Background
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equivalences in/of $(\infty,1)$-categories
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Models
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 $\mathcal{C}$ be a category with pullbacks. Then there is a (pseudo) 2-functor
defined by $S(x) \,\coloneqq\, \mathcal{C}/x$ (the slice category), called the self-indexing of $\mathcal{C}$. Its Grothendieck construction is the codomain fibration.
A colimit in $\mathcal{C}$ is van Kampen if it is preserved by the functor $S$ (1), i.e. it is taken to a (weak) 2-limit in $Cat$.
We need the following notation:
For $\mathcal{D}$ any category, write $\mathcal{D}^{\triangledown}$ for the result of adjoining to it a terminal object. This comes with a canonical full subcategory inclusion $\mathcal{D} \hookrightarrow \mathcal{D}^{\triangledown}$
For $\mathcal{D}$ a small category, and $G \,\colon\, \mathcal{D} \to \mathcal{C}$ a $\mathcal{D}$-shaped diagram with colimit $\underset{\longrightarrow}{\lim} G \,\in\, \mathcal{C}$, write $G^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \longrightarrow \mathcal{C}$ the extension of $G$ to $\mathcal{D}^{\triangledown}$ by assigning $\underset{\longrightarrow}{\lim} G$ to the adjoined terminal object.
(equifibrancy characterization of van Kampen colimits)
If $\mathcal{C}$ has all colimits of $\mathcal{D}$-shaped diagrams, then the colimit $\underset{\longrightarrow}{\lim}G$ of $G \colon \mathcal{D}\to \mathcal{C}$ is van Kampen (Def. ) if and only if the following condition holds:
For any diagram $F^{\triangledown} \,\colon\, \mathcal{D}^{\triangledown} \to \mathcal{C}$ and natural transformation $\alpha^{\triangledown} \,\colon\, F^{\triangledown} \Rightarrow G^{\triangledown}$ whose restriction $\alpha \colon F \Rightarrow G$ along $\mathcal{D} \hookrightarrow \mathcal{D}^{\triangledown}$ is equifibered, the following are equivalent:
$\alpha^{\triangledown}$ is equifibered.
$F^{\triangledown}$ is a colimiting cocone.
First note that the 2-limit of $S\circ G$ is equivalent to the full subcategory of the slice category of the functor category
consisting of the equifibered transformations. 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. 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 $G$ into the pullback of its colimit. The latter underlies an equifibered $\alpha'$ by construction, so the unit is an isomorphism just when $\text{(2)}\Rightarrow \text{(1)}$. Similarly, the counit is the map into an object over $x$ from the colimit of its pullback. Thus, it is an isomorphism just when $\text{(1)}\Rightarrow\text{(2)}$.
The condition $\text{(1)}\Rightarrow\text{(2)}$ is precisely the statement that the colimit of $G$ is universal, i.e. preserved by pullback. The condition $\text{(2)}\Rightarrow\text{(1)}$ 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 $C$.
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.
A regular category with van Kampen quotients of congruences is exact.
Let $p_1, p_2 : R \rightrightarrows A$ be a congruence. We need to show that $R$ is the kernel pair of the quotient map $A \to A/R$.
Let the transitivity of the congruence be witnessed by $t \,\colon\, R \times_A R \to R$. Then, in the following diagram
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 $R$ is the kernel pair of $A \to A/R$.
is not van Kampen, neither is the coequalizer diagram $\mathbf 1 \rightrightarrows \mathbf 1 \to \mathbf 1$.
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 $\mathrm{id}$ and $s$ 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.
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 $\mathbb{Z}$.
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 (specifically, the representable 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 December 14, 2023 at 12:20:43. See the history of this page for a list of all contributions to it.