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I like the following version of SvKT. If $\Pi_1$ is the functor of fundamental groupoid and $(X_i)_{i\in I}$ is a diagram of spaces then $$\Pi_1({\sf hocolim}\: X_i)\simeq {\sf hocolim}\: \Pi_1(X_i).$$

Question: Is there a similar statement for higher homotopy? For example, if we replace $\Pi_1$ by some version of the infinity-groupoid $\Pi_\infty$. But it should be tricky because the homotopy pushout of the diagram $* \leftarrow S^1 \to *$ is $S^2$ whose homotopy groups are complicated.

I know about Brown's version of this theorem but filtered spaces is not a convenient setting for me.

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    $\begingroup$ This is closely related to the homotopy hypothesis, and some definitions of higher groupoids make it essentially a tautology. Of course, this is not much use for computations unless you have a definition of higher groupoid from which you can actually compute the homotopy groups of a colimit. $\endgroup$ Jun 18 '15 at 9:25
  • $\begingroup$ I thought Lurie had a version of this, but I recall offhand where. $\endgroup$ Jun 18 '15 at 9:50
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    $\begingroup$ I totally agree with @EricWofsey Morally, the fundamental infinity groupoid is the space itself, so you find the same thing at both sides of the equation. Brown's version need not be for filtered spaces, I mean, the skeletal filtration is fine and canonical. I'd say that such a result really represents a simplification when you replace $\Pi_1$ with something which is easy enough, such that the fundamental crossed module, categorical group, etc. $\endgroup$ Jun 18 '15 at 10:02
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    $\begingroup$ @DavidRoberts, Lurie's version is discussed here: ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem $\endgroup$
    – AAK
    Jun 18 '15 at 12:07
  • $\begingroup$ Thanks, @Adeel. I was on my phone and in a rush, and didn't do the obvious google search! $\endgroup$ Jun 18 '15 at 21:05
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The higher version of this statement is the following: taking the fundamental $n$-groupoid $\Pi_{\le n}(-)$, or equivalently $n$-truncating, is (higher) left adjoint to the inclusion of $n$-truncated spaces into spaces. Hence it sends homotopy colimits to homotopy colimits. This implies, for example, that if you want to compute $\pi_n$ of a homotopy colimit it suffices to remember the $n$-truncations of all of the spaces involved.

This isn't too useful as a computational tool because for $n \ge 2$ it's hard to compute homotopy colimits of $n$-groupoids, or equivalently $n$-truncated spaces. It's also worth pointing out that higher homotopy groups $\pi_n$ really behave nicely with respect to homotopy limits, not homotopy colimits, so for $n \ge 2$ the gap between knowing $\Pi_{\le n}(-)$ of a homotopy colimit and knowing $\pi_n(-)$ is much larger than for $n = 1$.

I also hesitate to call this a version of the Seifert-van Kampen theorem because the computational power of the Seifert-van Kampen theorem comes from the fact that it guarantees that certain homotopy pushouts can be computed as ordinary pushouts. This is a model-dependent kind of statement, whereas what I've said above is model-independent.

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The subtlety of the classical SvK theorem can be interpreted as being the passage from spaces to simplicial sets. Beyond that it relates more to behaviour of colimits under adjoint functors. In this I would disagree with Fernando as to identify the space and the corresponding infinity groupoid misses the main point. It thus looks, from that viewpoint, as if your favorite form of the statement misses the key intuition of the classical result. That then becomes a comparison result on homotopy colimits and colimits. More to the point if you take the singular complex as the fundamental infinity groupoid of the space, you have to look at subdivisions of simplices and their `reintegration' from subdivisions. (There is a tentative solution to this in all generality made by Phil Ehlers in his thesis.)

I agree with Fernando that if you want to do calculations then one key decision is deciding what `calculation' means. If you work with crossed modules as small models of some homotopy types then you can work out quite a lot on problems like your one on $S^2$ using the Brown-Loday tensor product which is an offshoot of a form of their generalised SvK result. Their theory using crossed n-cubes extends this.

The final point to make is that the alternative interpretation of the SvK in terms of covering spaces does have a generalisation to higher order groupoids and this relates to the Pursuing Stacks program of Grothendieck.

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This answer relates mainly to the term "Higher refinement of Seifert-van Kampen Theorem", rather than homotopy colimit.

Last week I gave a talk to the 2015 Category Theory Meeting in Aveiro, entitled "A philosophy of modelling and computing homotopy types". The Abstract and full and handout versions (slightly refined) are available on my preprint page. Homotopy colimits enter from the side in order to get a "nice" pushout of spaces, but the aim is

explicit nonabelian pushout computations of certain **homotopy types.

It is often hard work to compute from this homotopy groups and $k$-invariants! (There was also no time for indications in the talk.)

I do not know of an account of a homotopy theory, and so of "homotopy colimits", for the crossed squares used in this presentation for modelling, and some computations of, homotopy $3$-types.

Later: I'll add that in 1965 I found the nice statement and proof for the fundamental groupoid $\pi_1(X)$ of a space which was a union of open sets. I thought: great! One can get rid of base points!

Then I found this was not enough for my first aim, getting as a Corollary the fundamental group of the circle. So then I discovered the applications of the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, chosen according to the geometry of the given situation. In particular, the determination of the fundamental group of the circle required at least $2$ base points. These results were published in 1967.

This simple extension of the theorem and proof of the usual van Kampen theorem is, to my knowledge, given in topology texts in English only in the 1968, 1988, 2006 editions of what is now titled "Topology and Groupoids".

As for higher versions of the nonabelian fundamental group, these were sought by the topologists of the early 20th century topologists (Dehn, Cech, Hopf, ...) but I think did require for the proof a shift from groups to groupoids, and a 2-dim Seifert-van Kampen Theorem was published by Philip Higgins and I in 1978. That was a theorem on second relative homotopy groups.

June 24: The updated Abstract of the Aveiro presentation is as follows, and I hope shows some special features of this approach to Higher Seifert-van Kampen Theorems:

ABSTRACT

This philosophy involves functors $\mathbb H$ from (Topological Data) to (Algebraic Data), and conversely "classifying space" functors $ \mathbb B$ from (Algebraic Data) to (Topological Data). These should satisfy:

  1. $\mathbb H$ is homotopically defined.
  2. $\mathbb{ HB}$ is naturally equivalent to 1.
  3. The Topological Data has a notion of connected.
  4. For all Algebraic Data $A$, we have $\mathbb BA$ is connected.
  5. $\mathbb H$ preserves certain colimits of connected Topological Data.

The Algebraic Data splits into several equivalent kinds, ranging from "broad" to "narrow", related by Dold-Kan type equivalences. The broad data is used for conjecturing and proving theorems; the narrow data is used for calculations and relating to classical methods.

As examples of Algebraic Data we give groupoids, crossed modules and crossed squares. We give a sample computation, using crossed squares, of the homotopy 3-type of the mapping cone of the classifying space of a morphism of crossed modules.

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