Re: when does preservation of monos imply left exactness?

[Dmitry Roytenberg <starrgazerr@gmail.com>]

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

On Fri, Oct 28, 2011 at 12:08 AM, Steve Lack <steve.lack@mq.edu.au> wrote:
> Dear All,
>
> While I agree that category theory is not just generalized abelian
> category theory, there are nonetheless some theorems along the
> lines that Dmitry suggests. Richard mentioned one; here is another.
> Of course neither is nearly as simple as the abelian case.
>
> If the opposite of the categories A and B are exact Mal'cev, then any
> functor f:A->B which preserves finite colimits and regular monos preserves equalizers
> of coreflexive pairs. Thus any functor f:A->B which preserves finite colimits, finite products
> and regular monos preserves all finite limits.
>
> In particular, A and B could be toposes, then if f:A->B preserves finite colimits, finite
> products, and monomorphisms, it preserves all finite limits.
>
> From the point of view of George's example, the problem is that in the category 2, the
> map 0->1 is mono but not regular mono.
>
> Best wishes,
>
> Steve Lack.

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