# Counterexample of pullback lemma.

I'm looking for a counterexample of the pullback lemma, i.e., a diagram

$$\require{AMScd} \begin{CD} A @>{}>> B @>>> C\\ @VVV @VVV @VVV\\ D @>>> E @>>> F \end{CD}$$

such that left square and outer square are pullbacks but right square is not a pullback. I have tried hard but I can't find a counterexample.

Any hint would be appreciated.

• Reference for "pullback lemma": proofwiki.org/wiki/Pullback_Lemma Apr 17, 2017 at 7:52
• You don't look for a counterexample of the pullback lemma (it is a proven lemma, it does not have counterexamples), but rahter of the variant that you have mentioned. Apr 17, 2017 at 7:53
• Hint: Look for a diagram of vector spaces where most of the objects are zero. Apr 17, 2017 at 8:11

Here's a counterexample that works in (almost) any category : let $p:X\to Y$ be a split epimorphism that is not an isomorphism, with section $s:Y\to X$. Then in the diagram $$\require{AMScd} \begin{CD} Y @>{id_Y}>> Y @>{id_Y}>> Y\\ @V{id_Y}VV @VV{s}V @VV{id_Y}V\\ Y @>>{s}> X @>>_p> Y, \end{CD}$$ the outer rectangle is a pullback since all its sides are identity maps, the left square is a pullback since $s$ is a monomorphism, but the right square cannot be a pullack if $s$ and $p$ are not isomorphisms.
Alternatively, you can take any monomorphism $m:A\to B$, and form the diagram $$\require{AMScd} \begin{CD} A @>{id_A}>> A @>{m}>> B\\ @V{id_A}VV @VV{m}V @VV{id_B}V\\ A @>>{m}> B @>>_{id_B}> B. \end{CD}$$ Here the left square is a pullback since $m$ is a mono, the rectangle is a pullback, but the right-hand square is not a pullback if $m$ is not an isomorphism.