equivalences in/of $(\infty,1)$-categories
A colimit in a category or higher category is van Kampen if it is both universal (i.e. stable under pullback) and satisfies descent. Many exactness properties can be phrased in terms of certain colimits being van Kampen.
Let $C$ be a category with pullbacks. Then there is a (pseudo) 2-functor
defined by $S(x) \coloneqq C/x$ (the slice category), called the self-indexing of $C$. Its Grothendieck construction is the codomain fibration.
A colimit in $C$ is van Kampen if it is preserved by the functor $S$, i.e. it is taken to a (weak) 2-limit in $Cat$.
Let $G:D\to C$ be a diagram with colimit $x$. Let $D'$ denote the category $D$ with a new terminal object adjoined, and $G':D'\to C$ the extension of $D$ by the colimiting cocone with vertex $x$.
Suppose $C$ has all colimits of $D$-shaped diagrams. Then the colimit $x$ of $G:D\to C$ is van Kampen if and only if the following condition holds: for any diagram $F':D'\to C$ and natural transformation $\alpha':F'\to G'$ whose restriction $\alpha:F\to G$ to $D\subset D$ is equifibered, the following are equivalent:
First note that the 2-limit of $S\circ G$ is equivalent to the full subcategory of $[D,C]/G$ consisting of the equifibered transformations. We denote this category by $([D,C] \Downarrow G)$. Moreover, under this equivalence, the comparison map $S(x) \to \lim (S\circ G)$ is identified with the pullback functor
Now this functor has a left adjoint given by taking colimits. Thus, it is an equivalence if and only if the unit and counit of the adjunction are isomorphisms.
The unit is the map from an equifibered transformation over $G$ into the pullback of its colimit. The latter underlies an equifibered $\alpha'$ by construction, so the unit is an isomorphism just when (2)$\Rightarrow$(1). Similarly, the counit is the map into an object over $x$ from the colimit of its pullback. Thus, it is an isomorphism just when (1)$\Rightarrow$(2).
The condition (1)$\Rightarrow$(2) is precisely the statement that the colimit of $G$ is universal, i.e. preserved by pullback. The condition (2)$\Rightarrow$(1) is a form of descent.
A colimit in $C$ is van Kampen if and only if it is preserved by the inclusion $C\to Span(C)$ into the bicategory of spans in $C$.
See (SH).
A category with pullbacks is lextensive just when coproducts are van Kampen.
A category with pullbacks is adhesive just when pushouts of monomorphisms are van Kampen.
A category with pullbacks is exhaustive just when transfinite unions of monomorphisms are van Kampen.
In $Set$, the pushout square
is not van Kampen.
There is an evident generalization of the definition to higher categories, and in particular to (∞,1)-categories. In the latter case, van Kampen colimits exactly characterize descent. In particular,we have:
If we take Theorem as the definition of “van Kampen colimit”, this follows from Theorem 6.1.3.9 of HTT, see also around (Lurie 2Cats+Goodwillie, example 1.2.3). The $(\infty,1)$-categorical version of Theorem may not exactly be in the literature, however. The cited theorem of HTT essentially gives Theorem under the additional global hypothesis that all colimits in $C$ are universal.
In this case, being van Kampen is also equivalent to being preserved by the composite of $S$ with the core functor $Core : (\infty,1)Cat \to \infty Gpd$. Thus, the adjoint functor theorem implies that (assuming all colimits to be universal), all colimits are van Kampen just when all small versions of $Core \circ S$ are representable, i.e. when object classifiers exist.
Last revised on June 3, 2021 at 21:08:57. See the history of this page for a list of all contributions to it.