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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 19th 2012
   • (edited Mar 19th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexReaders here may have noticed that recently I was getting interested in _internal_ higher category theory in an $(\infty,1)$-topos. Underlying this interest is some big question at the horizon, that I would like to reach eventually. For my own sake, I have started to make some preliminary notes on what that question is, mainly so as to organize my own thoughts. After writing it I thought that maybe somebody around here might (or might not) be interested in chatting about this. In any case, the notes are now * [here](http://ncatlab.org/schreiber/show/next). (Just a small Wiki-page on my personal web for the moment.)

   Readers here may have noticed that recently I was getting interested in internal higher category theory in an $(\infty,1)$-topos. Underlying this interest is some big question at the horizon, that I would like to reach eventually.

   For my own sake, I have started to make some preliminary notes on what that question is, mainly so as to organize my own thoughts. After writing it I thought that maybe somebody around here might (or might not) be interested in chatting about this. In any case, the notes are now

   • here.

   (Just a small Wiki-page on my personal web for the moment.)

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 20th 2012
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   Author: David_Corfield
   Format: MarkdownItexSo the idea is that there's still much more common internal structure to be located in these cohesive $(\infty, 1)$-toposes? At some point presumably you have to deal with what is specific to different kinds of cohesiveness. Are these specifics likely to be of considerable importance, as one might claim the specifics of Lie groups or topological groups or abelian groups are considerable relative to what is common to them as groups internal to different categories?

   So the idea is that there’s still much more common internal structure to be located in these cohesive $(\infty, 1)$-toposes? At some point presumably you have to deal with what is specific to different kinds of cohesiveness. Are these specifics likely to be of considerable importance, as one might claim the specifics of Lie groups or topological groups or abelian groups are considerable relative to what is common to them as groups internal to different categories?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2012
   • (edited Mar 22nd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> So the idea is that there's still much more common internal structure to be located in these cohesive (∞,1)-toposes? Yes, so far we looked at the $\infty$-groups and $\infty$-groupoids in the cohesive $\infty$-topos. This is only the first step in a tower of notions. Notably every cohesive $\infty$-topos will come with its notion of an internal category of quasi-coherent $\infty$-stacks that plays the role of the intrinsic category of cohesive modules / vector spaces. It's pretty clear how this will work, but a comprehensive account is missing at the moment. But there is much more. As I try to indicate in [those notes](http://ncatlab.org/schreiber/show/next), the fact that we find that every cohesive $\infty$-topos comes with its intrinsic notions of "extended action functionals" (refined higher Chern-Weil homomorphisms) makes it compelling to think about their quantization, which needs these higher categorical structures. The need for this is clear already if you simply view the [[cobordism hypothesis]] in its role in physics. Marvelous as it is, at the moment it only gives categories of "topological" cobordisms (actually smooth ones, but that's the way the terminology goes) where the extra structure is restricted, currently, to be, essentially, given by homotopy classes of maps into a topological space $X$: $Bord_n(X)$ is the free symmetric monoidal $(\infty,n)$-category with duals on the topological space $X$. which is really here to be regarded as a [[discrete infinity-groupoid]]. This gives [[HQFT]], but not more. We know that there are / we need categories of cobordisms with more structure than that, for instance with metric structure, or equipped with the structure of non-flat smooth $G$-bundles with connection. All such structures are obtained by homming not into a discrete $\infty$-groupoid, but by homming into a cohesive $\infty$-groupoid. This was already amplified in the thesis of David Ayala, linked to at _[[category of cobordisms]]_. But just for the 1-category of cobordisms. Looking at the situation, I think it is clear that a compelling task is to consider the free-forgetful adjunction between symmetric-monoidal-with-duals and bare $(\infty,n)$-categories not just internal to discrete $\infty$-groupoids / topological spaces, but internal to general cohesive $\infty$-toposes. Not that I am claiming I can do that. But I am interested in working my way in that rough direction.

   So the idea is that there’s still much more common internal structure to be located in these cohesive (∞,1)-toposes?

   Yes, so far we looked at the $\infty$-groups and $\infty$-groupoids in the cohesive $\infty$-topos. This is only the first step in a tower of notions.

   Notably every cohesive $\infty$-topos will come with its notion of an internal category of quasi-coherent $\infty$-stacks that plays the role of the intrinsic category of cohesive modules / vector spaces. It’s pretty clear how this will work, but a comprehensive account is missing at the moment.

   But there is much more. As I try to indicate in those notes, the fact that we find that every cohesive $\infty$-topos comes with its intrinsic notions of “extended action functionals” (refined higher Chern-Weil homomorphisms) makes it compelling to think about their quantization, which needs these higher categorical structures.

   The need for this is clear already if you simply view the cobordism hypothesis in its role in physics. Marvelous as it is, at the moment it only gives categories of “topological” cobordisms (actually smooth ones, but that’s the way the terminology goes) where the extra structure is restricted, currently, to be, essentially, given by homotopy classes of maps into a topological space $X$: $Bord_n(X)$ is the free symmetric monoidal $(\infty,n)$-category with duals on the topological space $X$. which is really here to be regarded as a discrete infinity-groupoid.

   This gives HQFT, but not more. We know that there are / we need categories of cobordisms with more structure than that, for instance with metric structure, or equipped with the structure of non-flat smooth $G$-bundles with connection. All such structures are obtained by homming not into a discrete $\infty$-groupoid, but by homming into a cohesive $\infty$-groupoid. This was already amplified in the thesis of David Ayala, linked to at category of cobordisms. But just for the 1-category of cobordisms. Looking at the situation, I think it is clear that a compelling task is to consider the free-forgetful adjunction between symmetric-monoidal-with-duals and bare $(\infty,n)$-categories not just internal to discrete $\infty$-groupoids / topological spaces, but internal to general cohesive $\infty$-toposes.

   Not that I am claiming I can do that. But I am interested in working my way in that rough direction.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 22nd 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexHmm, so when is one just working on a generalized cobordism hypothesis, or more generally, [generalized tangle hypothesis](http://ncatlab.org/nlab/show/generalized+tangle+hypothesis) in the standard setting? >The $k$-tuply monoidal $n$-category of $G$-structured $n$-tangles in the $(n + k)$-cube is the [[fundamental category|fundamental]] $(n + k)$-category with duals of $(M G,Z)$. >* $M G$ is the Thom space of group $G$. >* $G$ can be any group equipped with a homomorphism to $O(k)$. And when would it be better to say one is working on an internal cobordism hypothesis in a different cohesive setting? I guess that's what's being asked [here](http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c020220). In which case the difference would be when "the structures at hand are geometric and not just tangential".

   Hmm, so when is one just working on a generalized cobordism hypothesis, or more generally, generalized tangle hypothesis in the standard setting?

   The $k$-tuply monoidal $n$-category of $G$-structured $n$-tangles in the $(n + k)$-cube is the fundamental $(n + k)$-category with duals of $(M G,Z)$.

   • $M G$ is the Thom space of group $G$.
   • $G$ can be any group equipped with a homomorphism to $O(k)$.

   And when would it be better to say one is working on an internal cobordism hypothesis in a different cohesive setting?

   I guess that’s what’s being asked here. In which case the difference would be when “the structures at hand are geometric and not just tangential”.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2012
   • (edited Mar 22nd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> so when Essentially everything that has been said in print so far, with the exception of Ayala's thesis, concerns the case "internal to discrete $\infty$-groupoids" only. (With "only" to be read in quotation marks, of course.) In these setups, the extra structure on the cobordisms is always "topological structure". In as far as it is expressed in terms of tangent bundles, it is always structure on tangent bundles that can equivalently be formulated in terms of structure groups that the tangent bundle is associated to. I didn't remember your old question [here](http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c020220) that you point to. Fascinating that you were already pushing in this direction back then. I didn't realize this before. So I think the answer to your final question there is "yes": there should certainly be $(\infty,n)$-categories of cobirdisms that are equipped with a map not just into a topological space / discrete $\infty$-groupoid, but with a map into a more general $\infty$-stack over some given site. And when the context is well-behaved enough, then I expect that these will again be free symmetric symmetric monoidal with duals on that $\infty$-stack.

   so when

   Essentially everything that has been said in print so far, with the exception of Ayala’s thesis, concerns the case “internal to discrete $\infty$-groupoids” only. (With “only” to be read in quotation marks, of course.)

   In these setups, the extra structure on the cobordisms is always “topological structure”. In as far as it is expressed in terms of tangent bundles, it is always structure on tangent bundles that can equivalently be formulated in terms of structure groups that the tangent bundle is associated to.

   I didn’t remember your old question here that you point to. Fascinating that you were already pushing in this direction back then. I didn’t realize this before.

   So I think the answer to your final question there is “yes”: there should certainly be $(\infty,n)$-categories of cobirdisms that are equipped with a map not just into a topological space / discrete $\infty$-groupoid, but with a map into a more general $\infty$-stack over some given site. And when the context is well-behaved enough, then I expect that these will again be free symmetric symmetric monoidal with duals on that $\infty$-stack.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 23rd 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexDo Ayala's geometric structures match up with your cohesive environments?

   Do Ayala’s geometric structures match up with your cohesive environments?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 23rd 2012
   • (edited Mar 23rd 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Do Ayala's geometric structures match up with your cohesive environments? He considers sheaves on the site which is the subcategory of the category of differentiable manifolds on the embedding morphisms. That is cohesive, yes. (A [[dense subsite]] which is an [[infinity-cohesive site]] is the category [[CartSp]]${}_{emb}$ of Cartesian spaces and embeddings between them.) I haven't gone through the details of his proofs, but I expect that the restriction to inclusions is mainly there to make objects like the "sheaf of symplectic structures" exist, which, due to the non-degeneracy condition involved in symplectic structure, is not a presheaf on all smooth maps between manifolds. For the example that I find most interesting, namely $\infty$-sheaves of smooth $\infty$-connections, this is not an issue.

   Do Ayala’s geometric structures match up with your cohesive environments?

   He considers sheaves on the site which is the subcategory of the category of differentiable manifolds on the embedding morphisms. That is cohesive, yes. (A dense subsite which is an infinity-cohesive site is the category CartSp${}_{emb}$ of Cartesian spaces and embeddings between them.)

   I haven’t gone through the details of his proofs, but I expect that the restriction to inclusions is mainly there to make objects like the “sheaf of symplectic structures” exist, which, due to the non-degeneracy condition involved in symplectic structure, is not a presheaf on all smooth maps between manifolds. For the example that I find most interesting, namely $\infty$-sheaves of smooth $\infty$-connections, this is not an issue.

8. • CommentRowNumber8.
   • CommentAuthorStephan A Spahn
   • CommentTimeMar 25th 2012
   • (edited Mar 25th 2012)
   • PermaLink
   Author: Stephan A Spahn
   Format: MarkdownItex> Notably every cohesive ∞-topos will come with its notion of an internal category of quasi-coherent ∞-stacks In a cohesive ∞-topos $H$ equipped with infinitesimal cohesion e.g. the class of formally étale morphisms in $H$ satisfies three axioms -namely 2,3,5 in the definition of a class of [[open map]]s. By means of any class $S$ of $H$-morphisms satisfying these axioms one can define a stack $S_{(-)}$ on $H$ defined by: $S_X$ is the sub ∞-category of $H/X$ for $X\in H$ on the morphisms in $S$. This is explained (in the 1-dimensional case) in Remark 3.2 [here](http://www2.math.uu.se/research/pub/Palmgren1.pdf). As indicated there this construction has a component related to [[internal subcategory|internal subcategories]] of $H$ (and the internal logic of $H$). But I don't know if this is appropriately put into perspective of [next](http://ncatlab.org/schreiber/show/next).

   Notably every cohesive ∞-topos will come with its notion of an internal category of quasi-coherent ∞-stacks

   In a cohesive ∞-topos $H$ equipped with infinitesimal cohesion e.g. the class of formally étale morphisms in $H$ satisfies three axioms -namely 2,3,5 in the definition of a class of open maps. By means of any class $S$ of $H$-morphisms satisfying these axioms one can define a stack $S_{(-)}$ on $H$ defined by: $S_X$ is the sub ∞-category of $H/X$ for $X\in H$ on the morphisms in $S$. This is explained (in the 1-dimensional case) in Remark 3.2 here. As indicated there this construction has a component related to internal subcategories of $H$ (and the internal logic of $H$).

   But I don’t know if this is appropriately put into perspective of next.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMar 26th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat's a very good point. What I was alluding to with "quasicoherent stacks" was the construction via the [[tangent (infinity,1)-category]], (which is in fact not specific to cohesive contexts). There one fixes a site of definition $C$, so that $\mathbf{H} ) Sh_{(\infty,1)}(C)$. Then one forms the [[codomain fibration]] over $C^{op}$, classified by $A \mapsto C^{op}/A$ and then further passes to the fiberwise [[stabilization]] of this $A \mapsto Stab(C^{op}/A)$. As discussed at _[[tangent (infinity,1)-category]]_ this may be understood as being the $(\infty,1)$-category of [[quasicoherent infinity-stacks]] on $Spec A \in C$. By [[Yoneda extension]] along $C \to Sh_{(\infty,1)}(C)$ this produces an $(\infty,2)$-presheaf on $\mathbf{H}$ itself, which under good conditions will be an [[(infinity,2)-sheaf]]. This construction may well have some nice relation to what you, Stephan, just mentioned, but that may involve some dualization. Here one considers the stabilization of the codomain fibration of the opposite category, whereas in what you mention one consideres the sub-$(\infty,1)$-category of the codomain fibration of the original category.

   That’s a very good point.

   What I was alluding to with “quasicoherent stacks” was the construction via the tangent (infinity,1)-category, (which is in fact not specific to cohesive contexts).

   There one fixes a site of definition $C$, so that $\mathbf{H} ) Sh_{(\infty,1)}(C)$. Then one forms the codomain fibration over $C^{op}$, classified by $A \mapsto C^{op}/A$ and then further passes to the fiberwise stabilization of this $A \mapsto Stab(C^{op}/A)$. As discussed at tangent (infinity,1)-category this may be understood as being the $(\infty,1)$-category of quasicoherent infinity-stacks on $Spec A \in C$. By Yoneda extension along $C \to Sh_{(\infty,1)}(C)$ this produces an $(\infty,2)$-presheaf on $\mathbf{H}$ itself, which under good conditions will be an (infinity,2)-sheaf.

   This construction may well have some nice relation to what you, Stephan, just mentioned, but that may involve some dualization. Here one considers the stabilization of the codomain fibration of the opposite category, whereas in what you mention one consideres the sub-$(\infty,1)$-category of the codomain fibration of the original category.