In a bicategory a pseudoinverse to a 1-arrow is a 1-arrow such that there are invertible 2-cells , . Clearly a pseudoinverse exists if and only if is an equivalence.
A lax-inverse to a 1-arrow in a bicategory is the same except and 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 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.
Created on February 27, 2009 at 04:00:41. See the history of this page for a list of all contributions to it.