nLab inverse functor

Idea

The concept of inverse functor is the generalization of that of inverse function from sets to categories. Where the existence of an inverse function exhibits a bijection of sets, the existence of an inverse functor exhibits an equivalence of categories.

Definition

Given a functor

$$F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$$

then an inverse to $F$ is a functor going the other way around

$$G \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$$

together with natural isomorphisms relating their composites to the respective identity functors:

$$G\circ F \simeq id_{\mathcal{C}} \phantom{AAAA} F \circ G \simeq id_{\mathcal{D}} \,.$$

Related concepts

• inverse, inverse morphisms