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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 21st 2016
   • (edited Jan 21st 2016)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI would like to understand naive-equivariant Goodwillie calculus, for global equivariance with respect to finite groups. This immediately leads one to the following curious situation, which looks suggestive of something. Namely: 1. Goodwillie calculus is about localizations of the $\infty$-topos over the opposite of pointed [[finite homotopy types]]; 1. global finite-group-equivariant homotopy theory is about the $\infty$-topos over the pointed connected [[homotopy types with finite homotopy groups]]. These two concepts of finiteness of homotopy types are of course different, in fact pretty much orthogonal. But that makes it all the more suggestive, maybe: for naive-equivariant Goodwillie calculus we want the $\infty$-topos over the product of these two sites. Now, does that product site containing products of finite homotopy types for both notions of finiteness, maybe want to be understood as being a sub-category of something? Or generally: is there anything secret going on here with these two sites of homotopy types with finiteness condition appearing both, or is it just a meaningless coincidence?

   I would like to understand naive-equivariant Goodwillie calculus, for global equivariance with respect to finite groups.

   This immediately leads one to the following curious situation, which looks suggestive of something.

   Namely:

   1. Goodwillie calculus is about localizations of the $\infty$-topos over the opposite of pointed finite homotopy types;

   2. global finite-group-equivariant homotopy theory is about the $\infty$-topos over the pointed connected homotopy types with finite homotopy groups.

   These two concepts of finiteness of homotopy types are of course different, in fact pretty much orthogonal. But that makes it all the more suggestive, maybe:

   for naive-equivariant Goodwillie calculus we want the $\infty$-topos over the product of these two sites. Now, does that product site containing products of finite homotopy types for both notions of finiteness, maybe want to be understood as being a sub-category of something?

   Or generally: is there anything secret going on here with these two sites of homotopy types with finiteness condition appearing both, or is it just a meaningless coincidence?