Re: when does preservation of monos imply left exactness?

["George Janelidze" <janelg@telkomsa.net>]

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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