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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