Re: when does preservation of monos imply left exactness?

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

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commonly known as an unpublished theorem of A. Joyal and M. Tierney, was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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