Re: when does preservation of monos imply left exactness?

[Steve Lack <steve.lack@mq.edu.au>]

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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