to covariant isomorphism) object contravariantly isomorphic to $A$", which is also a copower of $A$ in the sense of enriched category theory.

This structure is actually closely related to the construction in Simon's answer; the monoidal structure on $\bf Cat\times Cat$ is constructed using the $\mathbb{Z}/2\mathbb{Z}$ action on $\bf Cat$, and can be generalized to other group actions.

Also I should note that when I say $\bf Cat'$ "is" enriched over $\bf Cat\times Cat$, I don't mean in the precise sense that $\bf Cat'$ is the "underlying ordinary category" of an enriched category in the standard sense of enriched category theory; the underlying ordinary category of the enriched form of $\bf Cat'$ is actually just $\bf Cat$. This is similar to the situation of group actions, where a category enriched over $G$-sets is a category with $G$-actions on its hom-sets, but the "underlying ordinary category" takes the fixed-point sets of those hom-sets (rather than, for instance, forgetting the $G$-actions).