Preliminary thoughts: 1. Certainly for any category C and any set of morphisms in it you can freely freely adjoin splittings for all those morphisms. In particular you might start with the set of all epis in C. (They're going to become epi anyway.) This construction is just universal algebra, but the universal algebra dooen't answer questions like (a) Is the functor from C faithful? (b) Does the new category inherit nice properties of C? (c) Do all epis split in the new category? 2. Suppose you are willing to restrict to categories with finite colimits. Then epis can be characterized equationally, so there is a cartesian theory of finite colimit categories in which every epi has a chosen splitting. Then the forgetful functor from such categories to finite colimit categories has a left adjoint. You can extend this to deal with categories with additional structure on top of the finite colimits, so long as the additional structure can be expressed as cartesian theory. Happy New Year, Steve. On 3 Jan 2013, at 12:36, Andrej Bauer <andrej.bauer@andrej.com> wrote:
On Mathoverflow there is a discussion (see http://mathoverflow.net/questions/117921/relative-consistency-of-etcs-over-t...) which got me thinking.
Is there a construction which "freely" splits all epis in a category C? Something like: we add sections to every epi and wish we are done?
The question is somewhat lose, but I think it is clear nontheless.
With kind regards,
Andrej
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]