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. On 27/10/2011, at 9:32 PM, George Janelidze wrote:

All right, somebody should answer...

Dear Dmitry,

First, may I suggest that looking at category theory as merely a 'generalized abelian category theory' is not really a good idea?

Specifically, in the abelian context

'right exact + preserves monos => left exact'

is simple but extremely important, while for general categories it is indeed wrong, and adding conditions to make this work seems to be a strange thing to do!

Now, more specifically, concerning Barr exact:

Let 2 be the ordered set {0,1} considered as a category, let S be the category of sets, and let F : S --> 2 be the left adjoint of the canonical embedding 2 --> S (hence, for a set X, F(X) = 0 if and only if X is empty). Then:

(a) 2 and S are Barr exact.

(b) F preserves all colimits since it is a left adjoint.

(c) F preserves products (moreover, the fact that it preserves all products is one of the standard forms of the Axiom of Choice).

(d) F preserves monomorphisms since every morphism in 2 is a monomorphism.

(e) F does not preserve all equalizers since two parallel morphisms between non-empty sets might have the empty equalizer.

Best regards

George Janelidze

-------------------------------------------------- From: "Dmitry Roytenberg" <starrgazerr@gmail.com> Sent: Wednesday, October 26, 2011 1:59 PM To: "Categories list" <categories@mta.ca> Subject: categories: Re: when does preservation of monos imply left exactness?

...

I'll try again...

On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg <starrgazerr@gmail.com> wrote:

...

Dear category theorists,

It is well known that any functor that preserves finite limits preserves monomorphisms, and that for an additive right-exact functor between abelian categories, the converse is also true. Is it known how far this extends to the non-additive setting? In other words, what exactness properties of two categories and a functor between them would suffice to conclude that the functor preserves finite limits if and only if it preserves monos? For instance, is it enough to assume that the categories be Barr-exact and that the functor preserve all colimits, finite products and monos to conclude that it also preserves equalizers?

Any references would be extremely helpful.

Thanks, Dmitry

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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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