Re: Splitting epis by wishful thinking

["Fred E.J. Linton" <fejlinton@usa.net>]

On Thu, 03 Jan 2013 08:55:07 PM EST Andrej Bauer <andrej.bauer@andrej.com>
asked

> Is there a construction which "freely" splits all epis in a category ... ?

To the responses already received I thought it perhaps worth adding the idea
of 
freely (or generically) splitting everything, after a fashion I first heard
described by
Bill Lawvere -- that is, freely adjoining, for each map e (epi or not), a  map
f with
  efe = e (and perhaps, if you like, fef = f) .

Note that, with e epi, efe = e will entail ef = id, i.e., f will be a  section
for e.
Likewise, for e mono, efe = e will entail fe = id, i.e., f will be a
retraction for e.
(In these two cases, of course, it will follow that fef = f. In general, tho',
... .)

One should, of course, ask oneself whether one should really be wanting
sections for 
quite all epimorphisms -- in the category [R] of unital rings R, for example,
should one 
really ever want the trivializing homomorphisms !: R --> 1 to the terminal
ring to split,
anywhere?

Cheers, -- Fred 





[For admin and other information see: http://www.mta.ca/~cat-dist/ ]