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/ ]