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Stream: theory: category theory

Topic: Van Kampen colimit wrt a fibration

Patrick Nicodemus (Jun 10 2023 at 01:21):

I was reading about van Kampen colimits today and the idea seems pretty plausible and generally useful in descent theory. It seems like the definition is easy to adapt to the context of an arbitrary fibration, not necessarily the codomain fibration.

If $p : E\to B$ is a Grothendieck fibration preserving colimits, and $X : A\to B$ is a diagram with colimit $b$, then call the colimit diagram van Kampen with respect to $p$ if for any lift $F: A\to E$ of $X$ along $p$, $F$ sending all maps to Cartesian morphisms, the following are equivalent for a cocone lying over the colimit diagram:

• the cocone morphisms are Cartesian
• the cocone is a colimit Cocone

Or we could say that $p^{-1}(b)$ is the weak 2-limit of the pseudofunctor $p^{-1}\circ X$.

So has this been studied in relationship to usual notions of descent? Where can I read more about this?

I also noticed that this could be adapted to weighted colimits. If $P$ is a presheaf over $X$ then we would consider "descent data" as morphisms of fibrations from $\pi : \operatorname{el}(P)\to p$ that extend $A$.