Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorKevinCarlson
   • CommentTimeSep 28th 2018
   • PermaLink
   Author: KevinCarlson
   Format: TextI've just run into a nice application of the fact that abelian groups satisfy (AB4*), which says that arbitrary products are exact, i.e. (do the obvious things and) commute with finite colimits. (AB5*) would say that cofiltered limits are exact, which will more or less never hold in the presence of (AB5), so this is a nice intermediate exactness property. It's used in proving that the Yoneda embedding of an additive category into its free completion under finite colimits (equivalently, under cokernels) preserves coproducts, even infinite ones, which I think is pretty striking. I'm thinking about some nonabelian analogues of this process-in particular, in what sense are products exact in sets? Products won't commute with all finite colimits, since for instance coequalizers aren't even sifted. This indicates that the abelian result above is really about something like the free completion under quotients of congruences, rather than under finite colimits-these just happen to coincide in the additive world. And indeed, infinite products commute with colimits of equivalence relations in sets! However, infinite products don't commute with reflexive coequalizers. The transitivity is necessary to make this work. Anyway, this seems like a natural formulation of an (AB4*)-type axiom for Barr-exact categories, which could have some handy consequences. It'd be interesting to know whether people have thought about various related questions-for instance, which colimits commute with arbitrary products? I can't find anything on this in the literature: arbitrary small discrete categories don't form the right kind of doctrine for the work of Adamek-Borceux-Lack-Rosicky on generalized accessibility, for instance. Any idea whether any of this fits in with a wider story somewhere?

   I've just run into a nice application of the fact that abelian groups satisfy (AB4*), which says that arbitrary products are exact, i.e. (do the obvious things and) commute with finite colimits. (AB5*) would say that cofiltered limits are exact, which will more or less never hold in the presence of (AB5), so this is a nice intermediate exactness property. It's used in proving that the Yoneda embedding of an additive category into its free completion under finite colimits (equivalently, under cokernels) preserves coproducts, even infinite ones, which I think is pretty striking.
   
   I'm thinking about some nonabelian analogues of this process-in particular, in what sense are products exact in sets? Products won't commute with all finite colimits, since for instance coequalizers aren't even sifted. This indicates that the abelian result above is really about something like the free completion under quotients of congruences, rather than under finite colimits-these just happen to coincide in the additive world. And indeed, infinite products commute with colimits of equivalence relations in sets! However, infinite products don't commute with reflexive coequalizers. The transitivity is necessary to make this work.
   
   Anyway, this seems like a natural formulation of an (AB4*)-type axiom for Barr-exact categories, which could have some handy consequences. It'd be interesting to know whether people have thought about various related questions-for instance, which colimits commute with arbitrary products? I can't find anything on this in the literature: arbitrary small discrete categories don't form the right kind of doctrine for the work of Adamek-Borceux-Lack-Rosicky on generalized accessibility, for instance. Any idea whether any of this fits in with a wider story somewhere?
2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeSep 29th 2018
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexHave you checked out the literature on 'semi-abelian' categories etc., e.g. in the book by Borceux and Bourn? That theory looks at a wide range of non-abelian analogues of abelian conditions although I do not know it well enough (by far) to be able to answer your question directly.

   Have you checked out the literature on ’semi-abelian’ categories etc., e.g. in the book by Borceux and Bourn? That theory looks at a wide range of non-abelian analogues of abelian conditions although I do not know it well enough (by far) to be able to answer your question directly.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 29th 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThe general question of which limits commute with which colimits in $Set$ seems to be a complex one. There may be some unpublished results out there which would be relevant -- I recall a student of Johnstone had studied the question systematically for a PhD thesis, but the thesis was not completed. Maybe I can find something out. The observation about commutation with quotients of equivalence relations is a good one; I'm not sure this is recorded in the nLab, except in disguise.

   The general question of which limits commute with which colimits in $Set$ seems to be a complex one. There may be some unpublished results out there which would be relevant – I recall a student of Johnstone had studied the question systematically for a PhD thesis, but the thesis was not completed. Maybe I can find something out. The observation about commutation with quotients of equivalence relations is a good one; I’m not sure this is recorded in the nLab, except in disguise.