# When is a pullback also a pushout?

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!

• 1) Sure, a square that is both a pull-back (cartesian) and a push-out (cocartesian) is called a bicartesian square (Freyd called them push-me pull-you first, then later Doolittle diagrams...). 2) I don't think that there is a general characterization without further assumptions on the morphisms and the surrounding category. For example, in additive categories you can show that the push-out under a kernel is also a pull-back if and only if the push-out of the kernel is a monic.
– t.b.
Nov 12, 2011 at 13:08
• For the first question see also en.wikipedia.org/wiki/Pulation_square (and references give there.) Nov 12, 2011 at 13:14
• I know this question is old and you are asking about the direction "pullback$\implies$pushout", but I thought to myself I just comment here what I found out recently: $\require{AMScd}$ If the square $$\small\begin{CD} A @>i>>X \\ @VVV@VVV \\ Y@>j>>Z \end{CD}$$ is a pushout and $i$ is injective, then $j$ is injective and the square is a pullback. This also holds in the category of weak Hausdorff $k$-spaces: If $i$ is a closed embedding, then so is $j$, the pushout is constructed as in the category of sets, and the square is also a pullback. Sep 13, 2014 at 16:54
• @Stephan, How can I prove this. Can you take a look at this question here Nov 6, 2015 at 20:49

$\require{amsCD}$ 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.

• What about Top? Jan 7, 2015 at 23:14
• @Ivan: things already don't work out very well in $\text{Set}$. Looking at the cardinalities involved in a pushout vs. pullback diagram of finite sets you'll see that there are hardly any examples. Jan 8, 2015 at 3:30

$\require{amscd}$

There are some special cases of this phenomenon where things can be said. Consider the following diagram:

$$\begin{CD} A @>f>> B\\ @VVgV @VVhV\\ C @>k>> D \end{CD}$$

1. 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 $\mathsf{Ab}$.

2. 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 $B = 2$, $D=1$.

3. 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 $\mathsf{Set}$.

4. 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.