David Roberts pseudoinverse

In a bicategory $B$ a pseudoinverse to a 1-arrow $f:x \to y$ is a 1-arrow $g:y \to x$ such that there are invertible 2-cells $a:fg \Rightarrow id$, $b:gf \Rightarrow id$. Clearly a pseudoinverse exists if and only if $f$ is an equivalence.

More generally…

A lax-inverse to a 1-arrow in a bicategory is the same except $a$ and $b$ are not required to be invertible. For example, one half of an adjunction is lax-inverse to the other half, but not all lax inverses are of this form, as they are not required to satisfy the triangle identities.

Remark: If the bicategory in question is the 2-category $Cat$ of small categories, then the geometric realisation of a lax-inverse is a homotopy equivalence. There is also a version of this for categories internal to Top.