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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 13th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownAt [[category with weak equivalences]] we say that it is unclear whether every (oo,1)-category arises as the simplicial localization of a cat with weak equivalences, but that it seems plausible. At [[(infinity,1)-category]] we say that indeed every (oo,1)-category arises as the simplicial localization of a [[homotopical category]]. I had put in the paragraph that says this based on a message that Andre Joyal recently posted to the CatTheory mailing list. It would be good to harmonize this with the discussion at [[category with weak equivalences]] and maybe to add some references.

   At category with weak equivalences we say that it is unclear whether every (oo,1)-category arises as the simplicial localization of a cat with weak equivalences, but that it seems plausible.

   At (infinity,1)-category we say that indeed every (oo,1)-category arises as the simplicial localization of a homotopical category.

   I had put in the paragraph that says this based on a message that Andre Joyal recently posted to the CatTheory mailing list.

   It would be good to harmonize this with the discussion at category with weak equivalences and maybe to add some references.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 13th 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownI think it was me who had put in the comment at [[category with weak equivalences]]. But I recently discovered that Clark Barwick and Dan Kan claim to have put a model structure on the category of categories-with-weak-equivalences which is Quillen equivalent to complete Segal spaces (and hence is a model for (oo,1)-categories). I've been meaning to read their preprints and write about them on the Lab, but haven't found the time yet. If anyone else wanted to do it that would be awesome; the link is [here](http://www.math.harvard.edu/~clarkbar/relcat.html).

   I think it was me who had put in the comment at category with weak equivalences. But I recently discovered that Clark Barwick and Dan Kan claim to have put a model structure on the category of categories-with-weak-equivalences which is Quillen equivalent to complete Segal spaces (and hence is a model for (oo,1)-categories). I've been meaning to read their preprints and write about them on the Lab, but haven't found the time yet. If anyone else wanted to do it that would be awesome; the link is here.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeFeb 13th 2010
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   Author: zskoda
   Format: MarkdownIf I understood correctly Maltsiniotis' remark, he considers Cisinski's notion of "categories derivables" in some hierarchy most general setup for homotopy theory, wider and more fundamental than derivators which are in turn more general than model categories. But derivators have some additional structure enabling to prove some universality properties, so when equating with some higher categorical content one should be really careful. But I am here just delegating impressions, I am a novice to this level of generality.

   If I understood correctly Maltsiniotis' remark, he considers Cisinski's notion of "categories derivables" in some hierarchy most general setup for homotopy theory, wider and more fundamental than derivators which are in turn more general than model categories. But derivators have some additional structure enabling to prove some universality properties, so when equating with some higher categorical content one should be really careful. But I am here just delegating impressions, I am a novice to this level of generality.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2010
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   Author: Urs
   Format: MarkdownThanks, Mike. I downloaded the articles. Will see what I can do.

   Thanks, Mike. I downloaded the articles. Will see what I can do.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2010
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   Author: Urs
   Format: Markdownokay, I started a stub for [[model structure on categories with weak equivalences]]

   okay, I started a stub for model structure on categories with weak equivalences