Re: when does preservation of monos imply left exactness?
Dear Dmitry, Absolutely correct (although it does not change anything I said). Thank you for explaining "why". So your real question is about preservation of finite limits by functors of the form A+(-), in the case non-commutative algebras (of various kinds). Well, from this point of view the categories of algebras are 'difficult', and I don't recall any reasonable result at the moment. Moreover, I am surprised that Barr's theorem helps here (which does not mean that the theorem itself is not good of course!), and I would be very interested to learn, what exactly could you deduce from it? Best regards, George -------------------------------------------------- From: "Dmitry Roytenberg" <starrgazerr@gmail.com> Sent: Saturday, October 29, 2011 11:55 PM To: "George Janelidze" <janelg@telkomsa.net> Cc: "Steve Lack" <steve.lack@mq.edu.au>; <richard.garner@mq.edu.au>; <categories@mta.ca> Subject: Re: categories: Re: when does preservation of monos imply left exactness?
Dear George,
First, a small correction: A@- should be considered as a functor to A-Alg, not k-Alg, in order for what I said to be correct (I thank Steve Lack for pointing that out).
The square-zero extension is used to show that preservation of monomorphisms in k-Alg by A@- -- a priori a weaker condition than flatness -- actually implies preservation of monomorphisms in k-Mod, i.e flatness. After that it's the classical story you recalled.
As for why - fair enough: I'm interested to know whether this property of commutative algebras is shared by other types of algebras (e.g algebras over k-linear operads, or more general algebraic theories like analytic or C-infinity rings). The fact that the coproduct coincides with the tensor product of underlying modules is a very special property of commutative algebras, so the classical proof fails already for associative algebras. So I wonder what general exactness results are available. For instance, I find Michael Barr's theorem mentioned by Richard very useful.
Best,
Dmitry
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Dear George, Well, I find Barr's theorem useful insomuch as it highlights regular monos as the relevant ones and thereby brings the situation into sharper focus: proving the preservation of the equalizers of cokernel pairs should be easier than arbitrary ones. Of course, characterizing the regular monos and proving that they are preserved by cobase change (I've finally remembered what A@- is called!) could be a difficult matter, depending on the circumstances. So, I thank everyone for the feedback. I will post here if I manage to prove anything of interest. Best, Dmitry On Sun, Oct 30, 2011 at 1:10 AM, George Janelidze <janelg@telkomsa.net> wrote:
Dear Dmitry,
Absolutely correct (although it does not change anything I said).
Thank you for explaining "why". So your real question is about preservation of finite limits by functors of the form A+(-), in the case non-commutative algebras (of various kinds). Well, from this point of view the categories of algebras are 'difficult', and I don't recall any reasonable result at the moment. Moreover, I am surprised that Barr's theorem helps here (which does not mean that the theorem itself is not good of course!), and I would be very interested to learn, what exactly could you deduce from it?
Best regards,
George
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (2)
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Dmitry Roytenberg -
George Janelidze