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Circumstances: I'm studying Grothendieck's Galois Theory and recently encountered a proposition that discussed the stability of coproducts under pullback. And I found the page pullback-stable colimit in nLab.

On the mentioned nLab page, pullback-stability wasn't defined for individual colimits but for colimits with the same shape. However, I think it can be also defined for individual colimits like following.

For a functor $G: D \to C$, we say that $\mathrm{colim}_D\ G$ is stable under the pullback if for all pullback diagrams of $(\mathrm{colim}_{D}\ G) \times_Z Y,$ the canonical morphism

$$\mathrm{colim}_{d∈D}\ (G(d) \times_Z Y) \to (\mathrm{colim}_{d∈D}\ G(d))\times_Z Y$$

is an isomorphism.

So why isn't pullback-stability defined for individual colimits? Or are there any papers available that define pullback-stability for individual colimits?

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  • $\begingroup$ Why don't you show us how you think it should be done, then? $\endgroup$
    – Zhen Lin
    Commented Jul 17, 2023 at 13:51
  • $\begingroup$ Thank you for your comment. I thought it was obvious, but I should have written it. I'll write it. $\endgroup$
    – Linuxmetel
    Commented Jul 17, 2023 at 13:59
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    $\begingroup$ Well, $\mathrm{colim}_{d∈D}\ (G(d) \times_Z Y) \to (\mathrm{colim}_{d∈D}\ G(d))\times_Z Y$ is a different colimit than $\mathrm{colim}_D\ G$. So what you have defined is not a property of $\mathrm{colim}_D\ G$ because it depends on something else, namely, the functor $G(-) \times_Z Y$. $\endgroup$
    – Najib Idrissi
    Commented Jul 17, 2023 at 15:38
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    $\begingroup$ @NajibIdrissi: That’s a rather fallacious objection: many interesting properties of an object or diagram involve its relationships with other related objects and diagrams. As noted in my answer, OP’s definition is a very reasonable and fairly standard definition of pullback-stability of an individual colimit. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Jul 17, 2023 at 15:50

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Pullback-stability is sometimes considered for individual colimits, or at least, smaller classes than “all colimits of shape $D$”. However, it’s most often used in settings where it holds for large classes of colimits, so authors usually define it just at the level of generality they need. In particular, the nlab article you link seems to be based in large part on Lurie’s treatment in Higher Topos Theory, §6.1.1.1; there, Lurie’s motivation is presenting the Giraud conditions for $\infty$-toposes, so he defines pullback-stability of colimits in the form and generality he wants for that.

One place where it’s used for smaller classes of colimits is in the study of adhesive categories, as studied in e.g. Garner, Lack, On the axioms for adhesive and quasi-adhesive categories, 2011, particularly in the condition “pushouts along monomorphisms exist, and are stable under pullback”. But, as discussed in that paper, adhesive categories satisfy rather more: such pushouts are van Kampen. And this I think is the other reason why pullback-stability isn’t often considered for individual colimits: in the most-studied settings that have only some colimits pullback-stable, one wants not just stability, but the stronger condition of van Kampen-ness. And the nlab page for van Kampen colimits does define it for individual colimits, and notes that one half of that definition can be seen as a definition of pullback-stability:

The condition (1) ⇒ (2) is precisely the statement that the colimit of $G$ is universal, i.e. preserved by pullback.

Many sources, e.g. the above paper of Garner and Lack, talk about pullback-stability of colimits without explicitly defining it. This I think is because it’s essentially always considered in settings where enough pullbacks exist that they can be viewed as pullback functors between slices $f^* : \mathcal{C}/X \to \mathcal{C}/Y$; so preservation of colimits under pullback is viewed just as a special case of preservation by a functor (this is explicit in e.g. Lurie’s Def. 6.1.1.2) — which is defined for individual colimits in many standard references. And this again comes out equivalent to the “universality” implication in the definition of van Kampen, and to your proposed definition in the question.

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  • $\begingroup$ I’m surprised that such a high quality answer was available to such a seemingly minor question. Very nice! $\endgroup$
    – Kevin Carlson
    Commented Jul 18, 2023 at 3:34

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