nLab van Kampen colimit

Van Kampen colimits


Category theory

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

Limits and colimits

Van Kampen colimits


A colimit in a category or higher category is called van Kampen [Sobocinski & Heindel 2011] 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\mathcal{C} be a category with pullbacks. Then there is a (pseudo) 2-functor

(1)S:𝒞 opCat S \,\colon\, \mathcal{C}^{op} \to Cat

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


A colimit in 𝒞\mathcal{C} is van Kampen if it is preserved by the functor SS (1), i.e. it is taken to a (weak) 2-limit in CatCat.

Universality and descent

We need the following notation:

  • For 𝒟\mathcal{D} any category, write 𝒟 \mathcal{D}^{\triangledown} for the result of adjoining to it a terminal object. This comes with a canonical full subcategory inclusion 𝒟𝒟 \mathcal{D} \hookrightarrow \mathcal{D}^{\triangledown}

  • For 𝒟\mathcal{D} a small category, and G:𝒟𝒞G \,\colon\, \mathcal{D} \to \mathcal{C} a 𝒟\mathcal{D}-shaped diagram with colimit limG𝒞\underset{\longrightarrow}{\lim} G \,\in\, \mathcal{C}, write G :𝒟 𝒞G^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \longrightarrow \mathcal{C} the extension of GG to 𝒟 \mathcal{D}^{\triangledown} by assigning limG\underset{\longrightarrow}{\lim} G to the adjoined terminal object.


(equifibrancy characterization of van Kampen colimits)
If 𝒞\mathcal{C} has all colimits of 𝒟\mathcal{D}-shaped diagrams, then the colimit limG\underset{\longrightarrow}{\lim}G of G:𝒟𝒞G \colon \mathcal{D}\to \mathcal{C} is van Kampen (Def. ) if and only if the following condition holds:

  • For any diagram F :𝒟 𝒞F^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \to \mathcal{C} and natural transformation α :F G \alpha^{\triangledown} \,\colon\, F^{\triangledown} \Rightarrow G^{\triangledown} whose restriction α:FG\alpha \colon F \Rightarrow G along 𝒟𝒟 \mathcal{D} \hookrightarrow \mathcal{D}^{\triangledown} is equifibered, the following are equivalent:

    1. α \alpha^{\triangledown} is equifibered.

    2. F F^{\triangledown} is a colimiting cocone.


First note that the 2-limit of SGS\circ G is equivalent to the full subcategory of the slice category of the functor category

([𝒟,𝒞]G)[𝒟,𝒞]/G \big([\mathcal{D},\mathcal{C}] \Downarrow G\big) \hookrightarrow [\mathcal{D},\mathcal{C}]/G

consisting of the equifibered transformations. 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). \mathcal{C}/x \longrightarrow ([\mathcal{D},\mathcal{C}] \Downarrow G) \,.

Now, this functor has a left adjoint given by taking colimits. Therefore 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 (2)(1)\text{(2)}\Rightarrow \text{(1)}. Similarly, the counit is the map into an object over xx from the colimit of its pullback. Thus, it is an isomorphism just when (1)(2)\text{(1)}\Rightarrow\text{(2)}.


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

Colimits in Span


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.

(Sobocinski & Heindel 2011).




A regular category with van Kampen quotients of congruences is exact.


Let p 1,p 2:RAp_1, p_2 : R \rightrightarrows A be a congruence. We need to show that RR is the kernel pair of the quotient map AA/RA \to A/R.

Let the transitivity of the congruence be witnessed by t:R× ARRt \,\colon\, R \times_A R \to R. Then, in the following diagram

R× AR π 2t R p 2 A π 1 p 1 R p 2p 1 A A/R \array{ R \times_A R & \underoverset{\pi_2}{t}{\rightrightarrows} & R & \stackrel{p_2}{\to} & A \\ {{}^\mathllap{\pi_1}}\big\downarrow && \big\downarrow{{}^\mathrlap{p_1}} && \big\downarrow \\ R & \underoverset{p_2}{p_1}{\rightrightarrows} & A & \to & A/R }

the left squares are pullbacks and the top diagram is a split coequalizer (with the splitting maps given by reflexivity), hence a coequalizer. Now since the bottom coequalizer is assumed to be van Kampen, Thm. implies that also the right square is a pullback. But this is the desired statement that RR is the kernel pair of AA/RA \to A/R.


In Set Set , the pushout square

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

is not van Kampen, neither is the coequalizer diagram 111\mathbf 1 \rightrightarrows \mathbf 1 \to \mathbf 1.

On the other hand, in an (∞,1)-topos (say ∞Grpd), the descent property of the circle seen as a coequalizer gives a concise proof that the loop space of the circle is equivalent to the integers: in the diagram below, the two squares on the left are pullbacks because both id\mathrm{id} and ss are isomorphisms, and the top row is a homotopy coequalizer diagram because the line type is contractible. Furthermore, there is a higher coherence relating the two homotopies obtained by composing the top cofork with the back and right squares and by composing the front and right squares with the bottom cofork, ensuring that this diagram is homotopy coherent.

ids 1 1 1 S 1. \array{ \mathbb{Z} & \underoverset{\mathrm{id}}{s}{\rightrightarrows} & \mathbb{Z} & \to & \mathbf{1} \\ \big\downarrow && \big\downarrow && \big\downarrow \\ \mathbf{1} & \rightrightarrows & \mathbf{1} & \to & S^1 \mathrlap{\,.} }

Then, descent implies that the square on the right is a pullback, which says exactly that the path space between any two points of the circle is homotopy equivalent to the integers \mathbb{Z}.

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 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 (specifically, the representable 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.


Last revised on December 14, 2023 at 12:20:43. See the history of this page for a list of all contributions to it.