weak inverse

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

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

$\iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$

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 $\epsilon$ sastisfy the triangle identities.

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

$\iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$

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 $\vert -\vert: Cat \to Top$, 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 $[0,1]$. 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 $\epsilon$ 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.)

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