bo functor

Bijective-on-objects functors

A functor is called bijective on objects, or bo, if it is, well, bijective on objects. One reason bo functors are important is because together with full and faithful (ff) functors they form an orthogonal factorization system on Cat; see bo-ff factorization system. This factorization system can also be constructed using a generalized kernel.

To be more in accord with the principle of equivalence, one could require that the functor be bijective on objects only up to isomorphism; that is, it is essentially surjective and full on isomorphisms. However, from the point of view of factorization systems, the version of the concept of a bo functor which is in accord with the principle of equivalence is nothing more or less than an essentially surjective functor, since essentially surjective functors and ff functors form a bicategorical factorization system on the bicategory CatCat.

R. Street in Categorical and combinatorial aspects of descent theory proves

Proposition. A functor is bijective on objects if and only if it exhibits its codomain as the (2-categorical) codescent object of some simplicial category.

This can be generalized to any regular 2-category.

Revised on April 22, 2017 06:01:07 by Urs Schreiber (