[nLab](http://ncatlab.org/nlab/show/van+Kampen+colimit) uses the following definition of van Kampen colimits --- a colimit in a category $\mathbb{C}$ is called van Kampen iff it is preserved by the internal indexing functor $\mathbb{C}/(-) \colon \mathbb{C}^{op} \rightarrow \mathbf{Cat}$ defined as:
$$X \mapsto \mathbb{C}/X$$
$$X \overset{f}\rightarrow Y \mapsto \mathbb{C}/Y \overset{f^\*}\rightarrow \mathbb{C}/X$$
where $f^\*$ is the pullback-along-$f$ functor.


My question is --- where does this definition come from and why are such colimits called "van Kampen"?


In case of coproducts one may notice that the property of being van Kampen in the above sense is equivalent to the usual property of being extensive.


On the other hand, van Kampen pushouts in the above sense do not match the usual definition of van Kampen pushouts from the definition of an adhesive category. For example, if $\mathbb{C} = \mathbf{Set}$ then the internal indexing functor $\mathbf{Set}/(-)$ is equivalent to the usual exponential functor $\mathbf{Set}^{(-)}$, which (like every exponential functor) preserves all colimits; but not every pushout in $\mathbf{Set}$ is van Kampen (pushouts along monomorphisms are).



Van Kampen colimits
Here's a summary of the comments above (which does not answer the question on the origin of the term, that I have no idea).


A colimit in a category $C$ with pullbacks is van Kampen if the indexing functor $C/(-): C^{op} \to Cat$ transforms it into a *weak* 2-limit, or bilimit, or homotopy limit. These are sometimes called 2-limit or simply limit, but the latter also have stricter meanings which do not give the correct definition.


This is a very strong property: a locally presentable category (or even a locally presentable $(n,1)$-category for any finite $n$) in which all small colimits are universal and all pushouts are van Kampen is necessarily the terminal category, because it [must be](http://ncatlab.org/nlab/show/van+Kampen+colimit#InHigherCategories) an $\infty$-topos.


For $C=Set$, for example, the coequalizer of $\*\rightrightarrows \*$ is not van Kampen, because the weak 2-limit of $Set\rightrightarrows Set$ is the category of $\mathbb{Z}$-sets rather than the category of sets. A colimit in $Set$ is van Kampen iff its category of elements is simply connected. Pushouts have this property if one of the legs is injective, although that's not necessary.


