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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 20th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted [[(infinity,1)-quasitopos]]

   started (infinity,1)-quasitopos

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI just [blogged](http://golem.ph.utexas.edu/category/2012/10/object_classifiers_and_1quasit.html) about the recent Gepner-Kock paper which, among other things, proposes a more general definition of [[(infinity,1)-quasitopos]] than the one on the nLab. If no one objects, then in a little while I may modify the nLab's definition.

   I just blogged about the recent Gepner-Kock paper which, among other things, proposes a more general definition of (infinity,1)-quasitopos than the one on the nLab. If no one objects, then in a little while I may modify the nLab’s definition.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2012
   • (edited Oct 18th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> If no one objects, then in a little while I may modify the nLab’s definition I certainly won't object if we put in a comparative disussion as you do on the blog (And thanks for the summary! That's useful. I haven't found time to study the article in any detail). This all is related to a long standing question that I have been banging my head against with Dave Carchedi: is there a notion of _[[concrete object]]_ in a local $\infty$-topos which is _independent_ of some of these evident choices? We were looking at the $n$-connected/$n$-truncated factorization and trying to see if we could somehow "average over all $n$" or the like to have not just a notion of "$n$-concrete object" but just of "concrete object". The same issue shows up when $\infty$-categorifying Lawvere's axiom for cohesive 1-toposes that we call "discrete objects are concrete".

   If no one objects, then in a little while I may modify the nLab’s definition

   I certainly won’t object if we put in a comparative disussion as you do on the blog (And thanks for the summary! That’s useful. I haven’t found time to study the article in any detail).

   This all is related to a long standing question that I have been banging my head against with Dave Carchedi:

   is there a notion of concrete object in a local $\infty$-topos which is independent of some of these evident choices?

   We were looking at the $n$-connected/$n$-truncated factorization and trying to see if we could somehow “average over all $n$” or the like to have not just a notion of “$n$-concrete object” but just of “concrete object”.

   The same issue shows up when $\infty$-categorifying Lawvere’s axiom for cohesive 1-toposes that we call “discrete objects are concrete”.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeOct 18th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexWhat are you looking to get out of such a definition?

   What are you looking to get out of such a definition?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 30th 2012
   • (edited Oct 30th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSorry for the slow reply. > What are you looking to get out of such a definition? We never were really sure. It just somehow felt like the fact that in a local 1-topos there is a notion of concrete object -- with the geometric meaning this has -- should generalize to a local $\infty$-topos having a notion of concrete $\infty$-groupoids. Concretely, where in the local 1-topos over [[CartSp]] the concrete objects are just the [[diffeological spaces]], a natural question seemed to be: can we intrinsically characterize the full sub-$\infty$-category of $Sh_\infty(CartSp)$ on those objects that are presented by **simplicial diffeological spaces**? Are these maybe somehow related to totally concrete objects in $Sh_\infty(CartSp)$? These were the kinds of ideas that we were toying around with. But it may well be, I think now, that all these questions are going in the wrong direction. I did eventually find a major application for just $n$-concrete objects, and maybe that's all there is to it. (Namely just as 0-concreteness allows to formulate integration in differential cohomology/higher holonomy in a cohesive $\infty$-topos, so $n$-concreteness allows to similarly formulate fiber integration in differential cohomology/transgression. I still need to write out the details of that, though. There is another aspect kicking in here for $n \geq 1$ that makes this statement be more than a blind generalization of the case for $n = 0$.)

   Sorry for the slow reply.

   What are you looking to get out of such a definition?

   We never were really sure. It just somehow felt like the fact that in a local 1-topos there is a notion of concrete object – with the geometric meaning this has – should generalize to a local $\infty$-topos having a notion of concrete $\infty$-groupoids.

   Concretely, where in the local 1-topos over CartSp the concrete objects are just the diffeological spaces, a natural question seemed to be: can we intrinsically characterize the full sub-$\infty$-category of $Sh_\infty(CartSp)$ on those objects that are presented by simplicial diffeological spaces? Are these maybe somehow related to totally concrete objects in $Sh_\infty(CartSp)$?

   These were the kinds of ideas that we were toying around with.

   But it may well be, I think now, that all these questions are going in the wrong direction. I did eventually find a major application for just $n$-concrete objects, and maybe that’s all there is to it.

   (Namely just as 0-concreteness allows to formulate integration in differential cohomology/higher holonomy in a cohesive $\infty$-topos, so $n$-concreteness allows to similarly formulate fiber integration in differential cohomology/transgression. I still need to write out the details of that, though. There is another aspect kicking in here for $n \geq 1$ that makes this statement be more than a blind generalization of the case for $n = 0$.)

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeNov 29th 2012
   • (edited Nov 29th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo I have now a better answer to the above. It comes in two parts: * the first is about explicit structures in differential geometry/cohomology, * the second is about general abstract higher modalities. ## Differential geometry, differential cohomology If $G$ is a braided $\infty$-group in a cohesive $\infty$-topos, write $\mathbf{B}G_{conn}$, as usual, for the coefficients for differential $G$-cohomology and let $X$ be any other object. Then then mapping space object $$ [X, \mathbf{B}G_{conn}] \in \mathbf{H} $$ looks a bit like the "moduli stack of $G$-connections on $X$". Indeed, its global points are exactly the $G$-connections on $X$. But its cohesion is not quite what one needs for some applications: for some applications, we want the moduli $\infty$-stack of $G$-connections to have $U$-plots which are cohesive $U$-parameterized collections of $G$-connections on $X$, of $G$-gauge transformations on $X$, etc. But the $U$-plots of $[X, \mathbf{B}G_{conn}]$ are instead, of course, $G$-connections on $U \times X$, gauge transformations on $U \times X$. That's closely related, but is not quite the same. In a rough sense, the right moduli object should be obtained by _degreewise concretification_ of $[X, \mathbf{B}G_{conn}]$: it should be like $[X, \mathbf{B}G_{conn}]$ _pointwise_ but otherwise just vary cohesively. I'll write $G\mathbf{Conn}(X)$ for that genuine object of differential moduli. There will be a canonical projection $$ conc \colon [X, \mathbf{B}G_{conn}] \to G \mathbf{Conn}(X) \,. $$ In standard models, for $X$ an ordinary manifold etc. it is clear what this should be. The question is how to says this abstractly using just the cohesion, such as to know what this means for general $X$ and general $G$ and general ambient cohesive contexts. ## Higher modalities It was clear all along that the $\sharp$-modality of the cohesion should induce the construction of $G \mathbf{Conn}(X)$. Now I think that I have proven that it indeed comes out, like this: By the construction of the forgetful map $\mathbf{B}G_{conn} \to \mathbf{B}G$ (that forgets the connection and only remembers the underlying bundle), it factors canonically as a tower $$ \mathbf{B}G_{conn} \to \mathbf{B}G_{conn^{n-1}} \to \mathbf{B}G_{conn^{n-2 }} \to \cdots \to \mathbf{B}G_{conn^{0}} \simeq \mathbf{B}G \,. $$ This tower "forgets connection data degreewise", first the $n$-form pieces, then then $(n-1)$-form pieces, and so on. For each stage $[X, \mathbf{B}G_{conn^k}]$ we have the [[n-image]]/[[Postnikov system]] factorization of its $\sharp$-modality unit, which I write $$ [X, \mathbf{B}G_{conn^k}] \simeq \sharp_\infty [X, \mathbf{B}G_{conn^k}] \to \cdots \to \sharp_2 [X, \mathbf{B}G_{conn^k}] \to \sharp_1 [X, \mathbf{B}G_{conn^k}] \to \sharp_0 [X, \mathbf{B}G_{conn^k}] \simeq \sharp [X, \mathbf{B}G_{conn^k}] \,. $$ Now the claim is that the differential moduli object $G\mathbf{Conn}(X)$ is the following iterated homotopy fiber product of "modality-images" $$ G \mathbf{Conn}(X) \simeq \sharp_1[X, \mathbf{B}G_{conn^n}] \underset{\sharp_1 [X, \mathbf{B}G_{conn^{n-1}}]}{\times} \sharp_2[X, \mathbf{B}G_{conn^{n-1}}] \underset{\sharp_2 [X, \mathbf{B}G_{conn^{n-3}}]}{\times} \cdots \underset{\sharp_{n-1} [X, \mathbf{B}G_{conn^{1}}]}{\times} [X, \mathbf{B}G] \,. $$ For instance for 1-truncated $G$ and hence $n = 2$ this is the $\infty$-limit over $$ \array{ \sharp_1[X, \mathbf{B}G_{conn}] && && \sharp_2[X, \mathbf{B}G_{conn^1}] && && [X, \mathbf{B}G] \\ & \searrow && \swarrow && \searrow && \swarrow && \\ && \sharp_1 [X, \mathbf{B}G_{conn^1}] && && \sharp_2[X, \mathbf{B}G] } \,. $$ Clearly this construction is just a special case of a general construction that exists for any higher modality and a given tower of morphisms. A natural question seems to be: is this general construction something of more general relevance?

   So I have now a better answer to the above. It comes in two parts:

   • the first is about explicit structures in differential geometry/cohomology,

   • the second is about general abstract higher modalities.

   Differential geometry, differential cohomology

   If $G$ is a braided $\infty$-group in a cohesive $\infty$-topos, write $\mathbf{B}G_{conn}$, as usual, for the coefficients for differential $G$-cohomology and let $X$ be any other object.

   Then then mapping space object

   $$[X, \mathbf{B}G_{conn}] \in \mathbf{H}$$

   looks a bit like the “moduli stack of $G$-connections on $X$”. Indeed, its global points are exactly the $G$-connections on $X$. But its cohesion is not quite what one needs for some applications:

   for some applications, we want the moduli $\infty$-stack of $G$-connections to have $U$-plots which are cohesive $U$-parameterized collections of $G$-connections on $X$, of $G$-gauge transformations on $X$, etc. But the $U$-plots of $[X, \mathbf{B}G_{conn}]$ are instead, of course, $G$-connections on $U \times X$, gauge transformations on $U \times X$. That’s closely related, but is not quite the same.

   In a rough sense, the right moduli object should be obtained by degreewise concretification of $[X, \mathbf{B}G_{conn}]$: it should be like $[X, \mathbf{B}G_{conn}]$ pointwise but otherwise just vary cohesively.

   I’ll write $G\mathbf{Conn}(X)$ for that genuine object of differential moduli. There will be a canonical projection

   $$conc \colon [X, \mathbf{B}G_{conn}] \to G \mathbf{Conn}(X) \,.$$

   In standard models, for $X$ an ordinary manifold etc. it is clear what this should be. The question is how to says this abstractly using just the cohesion, such as to know what this means for general $X$ and general $G$ and general ambient cohesive contexts.

   Higher modalities

   It was clear all along that the $\sharp$-modality of the cohesion should induce the construction of $G \mathbf{Conn}(X)$. Now I think that I have proven that it indeed comes out, like this:

   By the construction of the forgetful map $\mathbf{B}G_{conn} \to \mathbf{B}G$ (that forgets the connection and only remembers the underlying bundle), it factors canonically as a tower

   $$\mathbf{B}G_{conn} \to \mathbf{B}G_{conn^{n-1}} \to \mathbf{B}G_{conn^{n-2 }} \to \cdots \to \mathbf{B}G_{conn^{0}} \simeq \mathbf{B}G \,.$$

   This tower “forgets connection data degreewise”, first the $n$-form pieces, then then $(n-1)$-form pieces, and so on.

   For each stage $[X, \mathbf{B}G_{conn^k}]$ we have the n-image/Postnikov system factorization of its $\sharp$-modality unit, which I write

   $$[X, \mathbf{B}G_{conn^k}] \simeq \sharp_\infty [X, \mathbf{B}G_{conn^k}] \to \cdots \to \sharp_2 [X, \mathbf{B}G_{conn^k}] \to \sharp_1 [X, \mathbf{B}G_{conn^k}] \to \sharp_0 [X, \mathbf{B}G_{conn^k}] \simeq \sharp [X, \mathbf{B}G_{conn^k}] \,.$$

   Now the claim is that the differential moduli object $G\mathbf{Conn}(X)$ is the following iterated homotopy fiber product of “modality-images”

   $$G \mathbf{Conn}(X) \simeq \sharp_1[X, \mathbf{B}G_{conn^n}] \underset{\sharp_1 [X, \mathbf{B}G_{conn^{n-1}}]}{\times} \sharp_2[X, \mathbf{B}G_{conn^{n-1}}] \underset{\sharp_2 [X, \mathbf{B}G_{conn^{n-3}}]}{\times} \cdots \underset{\sharp_{n-1} [X, \mathbf{B}G_{conn^{1}}]}{\times} [X, \mathbf{B}G] \,.$$

   For instance for 1-truncated $G$ and hence $n = 2$ this is the $\infty$-limit over

   $$\array{ \sharp_1[X, \mathbf{B}G_{conn}] && && \sharp_2[X, \mathbf{B}G_{conn^1}] && && [X, \mathbf{B}G] \\ & \searrow && \swarrow && \searrow && \swarrow && \\ && \sharp_1 [X, \mathbf{B}G_{conn^1}] && && \sharp_2[X, \mathbf{B}G] } \,.$$

   Clearly this construction is just a special case of a general construction that exists for any higher modality and a given tower of morphisms. A natural question seems to be: is this general construction something of more general relevance?

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeNov 30th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexFascinating! But no, I don't think I've ever seen anything like that before.

   Fascinating! But no, I don’t think I’ve ever seen anything like that before.