Re: Splitting epis by wishful thinking

[Richard Garner <richard.garner@mq.edu.au>]

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-the-theory-of-a-well-pointed-topos-with-nno
> )
> 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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