One form of a higher homotopy van Kampen theorem is a theorem that asserts that the homotopy type of a topological space can be computed by a suitable colimit or homotopy colimit over homotopy types of its pieces. Another form which allows specific computation deals with spaces with certain kinds of structure, for example filtered spaces or $n$-cubes of spaces.
This generalizes the van Kampen theorem, which only deals with the underlying 1-type (the fundamental groupoid).
Let $X$ be a topological space, write $Op(X)$ for its category of open subsets and let
be a functor out of a small category $C$ such that
Then:
the canonical morphism in sSet out of the colimit
into the singular simplicial complex of $X$ exhibits $Sing(X)$ as the homotopy colimit $hocolim Sing \circ \chi$.
This is theorem A.1.1 in (Lurie).
The following is a version of the above general statement restricted to a strict ∞-groupoid-version of the fundamental ∞-groupoid and applicable for topological spaces that are equipped with the extra structure of a filtered topological space.
Notice that these strict $\infty$-groupoids are equivalent to crossed complexes.
Suppose $X_*$ is a filtered space and $X$ is the union of the interiors of sets $U^i$, $i \in I$. Let $U^i_*$ be the filtered space given by the intersections $U^i \cap X_n$ for $n \geq 0$. If $d=(i,j) \in I^2$ we write $U^d$ for $U^i \cap U^j$. We then have a coequaliser diagram of filtered spaces
If the filtered spaces $U^f_*$ are connected filtered spaces for all finite intersections $U^f_*$ of the filtered spaces $U^i_*$, then
(Conn) The filtered space $X_*$ is connected; and
(Iso) The fundamental crossed complex functor $\Pi$ takes the above coequaliser diagram of filtered spaces to a coequaliser diagram of crossed complexes.
A full account is given in (Brown-Higgins-Sivera and the methodology is discussed in (Brown).
Remarks
Note that because $\Pi$ uses groupoids, it obviously takes disjoint unions $\bigsqcup$ of filtered spaces into disjoint unions (= coproducts) $\bigsqcup$ of crossed complexes.
The proof of the theorem is not direct but goes via the fundamental cubical $\omega$-groupoid with connections of the filtered spaces, as that context allows the notions of algebraic inverse to subdivision and of commutative cube. However the proof is a direct generalisation of a proof for the van Kampen theorem for the fundamental groupoid.
Applications of this theorem include many basic facts in algebraic topology, such as the Relative Hurewicz Theorem, the Brouwer degree theorem, and new nonabelian results on 2nd relative homotopy groups, not of course obtainable by the traditional wholly abelian methods. No use is made of singular homology theory or of simplicial approximation. Also included is a version of the “small simplex theorem”, see Theorem 10.4.20 of (Brown-Higgins-Sivera).
In a cohesive (∞,1)-topos (already in a locally ∞-connected (∞,1)-topos) higher van Kampen theorems hold in great generality.
See the section cohesive (∞,1)-topos – van Kampen theorem.
In particular for the cohesive $(\infty,1)$-topos ∞TopGrpd of topological ∞-groupoids this reproduces the topological higher van Kampen theorem discussed above.
Here is one application in dimension 2 not easily obtainable by traditional algebraic topology.
Let $0 \to P \to Q \to R \to 0$ be an exact sequence of abelian groups. Let $X$ be the mapping cone of the induced map $K(P,1) \to K(Q,1)$ of Eilenberg-Mac Lane spaces. Then a crossed module representing the homotopy 2-type of $X$ is $\mu: C \to Q$ where $C$ is abelian and is the direct sum $\oplus_{r \in R} P^r$ of copies of $P$ one for each $r \in R$ and the action of $Q$ is via $R$ and permutes the copies by $(p,r)^s=(p,r+s)$. Similar examples for $P,Q,R$ nonabelian are do-able, more complicated, and certainly not obtainable by traditional methods.
The version for topological spaces and the fundamental infinity-groupoid functor is discussed in Appendix A of
The version for filtered topological spaces and the strict homotopy $\infty$-groupoid functor is discussed in
while the general methodology is discussed in
One area of application of work of Brown and Loday is to a nonabelian tensor product of groups, see: