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1. • CommentRowNumber1.
   • CommentAuthorTobias Fritz
   • CommentTimeJan 31st 2013
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   Author: Tobias Fritz
   Format: HtmlThere is a notion of <a href="http://ncatlab.org/nlab/show/stable+%28infinity%2C1%29-category">stable (oo,1)-category</a> due to Lurie; is there also a notion of "stable (oo,n)-category"? Of stable oo-category? Would a potential notion of stable oo-category somehow correspond to taking the limit along the diagonal in the <a href="http://ncatlab.org/nlab/show/periodic+table">periodic table</a>? Am I making any sense at all?

   There is a notion of stable (oo,1)-category due to Lurie; is there also a notion of "stable (oo,n)-category"? Of stable oo-category? Would a potential notion of stable oo-category somehow correspond to taking the limit along the diagonal in the periodic table?
   
   Am I making any sense at all?
2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 1st 2013
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   Author: Mike Shulman
   Format: MarkdownItexIt's a good question, but I've never seen an answer. I do think the periodic table is about a different sort of "stability".

   It’s a good question, but I’ve never seen an answer. I do think the periodic table is about a different sort of “stability”.

3. • CommentRowNumber3.
   • CommentAuthorDylan Wilson
   • CommentTimeFeb 1st 2013
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   Author: Dylan Wilson
   Format: TextI've wondered this before, too. I can think of two possible ways in... 1. The canonical 2-category is the category of categories. So maybe we think that such things should look like the (oo,2)-category of stable oo-cats. 2. On the other hand, stable oo-cats are modeled after additive categories- i.e. categories with a 0 object and a relationship between the product and coproduct. So we could try to think of what the 2-categorical version of this should be and then "infinitize". Dunno. Guess it depends what you want to do with these gadgets.

   I've wondered this before, too. I can think of two possible ways in...
   
   1. The canonical 2-category is the category of categories. So maybe we think that such things should look like the (oo,2)-category of stable oo-cats.
   
   2. On the other hand, stable oo-cats are modeled after additive categories- i.e. categories with a 0 object and a relationship between the product and coproduct. So we could try to think of what the 2-categorical version of this should be and then "infinitize".
   
   Dunno. Guess it depends what you want to do with these gadgets.
4. • CommentRowNumber4.
   • CommentAuthorTobias Fritz
   • CommentTimeFeb 1st 2013
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   Author: Tobias Fritz
   Format: HtmlThanks, both, for the enlightening answers. I figured that the periodic table might be related to the idea of stable oo-categories for the following reason: in the situations with which I'm somewhat familiar (spaces/C*-algebras), defining the stable homotopy category (of finite spectra) corresponds to stabilizing the homotopy category with respect to suspensions. Similarly, moving down in the periodic table corresponds to suspension. I'm on shaky ground with making these statements, but I hope that it is nevertheless a sensible question to ask, "What's going on here?"

   Thanks, both, for the enlightening answers.
   
   I figured that the periodic table might be related to the idea of stable oo-categories for the following reason: in the situations with which I'm somewhat familiar (spaces/C*-algebras), defining the stable homotopy category (of finite spectra) corresponds to stabilizing the homotopy category with respect to suspensions. Similarly, moving down in the periodic table corresponds to suspension.
   
   I'm on shaky ground with making these statements, but I hope that it is nevertheless a sensible question to ask, "What's going on here?"
5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 1st 2013
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   Author: Mike Shulman
   Format: MarkdownItexI would say that moving down the periodic table actually corresponds to *delooping*, which is not quite the same as suspension. There is a relationship to stabilization: I would say that stabilizing an $(\infty,1)$-category (in the sense of Lurie) is similar to stablizing its *objects* (in the sense of the periodic table). Although you have to be careful what you mean by the latter, to ensure that you get all spectra and not just connective spectra. Maybe a good place to start is with a notion of directed spectrum. We could define an "$(\infty,n)$-spectrum" to be a sequence $(E_k)$ where $E_k$ is a pointed $(\infty,n+k)$-category and we have equivalences $E_k \simeq \Omega E_{k+1}$. What sort of category do these things form?

   I would say that moving down the periodic table actually corresponds to delooping, which is not quite the same as suspension. There is a relationship to stabilization: I would say that stabilizing an $(\infty,1)$-category (in the sense of Lurie) is similar to stablizing its objects (in the sense of the periodic table). Although you have to be careful what you mean by the latter, to ensure that you get all spectra and not just connective spectra.

   Maybe a good place to start is with a notion of directed spectrum. We could define an “$(\infty,n)$-spectrum” to be a sequence $(E_k)$ where $E_k$ is a pointed $(\infty,n+k)$-category and we have equivalences $E_k \simeq \Omega E_{k+1}$. What sort of category do these things form?