Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Forgot your password?
• Apply for membership

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 5th 2013
   • (edited Apr 5th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt seems clear that the process of forming [[groupoid convolution algebras]] wants to be a 2-functor from the (2,1)-category of differentiable stacks to a 2-category of $C^\ast$-algebras, and $C^\ast$-bimodules between them. Aspects of this have been discussed in the literature. A commented list of some of the relevant references that I am aware of is [here](http://ncatlab.org/nlab/show/category+algebra#ExtensionToBibundlesAndBimodules) in the above entry. Given that, it seems fairly clear how to combine the available literature to produce this 2-functor. But I would still like to know: has this been considered and published before? Furthermore, for applications it is desireable to refine this 2-functor to one whose domain also admits differentiable stacks equipped with a smooth 3-cocycle/circle 2-group-principal 2-bundle/circle-bundle gerbe and sends them to the corresponding twisted convolution algebras (of sections of the corresponding transition line bundles). Again, it seems fairly straightforward, if somewhat tedious, to write that out and check that it behaves as expected. But: has this not been done before somewhere? In either case, if anyone has further pointers to work that disusses aspects of this point, we should add it to that discussion in the nLab entry. I am also collecting some related references at _[[bibundle]]_.

   It seems clear that the process of forming groupoid convolution algebras wants to be a 2-functor from the (2,1)-category of differentiable stacks to a 2-category of $C^\ast$-algebras, and $C^\ast$-bimodules between them.

   Aspects of this have been discussed in the literature. A commented list of some of the relevant references that I am aware of is here in the above entry.

   Given that, it seems fairly clear how to combine the available literature to produce this 2-functor. But I would still like to know: has this been considered and published before?

   Furthermore, for applications it is desireable to refine this 2-functor to one whose domain also admits differentiable stacks equipped with a smooth 3-cocycle/circle 2-group-principal 2-bundle/circle-bundle gerbe and sends them to the corresponding twisted convolution algebras (of sections of the corresponding transition line bundles). Again, it seems fairly straightforward, if somewhat tedious, to write that out and check that it behaves as expected. But: has this not been done before somewhere?

   In either case, if anyone has further pointers to work that disusses aspects of this point, we should add it to that discussion in the nLab entry. I am also collecting some related references at bibundle.

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 7th 2013
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownA minor remark: if one expects a functor valued in C\*-algebras as opposed to mere C\*-categories, then the source bicategory should be the bicategory of _presented_ differential stacks. A presentation of a differential stack then gives an element in the Morita equivalence class of generators in the convolution C\*-category. Of course, there is also a functor from differential stacks to C\*-categories (without a canonical choice of a generator). Recently I did quite a bit of search on this topic, but was unable to find anything that is not mentioned in the [[bibundle]] article. (But I would consider adding Christian Blohmann's nicely written article “Stacky Lie groups” to the list of references of the [[bibundle]] article.) Conjecturally, the bicategory of Lie groupoids is actually equivalent to a full subbicategory of the bicategory of Hopfish algebras (like in Weinstein's paper, but appropriately weakened) via the equivalence functor given by the convolution construction in the same way as the category of smooth manifolds is equivalent to a full subcategory of the category of algebras via the equivalence functor given by the algebra of real valued functions construction.

   A minor remark: if one expects a functor valued in C*-algebras as opposed to mere C*-categories, then the source bicategory should be the bicategory of presented differential stacks. A presentation of a differential stack then gives an element in the Morita equivalence class of generators in the convolution C*-category. Of course, there is also a functor from differential stacks to C*-categories (without a canonical choice of a generator).

   Recently I did quite a bit of search on this topic, but was unable to find anything that is not mentioned in the bibundle article.

   (But I would consider adding Christian Blohmann's nicely written article “Stacky Lie groups” to the list of references of the bibundle article.)

   Conjecturally, the bicategory of Lie groupoids is actually equivalent to a full subbicategory of the bicategory of Hopfish algebras (like in Weinstein's paper, but appropriately weakened) via the equivalence functor given by the convolution construction in the same way as the category of smooth manifolds is equivalent to a full subcategory of the category of algebras via the equivalence functor given by the algebra of real valued functions construction.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 7th 2013
   • (edited Apr 7th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks for the reaction! I will add Christian's article to the references, good point. Concerning the presentation issue: I would think that this will be taken care of by the (2,1)-functoriality: the (2,1)-category of differentiable stacks is equivalent to a (2,1)-category of presented differentiable stacks, so we can define the construction there. But this relates to the following question, which I would love to know the answer to: what is the _intrinsic_ characterization of the resulting convolution algebra (2,1)-functor, one that does not require us to talk about Haar measures or half-densities and integration? In the case of discrete geometry there is a beautiful characterization of groupoid convolution algebras claimed in [[Topological Quantum Field Theories from Compact Lie Groups|FHLT]]: these are just the homotopy colimits over the 2-functor on the groupoid constant on the unit 2-vector space (2-module). And the twisted convolution algebras are just the homotopy colimits over the coresponding non-constant 2-functor which is the coefficient system/line 2-bundle. An abstract characterization along these lines but for the smooth case would be beautiful. It is tempting to speculate here a bit: following the references by Klaas Landsman cited at [[groupoid convolution algebra]] (and at [[geometric quantization by push-forward]]) one is naturally led to wonder what it takes to make the groupoid convolution (2,1)-functor lift from $C^\ast$-algebras with bimodules between them to cocycles in [[KK-theory]], hence to make the bimodules into Hilbert bimodules and equip them with a weak Fredholm module structure on the left. By the universal characterization of KK-theory by localization, in this case we'd be talking about a (2,1)-functor from differentiable stacks equipped with some extra "quantization" structure to a stabilized split-exact homotopy-invariant localization of "noncommutative topological spaces" at the duals of compact operators. This way we'd have a construction that clearly wants to be formulated abstractly without recourse to any measure theory, functional analysis etc. As Landsman rightly argues, such a (2,1)-functor and such a characterization seem to be "in the air" and be quite desireable.

   Thanks for the reaction!

   I will add Christian’s article to the references, good point.

   Concerning the presentation issue: I would think that this will be taken care of by the (2,1)-functoriality: the (2,1)-category of differentiable stacks is equivalent to a (2,1)-category of presented differentiable stacks, so we can define the construction there.

   But this relates to the following question, which I would love to know the answer to:

   what is the intrinsic characterization of the resulting convolution algebra (2,1)-functor, one that does not require us to talk about Haar measures or half-densities and integration?

   In the case of discrete geometry there is a beautiful characterization of groupoid convolution algebras claimed in FHLT: these are just the homotopy colimits over the 2-functor on the groupoid constant on the unit 2-vector space (2-module). And the twisted convolution algebras are just the homotopy colimits over the coresponding non-constant 2-functor which is the coefficient system/line 2-bundle.

   An abstract characterization along these lines but for the smooth case would be beautiful.

   It is tempting to speculate here a bit: following the references by Klaas Landsman cited at groupoid convolution algebra (and at geometric quantization by push-forward) one is naturally led to wonder what it takes to make the groupoid convolution (2,1)-functor lift from $C^\ast$-algebras with bimodules between them to cocycles in KK-theory, hence to make the bimodules into Hilbert bimodules and equip them with a weak Fredholm module structure on the left. By the universal characterization of KK-theory by localization, in this case we’d be talking about a (2,1)-functor from differentiable stacks equipped with some extra “quantization” structure to a stabilized split-exact homotopy-invariant localization of “noncommutative topological spaces” at the duals of compact operators. This way we’d have a construction that clearly wants to be formulated abstractly without recourse to any measure theory, functional analysis etc. As Landsman rightly argues, such a (2,1)-functor and such a characterization seem to be “in the air” and be quite desireable.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 10th 2013
   • PermaLink
   Author: Dmitri Pavlov
   Format: Markdown>An abstract characterization along these lines but for the smooth case would be beautiful. I once did a computation that seemed to indicate that reinterpreting everything in the bicategory of smooth stacks (sheaves of categories on the site of smooth manifold), in particular, reinterpreting left/right Kan extensions in this bicategory, and computing the relevant smooth right Kan extension (in our case the smooth homotopy limit) gives one the smooth convolution algebra. Similarly, smooth left Kan extensions seem to give the distributional convolution algebra. In the categorical dimension 1 this is also consistent with Anders Kock's paper [Commutative monads as a theory of distributions](http://www.tac.mta.ca/tac/volumes/26/4/26-04.pdf), in which (if I am not mistaken) he essentially proves (but phrases it in a different language) that the smooth vector space of distributions on X is the smooth colimit of the trivial functor on the smooth set X that sends everything to complex numbers. >what it takes to make the groupoid convolution (2,1)-functor lift from C∗-algebras with bimodules between them to cocycles in KK-theory, hence to make the bimodules into Hilbert bimodules and equip them with a weak Fredholm module structure on the left Such a lift would in particular require a strict model for cocycles in KK-theory, and I think Bram Mesland's paper is a step in the right direction. Essentially the point is that one needs a _connection_ on the bibundle to get a morphism in KK.

   An abstract characterization along these lines but for the smooth case would be beautiful.

   I once did a computation that seemed to indicate that reinterpreting everything in the bicategory of smooth stacks (sheaves of categories on the site of smooth manifold), in particular, reinterpreting left/right Kan extensions in this bicategory, and computing the relevant smooth right Kan extension (in our case the smooth homotopy limit) gives one the smooth convolution algebra. Similarly, smooth left Kan extensions seem to give the distributional convolution algebra.

   In the categorical dimension 1 this is also consistent with Anders Kock's paper Commutative monads as a theory of distributions, in which (if I am not mistaken) he essentially proves (but phrases it in a different language) that the smooth vector space of distributions on X is the smooth colimit of the trivial functor on the smooth set X that sends everything to complex numbers.

   what it takes to make the groupoid convolution (2,1)-functor lift from C∗-algebras with bimodules between them to cocycles in KK-theory, hence to make the bimodules into Hilbert bimodules and equip them with a weak Fredholm module structure on the left

   Such a lift would in particular require a strict model for cocycles in KK-theory, and I think Bram Mesland's paper is a step in the right direction. Essentially the point is that one needs a connection on the bibundle to get a morphism in KK.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 10th 2013
   • (edited Apr 10th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHey, wow, that's a nice picture that you are alluding to, there. Just to see if I understand what you mean: you are thinking of embedding smooth monoids $A$ as their smooth delooping stacks-of-categories $\mathbf{B}A$ into the [[2-topos]] over smooth manifolds. Then for $\mathcal{G}$ a smooth groupoid you consider (as the basic example) the map $const_{\mathbf{B}A} \colon \mathcal{G} \to \mathbf{Cat}$ from the smooth groupoid to the internal universe stack of small stacks (I suppose) constant on $\mathbf{B}k$, and then Kan-extend that to a map $ \lim (const_{\mathbf{B}K})\; \colon \;* \to \mathbf{Cat}$ out of the terminal object. You claim that this map then classifies $\mathbf{B}(-)$ of the smooth (distributional) convolution algebra. Is that what you mean? Roughly? That would be awesome. I need to think about this. Will also have to study in more detail the two references that you point out. Had only skimmed them so far. Thanks for that message.

   Hey, wow, that’s a nice picture that you are alluding to, there.

   Just to see if I understand what you mean: you are thinking of embedding smooth monoids $A$ as their smooth delooping stacks-of-categories $\mathbf{B}A$ into the 2-topos over smooth manifolds. Then for $\mathcal{G}$ a smooth groupoid you consider (as the basic example) the map $const_{\mathbf{B}A} \colon \mathcal{G} \to \mathbf{Cat}$ from the smooth groupoid to the internal universe stack of small stacks (I suppose) constant on $\mathbf{B}k$, and then Kan-extend that to a map $\lim (const_{\mathbf{B}K})\; \colon \;* \to \mathbf{Cat}$ out of the terminal object. You claim that this map then classifies $\mathbf{B}(-)$ of the smooth (distributional) convolution algebra.

   Is that what you mean? Roughly? That would be awesome.

   I need to think about this. Will also have to study in more detail the two references that you point out. Had only skimmed them so far.

   Thanks for that message.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 11th 2013
   • (edited Apr 11th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI looked at "Commutative monads as a theory of distributions" ([arxiv/1108.5952](http://arxiv.org/abs/1108.5952)) now and thought a bit about what you said, but I can't really see it yet. So Frölicher-Kriegl observed that forming "linear functions on a space of non-linear functions" has a natural expression in terms of a certain monad for vector spaces and the above article explores this monad construction further, more generally. Are you maybe thinking of that formula $$ (X \pitchfork B) \pitchfork_T B $$ which appears on pages 10, 12 as something to be reminiscent of parts of an end-formula for an internal Kan extension?? Or something like this? You may have to help me here, I don't see yet how there is an internal colimit over a constant map hidden in that article. Sorry for being dense.

   I looked at “Commutative monads as a theory of distributions” (arxiv/1108.5952) now and thought a bit about what you said, but I can’t really see it yet.

   So Frölicher-Kriegl observed that forming “linear functions on a space of non-linear functions” has a natural expression in terms of a certain monad for vector spaces and the above article explores this monad construction further, more generally.

   Are you maybe thinking of that formula

   $$(X \pitchfork B) \pitchfork_T B$$

   which appears on pages 10, 12 as something to be reminiscent of parts of an end-formula for an internal Kan extension?? Or something like this?

   You may have to help me here, I don’t see yet how there is an internal colimit over a constant map hidden in that article. Sorry for being dense.

7. • CommentRowNumber7.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 11th 2013
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownYes, the description is correct, except that limits give smooth functions / smooth sections (with convolution), whereas colimits give (smooth) distributions / distributional sections (with convolution). (In the finite situation there is no difference between colimits and limits or between left and right Kan extensions, but in our case there is a big difference, so we must distinguish the left quantization and left pushforward from the right quantization and right pushforward.)

   Yes, the description is correct, except that limits give smooth functions / smooth sections (with convolution), whereas colimits give (smooth) distributions / distributional sections (with convolution).

   (In the finite situation there is no difference between colimits and limits or between left and right Kan extensions, but in our case there is a big difference, so we must distinguish the left quantization and left pushforward from the right quantization and right pushforward.)

8. • CommentRowNumber8.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 11th 2013
   • PermaLink
   Author: Dmitri Pavlov
   Format: Markdown1) The ordinary algebraic dual of C^∞(X), where X is a smooth manifold, taken in the topos of sheaves on smooth manifolds, is precisely the space of compactly supported distributions on X. Similarly for compactly supported functions and arbitrary distributions. Applying the ordinary algebraic dual one more time gives us back the original spaces. All these fact follow from the simple lemma that a linear functional on a Fréchet space is continuous if and only if it sends smooth families of elements to smooth functions. 2) Thus the ordinary algebraic dual functor induces and adjunction and a monad. Anders Kock's theory covers this case, I think (his paper has a section with this particular example). 3) For example, we can compute the smooth limit of a constant functor (or rather the smooth product of a constant smooth family) defined on a smooth set (manifold) X by checking that the smooth vector space O(X) of smooth functions on X satisfies the universal property. Specifically, maps from a smooth vector space V to O(X) are the same as morphisms of from the trivial vector bundle with fiber V over X to the trivial line bundle over X.

   1) The ordinary algebraic dual of C^∞(X), where X is a smooth manifold, taken in the topos of sheaves on smooth manifolds, is precisely the space of compactly supported distributions on X. Similarly for compactly supported functions and arbitrary distributions. Applying the ordinary algebraic dual one more time gives us back the original spaces. All these fact follow from the simple lemma that a linear functional on a Fréchet space is continuous if and only if it sends smooth families of elements to smooth functions.

   2) Thus the ordinary algebraic dual functor induces and adjunction and a monad. Anders Kock's theory covers this case, I think (his paper has a section with this particular example).

   3) For example, we can compute the smooth limit of a constant functor (or rather the smooth product of a constant smooth family) defined on a smooth set (manifold) X by checking that the smooth vector space O(X) of smooth functions on X satisfies the universal property. Specifically, maps from a smooth vector space V to O(X) are the same as morphisms of from the trivial vector bundle with fiber V over X to the trivial line bundle over X.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 11th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexHi Dmitri, the statements in your items 1)-3) are clear, but what I am unsure about is which claim you think is being implied by them. Would you mind making explicit the formal statement that you think is or ought to be true? Is it exactly what I stated in #5? Your statements seem to be written with categories of _linear_ maps in mind.(?) Here is another point I was wondering about, somewhat orthogonal to these items: Of course the full sub-2-category of $\mathbf{Cat}$ (boldface for smoothness, but that's not the point for the folllowing) on the directed-delooped monoids $\mathbf{B}_\times A$ is not Algebras+Bimodules, but is just the sub-2-category Algebras+Homomorphisms+Intertwiners. So first one might think for the statement in #5 to have a chance to work out at all one would need $\mathbf{Prof}$ (categories and [[profunctors]] between them) instead of $\mathbf{Cat}$. But I suppose the lax (btw !) colimiting cocone under a functor constant on $\mathbf{B}_\times A$ in $\mathbf{Prof}$ (as [here](http://ncatlab.org/nlab/show/category+algebra#WeakColimit)) sits in the inclusion $\mathbf{Cat} \to \mathbf{Prof}$ and probably $\mathbf{B}_\times(-)$ of the convolution algebras is also lax colimiting there. I need to think about this. But not now. Now I need to run and do something else...

   Hi Dmitri,

   the statements in your items 1)-3) are clear, but what I am unsure about is which claim you think is being implied by them. Would you mind making explicit the formal statement that you think is or ought to be true? Is it exactly what I stated in #5? Your statements seem to be written with categories of linear maps in mind.(?)

   Here is another point I was wondering about, somewhat orthogonal to these items:

   Of course the full sub-2-category of $\mathbf{Cat}$ (boldface for smoothness, but that’s not the point for the folllowing) on the directed-delooped monoids $\mathbf{B}_\times A$ is not Algebras+Bimodules, but is just the sub-2-category Algebras+Homomorphisms+Intertwiners.

   So first one might think for the statement in #5 to have a chance to work out at all one would need $\mathbf{Prof}$ (categories and profunctors between them) instead of $\mathbf{Cat}$. But I suppose the lax (btw !) colimiting cocone under a functor constant on $\mathbf{B}_\times A$ in $\mathbf{Prof}$ (as here) sits in the inclusion $\mathbf{Cat} \to \mathbf{Prof}$ and probably $\mathbf{B}_\times(-)$ of the convolution algebras is also lax colimiting there.

   I need to think about this. But not now. Now I need to run and do something else…