# nLab weak inverse

### Context

#### Higher category theory

higher category theory

# Contents

## 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

$\iota :{\mathrm{id}}_{C}\to G\circ F,\phantom{\rule{thickmathspace}{0ex}}ϵ:F\circ G\to {\mathrm{id}}_{D}.$\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

$\iota :{\mathrm{id}}_{C}\to G\circ F,\phantom{\rule{thickmathspace}{0ex}}ϵ:F\circ G\to {\mathrm{id}}_{D}.$\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.

Given the geometric realization of categories functor $\mid -\mid :\mathrm{Cat}\to \mathrm{Top}$, 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 $\left[0,1\right]$. 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.)

Revised on June 1, 2011 09:37:08 by Urs Schreiber (131.211.238.83)