Dear Andrej, The following formulation of your question has negative answers: Let Cat be the category of small categories and functors between them, and SCat be its full sub-category of small categories in which all epis split. Let U : SCat -> Cat be the full inclusion. U does not have a left adjoint, nor a right adjoint. Proof: Let C be the following category: It has 3 objects: 0 - an initial object A and B. Apart from the identities and the initial maps, C has the following 3 morphisms: a parallel pair f, g : A -> B an endomorphism h : B -> B The non-trivial compositions are given by: x o h = x, for x = f, g, and h. Note that C is in SCat, as it has no trivial epimorphisms: the non-trivial arrow into A equalises f and g, which are different, and the non-trivial arrows into B equalise h and id. Let F : C -> C be the endofunctor that swaps f with g. If U had a left adjoint, then SCat was a full reflective subcategory of Cat, it was complete. Consider the equaliser of F, Id : C->C. Whatever it is in SCat, this coequaliser cannot be preserved by U, as the equaliser in Cat is given by dropping f and g. This resulting subcategory has an epi, the initial arrow into A, that doesn't split. Similarly, if U had a right adjoint, then SCat was cocomplete. Then the coequaliser of F, Id : C->C in Cat would be C with g dropped (=identified with f). But then the initial map into A becomes epi, without a section. Thus this colimit is not in SCat, and U doesn't preserve it. The same construction actually lies within the category Cat_ac that Richard described (with specified sections), as C has no non-trivial sections. Thus the same proof applies to his formulation, and his forgetful functor also isn't a right nor a left adjoint. Ohad. On 4 January 2013 04:04, Richard Garner <richard.garner@mq.edu.au> wrote:
Hi Andrej,
This isn't exactly what you want, but it's along the right lines. Given a small category with cokernel pairs, one can construct another category with cokernel pairs in which all epis have been "freely" split. By this, I mean that a chosen section has been freely added to every epi in the category, even the ones that already had a section; thus the construction is not idempotent. Basically one uses the small object argument.
Consider the category K of small categories with cokernel pairs, and functors preserving such. Let C be the free category with cokernel pairs containing an epi e: it can be obtained by first forming the free category with cokernel pairs on an arrow f, and then coinverting the codiagonal of f. Let D be the free category with cokernel pairs containing a section-retraction pair (i,p). There is an obvious map C --> D in K which sends e to p. Now some E in K satisfies the axiom of choice if and only if it is projective (has the weak right lifting property) with respect to this map C --> D.
K is locally finitely presentable, and so the map C --> D generates via the small object argument a weak factorisation system (L,R) on it. By the above argument, the fibrant objects for (L,R) are those small categories with cokernel pairs satisfying the axiom of choice. If one uses the algebraic version of the small object argument, the fibrant replacement for this w.f.s. is a monad, S, say. The action of this monad on objects freely adjoins sections for all epis; its algebras are precisely the small categories with cokernel pairs with a chosen section for each epi.
One can ask what happens if one drops the assumption of cokernel pairs. Consider the category Cat_ac, whose objects are small categories in which every epi comes equipped with a chosen section. There is an obvious forgetful functor Cat_ac ---> Cat, and a more precise formulation of your original question would be to ask if this functor has a left adjoint. This is unclear to me; at the moment I feel like it probably doesn't. What does seem clear is that, if it does have a left adjoint, then it can't possibly be monadic, so whatever construction one gives won't be entirely honest or straightforward.
Richard
On 3 January 2013 23:36, 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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