Dear all,
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?
For the case where it works (that is, where we really start with a *set* rather than a proper class of epis), a) is answered in the positive in my APCS paper "Free adjunction of morphisms", see http://www8.informatik.uni-erlangen.de/~schroeder/papers/freeadj.pdf The idea is basically to apply rewriting theory: Check that the rewrite relation that reduces, for each (non-isomorphic) epi e with newly added section s and each suitable morphism f, the term (fe)s to f (where (fe) and s are meant to be letters in a "category of words") is locally confluent. Since this implies a normal form result, I expect it would also help answer c). When we want to add sections for a proper class of epis in a large category, then we run into problems if we insist that the resulting category remain locally small: clearly, the "category of words" arising from adding a proper class of new sections will typically fail to be locally small, and I expect the above confluence result, which says that the result of freely adding sections has as many morphisms as there are irreducible words under the mentioned rewrite relation, can be applied easily to produce a counterexample (not just to faithfulness, but to existence of the free extension). One maybe interesting question is whether one gets "spurious" free extensions where the free extension by sections for a class of epis exists only because local smallness forces some morphisms to be identified for non-algebraic reasons. (A related example has I believe recently been found by Paul Levy and Nathan Bowler, who proved that there exist spurious tensors of monads in a similar sense.) Happy New Year, Lutz
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
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-- -------------------------------------- Prof. Dr. Lutz Schr?der Friedrich-Alexander-Universit?t Erlangen-N?rnberg Department of Computer Science Chair 8 -- Theoretical Computer Science Martensstr. 3 91058 Erlangen +49-9131-85-64059 lutz.schroeder@informatik.uni-erlangen.de lutz.schroeder@cs.fau.de http://www8.cs.fau.de/~schroeder/ -------------------------------------- [For admin and other information see: http://www.mta.ca/~cat-dist/ ]