The subject line says it all, but perhaps it would be more reasonable to split the question into two parts: 1) can a pullback diagram also be a pushout diagram?; if so, 2) can necessary and sufficient conditions be given for this to happen?
Thanks!
The subject line says it all, but perhaps it would be more reasonable to split the question into two parts: 1) can a pullback diagram also be a pushout diagram?; if so, 2) can necessary and sufficient conditions be given for this to happen?
Thanks!
$\require{amsCD}$ Yes. The most familiar setting where this happens is in abelian categories, where the commutative square
$\begin{CD} A @>f_b>> B\\ @VVf_cV @VVg_bV\\ C @>g_c>> D \end{CD}$
is a pullback square iff the corresponding sequence
$0 \to A \xrightarrow{f_b \oplus f_c} B \oplus C \xrightarrow{g_c - g_b} D$
is exact. Similarly, the square is a pushout square iff the sequence
$A \xrightarrow{f_b \oplus f_c} B \oplus C \xrightarrow{g_c - g_b} D \to 0$
is exact. Hence the square is both a pullback and a pushout square iff the sequence
$0 \to A \xrightarrow{f_b \oplus f_c} B \oplus C \xrightarrow{g_c - g_b} D \to 0$
is exact.
$\require{amscd}$
There are some special cases of this phenomenon where things can be said. Consider the following diagram:
$\begin{CD} A @>f>> B\\ @VVgV @VVhV\\ C @>k>> D \end{CD}$
Suppose that $f$ and $k$ are monomorphisms. Then in many categories, the implication (pushout $\Rightarrow$ pullback) holds. This holds in any topos, in any abelian category, and more generally in any adhesive category. According to Stefan Hamcke's comment, this holds in $k$-spaces. But we typically don't get the converse implication (pullback $\Rightarrow$ pushout). For example, in $\mathsf{Set}$, let $B,C$ be any subsets of $D$ such that $B \cup C \subsetneq D$, and let $A = B \cap C$. Similar counterexamples work in $\mathsf{Ab}$.
Suppose that $f$ and $k$ are monomorphisms and $g$ and $h$ are epimorphisms. The facts Qiaochu discusses can be used to show that in an abelian category, the equivalence (pushout $\Leftrightarrow$ pullback) holds. In $\mathsf{Set}$, though, the leftward implication fails -- e.g. let $A = C = \emptyset$ and $B = 2$, $D=1$.
Suppose that $g$ and $h$ are epimorphisms. Then in any abelian category, the dual of (1) tells us that the implication (pullback $\Rightarrow$ pushout) holds. But this is not the case in $\mathsf{Set}$.
As an special case of (3), suppose that $f,g,h,k$ are all epimorphisms. If I'm not mistaken, we get the implication (pullback $\Rightarrow$ pushout) in $\mathsf{Set}$ and so also in any topos. But the converse implication rarely holds.
Abelian categories and topoi are about as nice as it gets, so I would expect that in less nice categories, the best one can do is similar implications with more restrictive assumptions on the morphisms involved.