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. 



Higher refinement of Seifert-van Kampen theorem on the language of hocolim
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. 


