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?


  • 4
    $\begingroup$ 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. $\endgroup$
    – t.b.
    Nov 12, 2011 at 13:08
  • 2
    $\begingroup$ For the first question see also en.wikipedia.org/wiki/Pulation_square (and references give there.) $\endgroup$ Nov 12, 2011 at 13:14
  • 2
    $\begingroup$ 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. $\endgroup$ Sep 13, 2014 at 16:54
  • $\begingroup$ @Stephan, How can I prove this. Can you take a look at this question here $\endgroup$
    – qartal
    Nov 6, 2015 at 20:49

2 Answers 2


$\require{amsCD}$ 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.

  • $\begingroup$ What about Top? $\endgroup$ Jan 7, 2015 at 23:14
  • 5
    $\begingroup$ @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. $\endgroup$ Jan 8, 2015 at 3:30


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .