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van Kampen colimit

Van Kampen colimits

Context

Category theory

(,1)(\infty,1)-Category theory

Limits and colimits

Van Kampen colimits

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 CC be a category with pullbacks. Then there is a (pseudo) 2-functor

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

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

Definition

A colimit in CC is van Kampen if it is preserved by the functor SS, i.e. it is taken to a (weak) 2-limit in CatCat.

Universality and descent

Let G:DCG:D\to C be a diagram with colimit xx. Let DD' denote the category DD with a new terminal object adjoined, and G:DCG':D'\to C the extension of DD by the colimiting cocone with vertex xx.

Theorem

Suppose CC has all colimits of DD-shaped diagrams. Then the colimit xx of G:DCG:D\to C is van Kampen if and only if the following condition holds: for any diagram F:DCF':D'\to C and natural transformation α:FG\alpha':F'\to G' whose restriction α:FG\alpha:F\to G to DDD\subset D is equifibered, the following are equivalent:

  1. α\alpha' is equifibered.
  2. FF' is a colimiting cocone.
Proof

First note that the 2-limit of SGS\circ G is equivalent to the full subcategory of [D,C]/G[D,C]/G consisting of the equifibered transformations. We denote this category by ([D,C]G)([D,C] \Downarrow G). Moreover, under this equivalence, the comparison map S(x)lim(SG)S(x) \to \lim (S\circ G) is identified with the pullback functor

C/x([D,C]G). 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 GG into the pullback of its colimit. The latter underlies an equifibered α\alpha' by construction, so the unit is an isomorphism just when (1)\Rightarrow(2). Similarly, the counit is the map into an object over xx from the colimit of its pullback. Thus, it is an isomorphism just when (2)\Rightarrow(1).

The condition (1)\Rightarrow(2) is precisely the statement that the colimit of GG is universal, i.e. preserved by pullback. The condition (2)\Rightarrow(1) is a form of descent.

Colimits in Span

Theorem

A colimit in CC is van Kampen if and only if it is preserved by the inclusion CSpan(C)C\to Span(C) into the bicategory of spans in CC.

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 SetSet, the pushout square

    2 1 1 1\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:

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 (,1)(\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 CC are universal.

In this case, being van Kampen is also equivalent to being preserved by the composite of SS with the core functor Core:(,1)CatGpdCore : (\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 CoreSCore \circ S are representable, i.e. when object classifiers exist.

References

  • Pawel Sobocinski and Tobias Heindel, Being Van Kampen is a universal property, arXiv:1101.4594

Last revised on August 14, 2013 at 07:11:26. See the history of this page for a list of all contributions to it.