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In Lack and Sobociński - Adhesive categories, Counterexample 7, the authors claim without proof that Cat is not an adhesive category. Seeing as Cat has all small (co)limits, the property of being an adhesive category that Cat doesn't satisfy must be that pushouts along monomorphisms are van Kampen squares. Note that the nlab article on extensive categories notes that Cat is extensive, so pushouts along coproduct injections are vK-squares.

My question is: Why are pushouts along monomorphisms not vK-squares for Cat? More explicitly, could somebody construct an example of small categories and functors such that one has a pushout along monomorphisms that is not a vK-square?

By Lemma 29 in the same paper, we should equivalently be able to consider Cat's satisfaction of the axioms of a High-Level Replacement Category equivalent to its satisfaction of the axioms of an adhesive category as Cat has an initial object. Cat certainly satisfies axioms 1, 2, and 4 by what's already been said. My suspicion is that it satisfies axioms 3 and 5 as well based on working out a basic example with diagram categories. So my guess, since Cat is not adhesive and thus also not a HLR category, is that Cat doesn't satisfy axiom 6. So alternatively, could somebody construct an example of small categories and functors such that one has a rectangle that is a pushout and a right square that is a pullback but a left square that is not a pushout?

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    $\begingroup$ Seeing as the authors note that the category of posets is also not adhesive and seeing as posets are categories, I would imagine that would be a fruitful source of counterexamples. $\endgroup$
    – Functorial Nonsense
    May 27, 2023 at 20:04
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    $\begingroup$ The much longer journal version is: Lack, S., & Sobociński, P. (2005). Adhesive and quasiadhesive categories. RAIRO - Theoretical Informatics and Applications, 39(3), 511–545. doi:10.1051/ita:2005028. It includes Peter Haine's counterexample as Example 3.4. $\endgroup$
    – Paul Taylor
    May 28, 2023 at 19:11

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Consider the following commutative cube of posets and inclusions: enter image description here

The bottom face is a pushout in both $\mathbf{Cat}$ and $\mathbf{Poset}$. The vertical faces are all pullbacks in both $\mathbf{Cat}$ and $\mathbf{Poset}$. However, the top face is not a pushout in $\mathbf{Cat}$ or $\mathbf{Poset}$. Hence in both of these categories, pushouts along monomorphisms are not van Kampen.

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    $\begingroup$ @FunctorialNonsense then you should mark the answer as accepted. $\endgroup$
    – Fernando Muro
    May 27, 2023 at 22:26
  • $\begingroup$ @FunctorialNonsense: Even though Peter's example appears in the journal paper, and whether he knew this or not, he still deserves credit for explaining the answer here. $\endgroup$
    – Paul Taylor
    May 29, 2023 at 8:08
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    $\begingroup$ I certainly wasn't aware that this example appears in the journal version of their paper (I had never heard of the paper until I saw the question, which links a different version). In any case, I think this is the "standard" example showing that pushouts in $\mathbf{Cat}$ are not universal. $\endgroup$
    – Peter Haine
    May 29, 2023 at 17:49

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