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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