nLab weak inverse



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A weak inverse or quasi-inverse is like an inverse, but weakened to work in situations where being an inverse on the nose would violate the principle of equivalence.


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

ι:id CGF,ϵ:FGid 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 ι\iota and ϵ\epsilon sastisfy the triangle identities.

More generally, given a 22-category \mathcal{B} and a morphisms F:CDF: C \to D in \mathcal{B}, a weak inverse of FF is a morphism G:DCG: D \to C with 22-isomorphisms

ι:id CGF,ϵ:FGid 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 22-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 ||:CatTop \vert -\vert: Cat \to 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 [0,1][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 ι\iota and ϵ\epsilon 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.)

Last revised on July 26, 2022 at 00:25:52. See the history of this page for a list of all contributions to it.