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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 11th 2010
   • (edited Sep 11th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexToday at ESI Christian Blohman gave a talk on an result he got with Chenchang Zhu, to appear soon. I missed almost all of the talk, but did get a personal lesson afterwards, so I am well informed now ;-) They have a crisp cool result, that should -- and that's their motivation -- serve to unify a bunch of constructions that are currently present in the literature, and a plethora of more such constructions that would certainly keep being invented until somebody gives a general statement such as they do now. Which is this: they construct a natural functor from the category of simplicial sets over $\Delta[1]$ to a 1-category of spans of simplicial sets $$ \left\{ \array{ K \\ \downarrow \\ \Delta[1] } \right\} \;\;\; \to \;\;\; \left\{ \array{ \hat K &\to& K_1 \\ \downarrow \\ K_0 } \right\} $$ where $K_0$ and $K_1$ are the fibers, and $\hat K$ is the simplicial set whose $k$-cells are those maps out of the [[join of simplicial set]] $$ [k] \star [k] \to K $$ such that the first copy of $[k]$ lands over $0$ and the second over $1$. Then they prove that **Proposition** If $K \to \Delta[1]$ is a [[left fibrationn]] then $\hat K \to K_0$ is an acyclic fibration, hence $K_o \leftarrow \hat K \to K_1$ an [[anafunctor]] of Kan complexes. Moreover, they show that left fibrations $K \to \Delta[1]$ that are 2-coskeletal sort of in both degrees are precisely the nerves of [[bibundle]]s, or rather are the [[action groupoid]]s of these. He ended by saying that they are still fiddling with how precisely to restate this with simplicial sets replaced by simplicial manifolds (which is the kind of case for which this would be of interest). But I think it is better to do the general abstract construction in the oo-topos first, and only later check -- if really necessary -- whether certain objects are representable in some way. And because the construction is functorial, it extends straighforwardly to the projective model structure on simplicial presheaves (over any site) and gives us the relation between $\infty$-bibundles and $\infty$-anafunctors there. Okay, we will want this for the local model structure, but if we assume the topos has enough points and we look at the hyperlocalization, then it is straightforward again, with taking left fibrations and acyclic fibrations in their statement to be stalkwise such.

   Today at ESI Christian Blohman gave a talk on an result he got with Chenchang Zhu, to appear soon. I missed almost all of the talk, but did get a personal lesson afterwards, so I am well informed now ;-)

   They have a crisp cool result, that should – and that’s their motivation – serve to unify a bunch of constructions that are currently present in the literature, and a plethora of more such constructions that would certainly keep being invented until somebody gives a general statement such as they do now.

   Which is this:

   they construct a natural functor from the category of simplicial sets over $\Delta[1]$ to a 1-category of spans of simplicial sets

   $$\left\{ \array{ K \\ \downarrow \\ \Delta[1] } \right\} \;\;\; \to \;\;\; \left\{ \array{ \hat K &\to& K_1 \\ \downarrow \\ K_0 } \right\}$$

   where $K_0$ and $K_1$ are the fibers, and $\hat K$ is the simplicial set whose $k$-cells are those maps out of the join of simplicial set

   $$[k] \star [k] \to K$$

   such that the first copy of $[k]$ lands over $0$ and the second over $1$.

   Then they prove that

   Proposition If $K \to \Delta[1]$ is a left fibrationn then $\hat K \to K_0$ is an acyclic fibration, hence $K_o \leftarrow \hat K \to K_1$ an anafunctor of Kan complexes.

   Moreover, they show that left fibrations $K \to \Delta[1]$ that are 2-coskeletal sort of in both degrees are precisely the nerves of bibundles, or rather are the action groupoids of these.

   He ended by saying that they are still fiddling with how precisely to restate this with simplicial sets replaced by simplicial manifolds (which is the kind of case for which this would be of interest). But I think it is better to do the general abstract construction in the oo-topos first, and only later check – if really necessary – whether certain objects are representable in some way.

   And because the construction is functorial, it extends straighforwardly to the projective model structure on simplicial presheaves (over any site) and gives us the relation between $\infty$-bibundles and $\infty$-anafunctors there. Okay, we will want this for the local model structure, but if we assume the topos has enough points and we look at the hyperlocalization, then it is straightforward again, with taking left fibrations and acyclic fibrations in their statement to be stalkwise such.