Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: reading & references

Topic: Colimits in Set are always "van Kampen"

Ambroise (Nov 14 2025 at 11:04):

The nlab page on van Kampen colimits mentions some counter-example of van Kampen colimits in the category of sets. But it seems that there is a refinement of the equifibrancy characterization of van Kampen colimits (mentioned in the nlab page) in sets that holds.

Given two families $(A_1 ∈𝐒𝐞𝐭, A_2 : A_1 → 𝐒𝐞𝐭)$ and $(B_1 ∈ 𝐒𝐞𝐭, B_2 : B_1 → 𝐒𝐞𝐭)$, a morphism $(A_1 \xrightarrow{f_1} B_1, f_2 : ∏_{a ∈ A_1} A_2(a) → B_2(f_1(a))$ is cartesian if $f_2(a,-):A_2(a) → B_2(f_1(a))$ is always an identity for all $a ∈ A_1$. Equivalently, this morphism is in the cleaveage of the split fibration Fam → 𝐒𝐞𝐭.

Now, consider a diagram in the category of families and cartesian morphisms between them. Then I think this diagram has a colimit, which is preserved by the forgetful functor to the category of families. With the equivalence between $Fam$ and $Set^→$ sending cartesian morphisms to pullbacks in mind, this looks similar to the equifibrancy characterisation of Van Kampen colimits. Of course, I am not sure that this connection is the most relevant one. Anyway, if you know of any reference and/or generalisations of this result, I am happy to hear about it!