Dear Dmitry, Let me write then F(X) = A@X, denoting the tensor product over k by @. Hence, briefly, @ is + (coproduct) in k-Alg. The fact that, when A is flat as a k-module, A+(-) : k-Alg --> k-Alg preserves finite limits is absolutely classical, as well as its proof via modules. However, you do not need the fully faithful functor from k-Mod to k-Alg given by the square-zero extension for that: simply use the fact the forgetful functor k-Alg --> k-Mod not only preserves limits, but also reflects isomorphisms (Some people prefer to say/use "creates isomorphisms"). The preservation of finite limits by A+(-) : k-Alg --> k-Alg (already for modules) implies, for example, Grothendieck's Descent Theorem saying that if A is flat, then k --> A is an effective descent morphism. However, much stronger result is known now: k --> A is an effective descent morphism if and only if it is pure as a monomorphism of k-modules. This stronger result, commonly known as an unpublished theorem of A. Joyal and M. Tierney, was claimed several times by various authors, and I can tell you more if you are interested. Anyway, since the category k-Alg has very bad exactness properties in a sense, modules were used by category-theorists themselves. And, in spite of nice results mentioned by Richard Garner and Steve Lack, I still suggest not to go through a generalization of the abelian case. Well, such a suggestion surely cannot be "universal", and if you need a better suggestion, you should tell us why are you doing this - if I may say so. Best regards, George Janelidze -------------------------------------------------- From: "Dmitry Roytenberg" <starrgazerr@gmail.com> Sent: Friday, October 28, 2011 2:27 PM To: "Steve Lack" <steve.lack@mq.edu.au> Cc: "George Janelidze" <janelg@telkomsa.net>; <richard.garner@mq.edu.au>; "Categories list" <categories@mta.ca> Subject: categories: Re: when does preservation of monos imply left exactness?
Dear colleagues,
First, let me thank everyone who has replied so far: I am learning a lot from this discussion (not being a category theorist myself). Far be it from me to make sweeping statements about general category theory, I am merely interested in particular functors between particular categories. It just seems that preservation of monos would be easier to check than preservation of equalizers, which is why I am looking for general criteria.
Consider the following example. Let k-Alg denote the category of commutative k-algebras over some ground ring k, and let
F:k-Alg-->k-Alg
denote taking coproduct with an object A. Then F preserves all colimits and finite products; furthermore, if F preserves monos, it preserves all finite limits (which is the case iff A is flat as a k-module). The only proof I know takes advantage of the convenient fact that the coproduct in k-Alg extends to the tensor product in the abelian category k-Mod, as well as the existence of a fully faithful functor from k-Mod to k-Alg given by the square-zero extension. If anyone knows a proof which does not involve going to k-Mod, just using some general properties of k-Alg, I'd be very happy to learn about it. Notice that k-Alg is not co-Mal'cev, nor is every monomorphism regular.
Thanks again,
Dmitry
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