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1. • CommentRowNumber1.
   • CommentAuthorJohnCohn
   • CommentTimeDec 11th 2016
   • (edited Dec 12th 2016)
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   Author: JohnCohn
   Format: TextI'm sorry if anyone will dislike my question, but it is the one that interests me. I'm by no means an expert in (∞,n)-categories or even (∞,1)-categories, so if some of my remarks would sound silly, I hope you forgive me. I'm intrigued by higher mathematics. By higher mathematics I mean (∞,1)-categorical mathematics, namely, higher algebra (as in Lurie's "Higher Algebra") and higher geometry (as in Lurie's "Spectral Algebraic Geometry", numerous Toen's and Vezzossi's articles on derived and homotopical algebraic geometry). Thanks to Lurie and others, theory of (∞,1)-categories is well-developed as of now. So well we can also develop the aformentioned "higher mathematics". But as I understand, theory of (∞,n)-categories is not as well-researched as the one of (∞,1)-categories. It also doesn't have as many applications. For example, the only ones I know of are the proof of Cobordism Hypothesis which requires (∞,n)-categories to state and the use of (∞,2)-categories in Goodwillie calculus of functors. Both topics are researched by Jacob Lurie. However, dear nCatLab's frequenters, what do you think about potential (∞,n)-categorical mathematics? About potential (∞,n)-topos on which we would potentially model some geometric object? About hypothetical (∞,n)-stacks? (∞,n)-operads? (∞,n)-algebras over (∞,n)-operads? And so on? What I'm talking about is the possibility of (∞,n)-algebra and (oo,n)-geometry. What do you think about it? Could it be useful? Could it be interesting? Could it be possible? Could it have interesting connections with (∞,1)-categorical algebra and geometry? What potential does these "(∞,n)-mathematics" would have?

   I'm sorry if anyone will dislike my question, but it is the one that interests me. I'm by no means an expert in (∞,n)-categories or even (∞,1)-categories, so if some of my remarks would sound silly, I hope you forgive me.
   
   I'm intrigued by higher mathematics. By higher mathematics I mean (∞,1)-categorical mathematics, namely, higher algebra (as in Lurie's "Higher Algebra") and higher geometry (as in Lurie's "Spectral Algebraic Geometry", numerous Toen's and Vezzossi's articles on derived and homotopical algebraic geometry).
   Thanks to Lurie and others, theory of (∞,1)-categories is well-developed as of now. So well we can also develop the aformentioned "higher mathematics".
   But as I understand, theory of (∞,n)-categories is not as well-researched as the one of (∞,1)-categories. It also doesn't have as many applications. For example, the only ones I know of are the proof of Cobordism Hypothesis which requires (∞,n)-categories to state and the use of (∞,2)-categories in Goodwillie calculus of functors. Both topics are researched by Jacob Lurie.
   
   However, dear nCatLab's frequenters, what do you think about potential (∞,n)-categorical mathematics? About potential (∞,n)-topos on which we would potentially model some geometric object? About hypothetical (∞,n)-stacks? (∞,n)-operads? (∞,n)-algebras over (∞,n)-operads? And so on? What I'm talking about is the possibility of (∞,n)-algebra and (oo,n)-geometry. What do you think about it?
   
   Could it be useful? Could it be interesting? Could it be possible? Could it have interesting connections with (∞,1)-categorical algebra and geometry? What potential does these "(∞,n)-mathematics" would have?
2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeDec 11th 2016
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   Author: David_Corfield
   Format: MarkdownItexThere's plenty of information on the nLab, e.g., start at [[(infinity,n)-category]].

   There’s plenty of information on the nLab, e.g., start at (infinity,n)-category.

3. • CommentRowNumber3.
   • CommentAuthorJohnCohn
   • CommentTimeDec 12th 2016
   • (edited Dec 12th 2016)
   • PermaLink
   Author: JohnCohn
   Format: MarkdownItex> There's plenty of information on the nLab, e.g., start at [[(infinity,n)-category]]. I'm sorry, it's either you misunderstood me, or I'm misunderstanding you. For example, the article on (∞,n)-categories on nLab contains a great deal of technical information about constructions of (∞,n)-categories and such. There is little to none information about what I've asked: about (∞,n)-algebra and (∞,n)-geometry. The closest I could find is this: >While the subject is still young, visible at the horizon is its role in higher topos theory. Where (∞,1)-toposes regarded as (∞,1)-categories of (∞,1)-sheaves/∞-stacks are by now fairly well understood, it is clear that the (∞,2)-categories of (∞,2)-sheaves – such as the codomain fibration/self-indexing of an (∞,1)-topos – will form an (∞,2)-topos in generalization of the non-homotopic notion of 2-topos. And so on. And I absolutely understand that there isn't much to mention yet! As "the subject is still young". The purpose of this thread is to discuss and gather opinions on the matters. What do people think about possible (∞,n)-categorical algebra and geometry (subsuming topos theory)?

   There’s plenty of information on the nLab, e.g., start at (infinity,n)-category.

   I’m sorry, it’s either you misunderstood me, or I’m misunderstanding you. For example, the article on (∞,n)-categories on nLab contains a great deal of technical information about constructions of (∞,n)-categories and such. There is little to none information about what I’ve asked: about (∞,n)-algebra and (∞,n)-geometry. The closest I could find is this:

   While the subject is still young, visible at the horizon is its role in higher topos theory. Where (∞,1)-toposes regarded as (∞,1)-categories of (∞,1)-sheaves/∞-stacks are by now fairly well understood, it is clear that the (∞,2)-categories of (∞,2)-sheaves – such as the codomain fibration/self-indexing of an (∞,1)-topos – will form an (∞,2)-topos in generalization of the non-homotopic notion of 2-topos. And so on.

   And I absolutely understand that there isn’t much to mention yet! As “the subject is still young”.

   The purpose of this thread is to discuss and gather opinions on the matters. What do people think about possible (∞,n)-categorical algebra and geometry (subsuming topos theory)?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeDec 12th 2016
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   Author: Mike Shulman
   Format: MarkdownItex$(\infty,n)$-algebra should certainly exist and be interesting. I expect it to meld features of the fairly well-understood subjects of $(\infty,1)$-algebra and 2-algebra. For instance, an $(\infty,2)$-monad will have three $(\infty,2)$-categories of algebras and pseudo, lax, or colax morphisms, that can be assembled into enhanced $(\infty,2)$-categorical structures like "$\infty,F$-categories" and "$\infty$-double categories". I don't know of anyone who's written any of this down yet; there may not be a whole lot of overlap between the people interested in $(\infty,1)$-algebra and 2-algebra. For $n\gt 2$ there could in principle be "laxness at all dimensions", but I don't know whether to expect that to be interesting. Even just 3-algebra is not very well-developed. For geometry the situation is rather murkier; even "2-geometry" and 2-toposes (meaning $(2,2)$-toposes) are not very well-understood.

   $(\infty,n)$-algebra should certainly exist and be interesting. I expect it to meld features of the fairly well-understood subjects of $(\infty,1)$-algebra and 2-algebra. For instance, an $(\infty,2)$-monad will have three $(\infty,2)$-categories of algebras and pseudo, lax, or colax morphisms, that can be assembled into enhanced $(\infty,2)$-categorical structures like “$\infty,F$-categories” and “$\infty$-double categories”. I don’t know of anyone who’s written any of this down yet; there may not be a whole lot of overlap between the people interested in $(\infty,1)$-algebra and 2-algebra. For $n\gt 2$ there could in principle be “laxness at all dimensions”, but I don’t know whether to expect that to be interesting. Even just 3-algebra is not very well-developed.

   For geometry the situation is rather murkier; even “2-geometry” and 2-toposes (meaning $(2,2)$-toposes) are not very well-understood.

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeDec 13th 2016
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   Author: Tim_Porter
   Format: MarkdownItexThere are perhaps the beginning shoots of a combinatorial $(\infty,n)$-algebra in some ideas in rewriting theory. (I mean analogous to combinatorial group theory.) For example looking at presentations of monoidal categories in which the analogue of the free structures are in higher dimensions than in, say, the one or two dimensions that one might expect, so, for instance, there has to be an explicit interchanger and higher dimensional words involving that. There are some quite interesting complications that arise and this links with work on the homology and cohomology of monoids. My problem with this is to find ways of thinking about the higher algebra that results.

   There are perhaps the beginning shoots of a combinatorial $(\infty,n)$-algebra in some ideas in rewriting theory. (I mean analogous to combinatorial group theory.) For example looking at presentations of monoidal categories in which the analogue of the free structures are in higher dimensions than in, say, the one or two dimensions that one might expect, so, for instance, there has to be an explicit interchanger and higher dimensional words involving that. There are some quite interesting complications that arise and this links with work on the homology and cohomology of monoids. My problem with this is to find ways of thinking about the higher algebra that results.