## Pull-backs which are push-outs

Consider a commutative square in a category $\mathcal{C}$ $$\begin{array}{ccc} A&\rightarrow&B\\ \downarrow&&\downarrow\\ C&\rightarrow&D \end{array}$$ Suppose $\mathcal{C}$ is abelian. If this square is a pull-back and $B\rightarrow D$ or $C\rightarrow D$ is an epimorphism, then this square is also a push-out square. Dually, if this square is a push-out and $A\rightarrow B$ or $A\rightarrow C$ is a monomorphism, then this square is also a pull-back square. ¿Are there more general kinds of categories were such things happen?

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 Try to look for effective epimorphisms. – Martin Brandenburg Mar 9 2011 at 12:16

Yes! Pretoposes (and in particular toposes) also have this property. It is a remarkable fact that pretoposes (which you can think of as having the first-order exactness properties of toposes or $Set$-like categories) have "most" of the same exactness properties as abelian categories (see below).

In fact, this is the beginning of a remarkable set of observations due to Peter Freyd, and expounded by him in a discussion at the categories mailing list, which led to a sharp distinction between pretoposes and abelian categories as concentrated particularly in the behavior of the initial object. (In an abelian category, $A \times 0 \cong A$, whereas in a pretopos $A \times 0 \cong 0$. But this is practically the only essential difference.) In fact, Freyd showed that abelian categories and pretoposes are special cases of what he dubbed "AT categories", which contain the core exactness properties which are common to abelian categories and pretoposes. AT categories cut so close to the essence of each of these two special cases that in fact every AT category splits cleanly as a product of an abelian category and a pretopos!

I wrote up my own account of this in the nLab, here.

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The question could almost have been written as a deliberate feedline to give this wonderful answer a home on MO! – Peter LeFanu Lumsdaine Mar 9 2011 at 14:04
Heh, thanks Peter! I was delighted to have this question fall into my lap, as it were. The idea of AT category is very cool, but I'm not aware that much has been done with it since those discussions between Freyd and Pratt. – Todd Trimble Mar 9 2011 at 15:09
One might argue that the idea of AT category is so neat that it destroys itself as an object to be studied: since any AT category splits into an abelian category and a pretopos, why study AT categories rather than just abelian categories and pretopoi separately? – Mike Shulman Mar 9 2011 at 15:58
I've often wondered about that too, Mike. I don't have an answer to that. – Todd Trimble Mar 9 2011 at 16:16
Hi Todd, thanks for your answer. I've had a look at your links and I've found that pretoposes have the second property I mentioned. Do they also have the first one? – Fernando Muro Mar 10 2011 at 15:02
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