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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 8th 2013
   • (edited Apr 8th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to _[[groupoid convolution algebra]]_ the beginning of an Examples-section titled _[Higher groupoid convolution algebras and n-vector spaces/n-modules](http://ncatlab.org/nlab/show/category+algebra#HigherGroupoidConvolutionAlgebras)_. Conservatively, you can regard this indeed as just some examples of applications of the groupoid convolution algebra construction. But the way it is presented is supposed to be suggestive of a "higher C*-algebra" version of convolution algebras of higher Lie groupoids. I have labelled it as "under construction" to reflect the fact that this latter aspect is a bit experimental for the moment. The basic idea is that to the extent that we do have groupoid convolution as a (2,1)-functor $$ C \colon Grpd \to Alg_{b}^{op} \simeq 2Mod $$ (as do do for discrete geometry and conjecturally do for smooth geometry), then this immediately means that it sends [[double groupoids]] to convolution [[sesquialgebras]], hence to [[3-modules]] with basis ([[3-vector spaces]]). As the simplest but instructive example of this I have spelled out how the ordinary dual(commutative and non-co-commutative) Hopf algebra of a finite group arises this way as the "horizontally constant" double groupoid incarnation of $\mathbf{B}G$, while the convolution algebra of $G$ is the algebra of the "vertically discrete" double groupoid incarnation of $\mathbf{B}G$. But next, if we simply replace the bare $Alg_b^{op} \simeq 2 Mod$ with the 2-category $C^\ast Alg_b$ of $C^\ast$-algebras and [[Hilbert bimodules]] between them and assume (as seems to be the case) that $C^\ast$-algebraic groupoid convolution is a 2-functor $$ LieGrpd_{\simeq} \to C^\ast Alg_n^{op} $$ then the same argument goes through as before and yields convolution "$C^\ast$-2-algebras" that look like [[Hopf-C*-algebras]]. Etc. Seems to go in the right direction...

   I have added to groupoid convolution algebra the beginning of an Examples-section titled Higher groupoid convolution algebras and n-vector spaces/n-modules.

   Conservatively, you can regard this indeed as just some examples of applications of the groupoid convolution algebra construction. But the way it is presented is supposed to be suggestive of a “higher C*-algebra” version of convolution algebras of higher Lie groupoids.

   I have labelled it as “under construction” to reflect the fact that this latter aspect is a bit experimental for the moment.

   The basic idea is that to the extent that we do have groupoid convolution as a (2,1)-functor

   $$C \colon Grpd \to Alg_{b}^{op} \simeq 2Mod$$

   (as do do for discrete geometry and conjecturally do for smooth geometry), then this immediately means that it sends double groupoids to convolution sesquialgebras, hence to 3-modules with basis (3-vector spaces).

   As the simplest but instructive example of this I have spelled out how the ordinary dual(commutative and non-co-commutative) Hopf algebra of a finite group arises this way as the “horizontally constant” double groupoid incarnation of $\mathbf{B}G$, while the convolution algebra of $G$ is the algebra of the “vertically discrete” double groupoid incarnation of $\mathbf{B}G$.

   But next, if we simply replace the bare $Alg_b^{op} \simeq 2 Mod$ with the 2-category $C^\ast Alg_b$ of $C^\ast$-algebras and Hilbert bimodules between them and assume (as seems to be the case) that $C^\ast$-algebraic groupoid convolution is a 2-functor

   $$LieGrpd_{\simeq} \to C^\ast Alg_n^{op}$$

   then the same argument goes through as before and yields convolution “$C^\ast$-2-algebras” that look like Hopf-C*-algebras. Etc. Seems to go in the right direction…