surjective on objects functor

Surjective on objects functors


A functor F:CDF:C\to D is surjective on objects if for each object yDy\in D there is an object xCx\in C such that F(x)=yF(x)=y.

This is of course not an invariant notion; the corresponding invariant notion is essentially surjective functor. However, surjective-on-objects functors are sometimes useful when doing strict 2-category theory.


Surjective-on-objects functors are the left class in an orthogonal factorization system on Cat whose right class consists of the injective-on-objects and fully faithful functors.


  • Ross Street, Two-dimensional sheaf theory refers to surjective-on-objects functors as acute and their corresponding right class as chronic.

Created on April 13, 2016 at 13:00:57. See the history of this page for a list of all contributions to it.