Equimorphisms of categories

Question

I've made the following observation: Let $i : Z \subseteq X$ the inclusion of a closed subset of a topological space. Then the functor

$i_* : Sh(Z) \to \{F \in Sh(X) : supp(F) \subseteq Z\}$

is an equivalence of categories with quasi-inverse $i^{-1}$. Remark that here $i^{-1}$ admits a simple description (or rather realization of its universal property being adjoint to $i_*$), namely $(i^{-1} F)(W) = F(\tilde{W})$, where $\tilde{W}$ is the greatest open subset of $X$ such that $\tilde{W} \cap Z = W$. We have an isomorphism $id \cong i_* i^{-1}$, but we even have an equality $id = i^{-1} i_*$ (because $(i^{-1} i_* F)(W) = (i_* F)(\tilde{W}) = F(\tilde{W} \cap Z) = F(W)$, compatible with restrictions).

More generally, we could define an equimorphism (mix of equivalence and isomorphism) to be a functor $F : C \to D$ such that there exists a functor $G : D \to C$ such that $1_C = GF$ and $1_D \cong FG$.

Do equimorphisms appear in other contexts as well? Is there some literature about them? What is the intuition behined them, how can I see directly that a known equivalence is actually an equimorphism? Is it possible to turn an equivalence (perhaps even by changing the categories by pure set-thoretically means, leaving the structure the same) into an equimorphism?

Answer 1

I think most category theorists would not regard your notion of equimorphism as particularly interesting. Usually whenever two functors are equal, rather than merely isomorphic, it's just a coincidence of the particular set-theoretic definitions chosen. From a categorical point of view, anything isomorphic to a functor like $i_*$ would serve just as well as $i_*$, and then you'd no longer have an equality $i^{-1}i_* = \mathrm{id}$. So I would say that there is really no intuition behind equimorphisms, because they are not really categorically interesting.

(Some people would argue that for small categories, the notion of isomorphism does become interesting, and thus so presumably might that of "equimorphism". But I've never seen something like it defined anywhere.)

On the other hand, if you assume enough AC, then every category is equivalent to a skeletal one, and two skeletal categories are equivalent iff they are isomorphic, so in this sense any equivalence can even be turned into an isomorphism by modifying the categories involved.

Answer 2

The analogue of "equimorphism" in topology is a deformation retract: $A$ is a deformation retract of $X$ if there are maps $i : A \to X$, $r : X \to A$ such that $ri = \operatorname{id}_A$ and $ir$ is homotopic to $\operatorname{id}_X$. Apparently this concept is important enough to have a name, though like Mike says, from a higher-categorical point of view, a deformation retract is not interestingly different from an arbitrary homotopy equivalence.

Answer 3

There is a natural example of such a "equimorphism" in the theory of uniform spaces. Beside the definitions of a uniformity via covers / relations, there is the definition via pseudo metrics. Every set of pseudo metrics on $X$ induces a uniformity, but this is not injective. On the other hand one can assign to every uniformity a maximal set of pseudo metrics that induces this particular uniformity. This is injecitve.

The category of set with sets of pseudometrics and the category of uniform spaces are "equimorph" by this contructions.

Answer 4

Actually, the analogy to deformation retracts in Reid's answer can be made quite precise. In the canonical or categorical model structure on Cat (often problematically called the "folk model structure"), the acyclic cofibrations are the equivalences of categories which are injective on objects. These are evidently precisely the functors that occur as F in your equimorphisms.

So probably a more correct thing to say than my original answer would be that it depends on whether you are treating categories truly "categorically," or whether you are using "too-strict" notions for convenience in getting at the weaker "correct" notions. Model category theory, and homotopy theory more generally, is all about doing the latter. Category theorists tend to work directly with the "right" notions, because for ordinary categories doing so is pretty easy--but of course when you get up to higher categories, some model category theory frequently turns out to be useful.

Your original question, though, suggested that you were thinking about whether such equimorphisms "arise naturally" between the ordinary sort of large categories that appear in mathematical practice. I think I would still say that whenever that happens, it's an accident of the chosen set-theory rather than anything really interesting. The "too-strict" notions really only have technical, rather than fundamental, importance.