nLab
weak inverse

Idea

A weak inverse is like an inverse, but weakened to work in situations where being an inverse on the nose would be evil.

Definitions

Given a functor $F:C\to D$, a weak inverse of $F$ is a functor $G:D\to C$ with natural isomorphisms

$$\iota :{\mathrm{id}}_C\to G\circ F,ϵ:F\circ G\to {\mathrm{id}}_D.$$\iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .

If it exists, a weak inverse is unique up to natural isomorphism, and furthermore can be improved to form an adjoint equivalence, where $\iota$ and $ϵ$ sastisfy the triangle identities.

More generally, given a $2$-category $ℬ$ and a morphisms $F:C\to D$ in $ℬ$, a weak inverse of $F$ is a morphism $G:D\to C$ with $2$-isomorphisms

$$\iota :{\mathrm{id}}_C\to G\circ F,ϵ:F\circ G\to {\mathrm{id}}_D.$$\iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .

Weak inverses give the proper notion of equivalence of categories and equivalence in a $2$-category. Note that you must use anafunctors to get the weak notion of equivalence of categories here without using the axiom of choice.

Links with homotopy theory

Given the classifying space functor $B:\mathrm{Cat}\to \mathrm{Top}$, weak inverses are sent to homotopy inverses. This is because the product with the interval groupoid is sent to the product with the topological interval $[\mathrm{0,1}]$. In fact, less is needed for this to be true, because the classifying space of the interval category is also the topological interval. If we define a lax inverse to be given by the same data as a weak inverse, but with $\iota$ and $ϵ$ replaced by natural transformations, then the classifying space functor sends lax inverses to homotopy inverses. An example of a lax inverse is an adjunction, but not all lax inverses arise this way, as we do not require the triangle identities to hold.

(David Roberts: I’m just throwing this up here quickly, it probably needs better layout or even its own page.)