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I started at cohesive (infinity,1)-topos a section van Kampen theorem
In the cohesive $\infty$-topos itself the theorem holds trivially. The interesting part is, I think, to which extent it restricts to the concrete cohesive objects under the embedding $Conc(\mathbf{H}) \hookrightarrow \mathbf{H}$.
The page cohesive infinity-topos doesn’t exist! But your other link works.
Thanks, fixed.
I shouldn’t have gotten myself distracted by this, since I need to be doing something else, but nevertheless I did spend some time now on expanding and polishing my purported proof of the higher van Kampen theorem using cohesive $\infty$-topos technology here.
I worked a bit more to polish the argument that the pushout of topological spaces remains a homotopy pushout after the embedding of the spaces as 0-truncated topological $\infty$-groupoids.
One can see this directly with a bit of effort, I think, but one can also fall back to the 1-categorical van Kampen theorem to see this, which is maybe noteworthy:
for
$\array{ U_1 \cap U_2 &\to& U_1 \\ \downarrow && \downarrow \\ U_2 &\to& X }$a cover of the topological space $X$ by two open subsets, pick a good open cover of $X$, call the Cech nerve $Q X$ and write $Q U_i$ for the Cech nerves of the restriction to those open subsets that land in $U_i$. Then one can show that the ordinary pushout
$\array{ Q(U_1) \cap Q(U_2) &\to& Q(U_1) \\ \downarrow && \downarrow \\ Q(U_2) &\to& Q(U_1) \coprod_{Q(U_1) \cap Q(U_2)} Q(U_2) }$of simplicial presheaves over open balls is a homotopy pushout. So the question is if this is again equivalent to $X$.
Since all objects are 0-truncated, it suffices to check that the 1-truncation of the pushout is $X$. It is easy to see that $\pi_0$ agrees. To see that $\pi_1$ of the pushout vanishes (the categorical $\pi_1$ not the geometric one!!) pass objectwise to the geometric realization and then use the ordinary 1-van Kampen theorem.
I made it overly complicated by working in the projective model structure. Using the injective structure the whole statement becomes trivial. We can prove higher van Kampen in a single line.
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