Contractible Exact Squares

摘要

Exact squares in Cat are not necessarily absolute (i.e., preserved by any 2-functor Cat -> Cat), or even preserved by any 2-functor given by exponentiation (-)(a") : Cat -> Cat: if a square is preserved by exponentiation it will be called a contractible exact square. We will characterize diagrammatically these contractible squares, and among them the contractible categories, and the so called fibering and cofibering squares, with especially the comma squares and the adjunction squares. As an application we conclude with a diagrammatical characterization of absolutely absolute Kan extensions and especially of absolutely final functors and of absolutely absolute colimits.