On 03/01/13 09:36, Andrej Bauer 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
I found pertinent (or related) to this problem to recall the following beautiful construction construction due to Joyal. A key step in Joyal approach to (Godel type) completeness theorem consists in freely adding constants (Henkin's proof). Given a regular category CC, for any epi B --->> 1, we freely add a constant 1 --->> B (in the 2-category of regular categories and regular functors). This is easily done by talking the slice category CC/B. We have CC ---> CC/B. Product of epis is epi so this yields a filtered system of regular categories. Let CC' be its colimit. We have CC ---> CC'. Doing this denumerable times, we get CC ---> sCC, where sCC is regular, every epi B ---> 1 in sCC splits and sCC has the corresponding universal property. e.d. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]