Re: when does preservation of monos imply left exactness?

[Richard Garner <richard.garner@mq.edu.au>]

Dear Dmitry,

There is a result along these lines in Mike Barr's paper "On
categories with effective unions", Springer LNM 1348. His Theorem 4.1
says:

Suppose F: C --> D is a functor such that C has finite limits,
cokernel pairs and effective unions, and F preserves finite products,
regular monomorphisms and cokernel pairs. Then F preserves finite
limits.

In the statement of this result, a category is said to have effective
unions if the union of two regular subobjects always exists and is
calculated as the pushout over the intersection.

Best,

Richard


On 26 October 2011 22:59, Dmitry Roytenberg <starrgazerr@gmail.com> wrote:
> 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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