nLab van Kampen colimit

Idea

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.

Definition

Let $C$ be a category with pullbacks. Then there is a (pseudo) 2-functor

$$S : C^{op} \to Cat$$

defined by $S(x) \coloneqq C/x$ (the slice category), called the self-indexing of $C$. Its Grothendieck construction is the codomain fibration.

Definition

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 $Cat$.

Universality and descent

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 $G$ by the colimiting cocone with vertex $x$.

Theorem

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 $\alpha':F'\Rightarrow G'$ whose restriction $\alpha:F\Rightarrow G$ to $D\subset D'$ is equifibered, the following are equivalent:

1. $\alpha'$ is equifibered.
2. $F'$ is a colimiting cocone.

Proof

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

$$C/x \to ([D,C] \Downarrow G).$$

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.

Colimits in Span

Theorem

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 $C$.

Proof

See (SH).

Examples

• 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

  $$\array{ 2 & \to & 1 \\ \downarrow && \downarrow \\ 1 & \to & 1 }$$

  is not van Kampen.

In higher categories

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:

• A locally presentable (∞,1)-category is an (∞,1)-topos just when all colimits are van Kampen.

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.

Related pages

• exactness property

• object classifier in an (∞,1)-topos

• descent / universal colimit

• strict initial object

References

• Pawel Sobocinski, Tobias Heindel, Being Van Kampen is a universal property, Logical Methods in Computer Science, 7 1 (2011) [arXiv:1101.4594, doi:10.2168/LMCS-7(1:14)2011]