Re: when does preservation of monos imply left exactness?

["George Janelidze" <janelg@telkomsa.net>]

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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