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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2010
   • (edited Oct 26th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI started at [[cohesive (infinity,1)-topos]] a section <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#vanKampenTheorem">van Kampen theorem</a> 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 $Conc(\mathbf{H}) \hookrightarrow \mathbf{H}$.

   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 $Conc(\mathbf{H}) \hookrightarrow \mathbf{H}$.

2. • CommentRowNumber2.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 26th 2010
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   Author: DavidRoberts
   Format: MarkdownItexThe page [[cohesive infinity-topos]] doesn't exist! But your other link works.

   The page cohesive infinity-topos doesn’t exist! But your other link works.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2010
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   Author: Urs
   Format: MarkdownItexThanks, fixed.

   Thanks, fixed.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 <a href="http://ncatlab.org/nlab/show/cohesive+(infinity%2C1)-topos#vanKampenTheorem">here</a>.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2010
   • (edited Oct 26th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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 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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 26th 2010
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   Author: Urs
   Format: MarkdownItexI 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.

   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.