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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeApr 30th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWell, the proof I thought I had for modeling univalence in $(\infty,1)$-toposes has run into a snag, as I thought it might, and I don't know how to fix it right now. I'm still going to write up what I've got, though, in hopes that it might be useful. But since that makes the paper somewhat more of a "status report", I thought it would be nice to include a bit about the known prospects of other models. I know of the following model categories that present $(\infty,1)$-presheaf categories: * any of the [[global model structures on simplicial presheaves]] -- projective, injective, or (when it exists) Reedy * the [[model structure for left fibrations]] on a slice category of simplicial sets * [this model structure](http://mathoverflow.net/questions/82685/presheaves-on-a-complete-segal-space) on a slice category of bisimplicial sets Are there others known?

   Well, the proof I thought I had for modeling univalence in $(\infty,1)$-toposes has run into a snag, as I thought it might, and I don’t know how to fix it right now. I’m still going to write up what I’ve got, though, in hopes that it might be useful. But since that makes the paper somewhat more of a “status report”, I thought it would be nice to include a bit about the known prospects of other models. I know of the following model categories that present $(\infty,1)$-presheaf categories:

   • any of the global model structures on simplicial presheaves – projective, injective, or (when it exists) Reedy
   • the model structure for left fibrations on a slice category of simplicial sets
   • this model structure on a slice category of bisimplicial sets

   Are there others known?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2012
   • (edited Apr 30th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexProbably you are already counting the following in, but just for completeness: there is of course also the [[model structure on sSet-enriched presheaves]], i.e.: over general [[simplicial sites]].

   Probably you are already counting the following in, but just for completeness: there is of course also the model structure on sSet-enriched presheaves, i.e.: over general simplicial sites.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2012
   • (edited Apr 30th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexOf course there is a certain flexibility in generating more model structures at will , for instance by producing more model structures equivalent to $sSet_{Quillen}$. For instance start with a model structure $[SmthMfd^{op}, sSet]_{loc}$ presenting $Sh_\infty(SmthMfd)$ and then pass to its $\mathbb{A}^1$-localization $[SmthMfd^{op}, sSet]_{loc, \mathbb{A}^1}$ for $\mathbb{A}^1 = \mathbb{R}^1$. This is Quillen equivalent to $sSet_{Quillen}$, and so $[C^{op}, [SmthMfd^{op}, sSet]_{loc, \mathbb{A}^1}]_{loc}$ is another model for $PSh_\infty(C)$. This can be useful when one considers diagrams in $Sh_\infty(SmthMfd)$. Similarly one could consider presheaves with values in a localization of a model structure for $(\infty,1)$-categories, which will be useful if one wants to consider $PSh_\infty(C)$ in conjunction with the $\infty$-categories over $C$. Things like this. Of course this will probably not be of interest for what you are after.

   Of course there is a certain flexibility in generating more model structures at will , for instance by producing more model structures equivalent to $sSet_{Quillen}$.

   For instance start with a model structure $[SmthMfd^{op}, sSet]_{loc}$ presenting $Sh_\infty(SmthMfd)$ and then pass to its $\mathbb{A}^1$-localization $[SmthMfd^{op}, sSet]_{loc, \mathbb{A}^1}$ for $\mathbb{A}^1 = \mathbb{R}^1$. This is Quillen equivalent to $sSet_{Quillen}$, and so $[C^{op}, [SmthMfd^{op}, sSet]_{loc, \mathbb{A}^1}]_{loc}$ is another model for $PSh_\infty(C)$. This can be useful when one considers diagrams in $Sh_\infty(SmthMfd)$.

   Similarly one could consider presheaves with values in a localization of a model structure for $(\infty,1)$-categories, which will be useful if one wants to consider $PSh_\infty(C)$ in conjunction with the $\infty$-categories over $C$.

   Things like this. Of course this will probably not be of interest for what you are after.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexToo bad about the univalence. Do you happen to have a partial result that gives univalence for _some_ subclass of sites, more general than the "inverse diagrams"?

   Too bad about the univalence. Do you happen to have a partial result that gives univalence for some subclass of sites, more general than the “inverse diagrams”?

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeApr 30th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> Of course there is a certain flexibility in generating more model structures at will Yes, that's true. One could look at presheaves of topological spaces, or cubical sets, etc. etc. But I agree that those directions don't immediately seem to be helpful; simplicial sets are so well-behaved already that it's hard to imagine that simply replacing them with something equivalent would help. > Do you happen to have a partial result that gives univalence for *some* subclass of sites, more general than the "inverse diagrams"? At the moment, I think I can deal with simplicial presheaves on [[elegant Reedy categories]]. But that could change tomorrow.

   Of course there is a certain flexibility in generating more model structures at will

   Yes, that’s true. One could look at presheaves of topological spaces, or cubical sets, etc. etc. But I agree that those directions don’t immediately seem to be helpful; simplicial sets are so well-behaved already that it’s hard to imagine that simply replacing them with something equivalent would help.

   Do you happen to have a partial result that gives univalence for some subclass of sites, more general than the “inverse diagrams”?

   At the moment, I think I can deal with simplicial presheaves on elegant Reedy categories. But that could change tomorrow.