\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{category algebra} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory} [[!include category theory - contents]] \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{for_bare_categories_discrete_geometry}{For bare categories (discrete geometry)}\dotfill \pageref*{for_bare_categories_discrete_geometry} \linebreak \noindent\hyperlink{ForLieGroupoids}{For Lie groupoids}\dotfill \pageref*{ForLieGroupoids} \linebreak \noindent\hyperlink{equivalent_characterizations}{Equivalent characterizations}\dotfill \pageref*{equivalent_characterizations} \linebreak \noindent\hyperlink{WeakColimit}{As a weak colimit over a constant $2Vect$-valued functor}\dotfill \pageref*{WeakColimit} \linebreak \noindent\hyperlink{InTermsOfCompositionOfSpans}{In terms of composition of spans}\dotfill \pageref*{InTermsOfCompositionOfSpans} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{basic_examples}{Basic examples}\dotfill \pageref*{basic_examples} \linebreak \noindent\hyperlink{incidence_algebras_poset_convolution_algebras}{Incidence algebras (poset convolution algebras)}\dotfill \pageref*{incidence_algebras_poset_convolution_algebras} \linebreak \noindent\hyperlink{HigherGroupoidConvolutionAlgebras}{Higher groupoid convolution algebras and n-vector spaces/n-modules}\dotfill \pageref*{HigherGroupoidConvolutionAlgebras} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{RelationToTwistedKTheory}{Relation to (twisted) K-theory}\dotfill \pageref*{RelationToTwistedKTheory} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{for_partial_orders}{For partial orders}\dotfill \pageref*{for_partial_orders} \linebreak \noindent\hyperlink{for_discrete_geometry}{For discrete geometry}\dotfill \pageref*{for_discrete_geometry} \linebreak \noindent\hyperlink{ReferencesForSmoothGeometry}{For continuous/smooth geometry}\dotfill \pageref*{ReferencesForSmoothGeometry} \linebreak \noindent\hyperlink{convolution_algebras}{Convolution $C^\ast$-algebras}\dotfill \pageref*{convolution_algebras} \linebreak \noindent\hyperlink{ReferencesConvolutionHopfAlgebroids}{Convolution Hopf algebroids}\dotfill \pageref*{ReferencesConvolutionHopfAlgebroids} \linebreak \noindent\hyperlink{ReferencesModulesOverConvolutionAlgebra}{Modules over Lie groupoid convolution algebras and K-theory}\dotfill \pageref*{ReferencesModulesOverConvolutionAlgebra} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} The [[space]] of [[functions]] $\mathcal{C}_1 \to R$ on the [[space]] of [[morphisms]] $\mathcal{C}_1$ of a [[small category]] $\mathcal{C}_\bullet$ (with [[coefficients]] in some [[ring]] $R$) naturally inherits a [[convolution algebra]] structure from the [[composition]] operation on morphisms. This is called the \emph{category convolution algebra} or just \emph{category algebra} for short. Often this is considered specifically for [[groupoids]] and hence accordingly called \emph{groupoid convolution algebra} or just \emph{groupoid algebra} for short. (For one-object [[delooping]] groupoids of [[groups]], groupoid algebras reduce to [[group algebras]].) The [[inverse|inversion operation]] of the groupoid naturally makes its groupoid algebra into a [[star-algebra]] (this is generally the case for category algebras of [[dagger categories]]) and accordingly groupoid algebras play a role in [[C-star-algebra]] theory. More generally, if the groupoid carries a [[line 2-bundle]] then (in its incarnation as a [[bundle gerbe]]-like transition bundle) the space of morphisms carries a [[line bundle]] (satisfying some compatibility conditions) and one can consider convolution algebras not just of functions, but of [[sections]] of this line bundle. The resulting algebra is called the \emph{twisted groupoid convolution algebra}, twisted by the [[characteristic class]] of the [[line 2-bundle]] (\hyperlink{TXLG}{TXLG}). For ``bare'' categories/groupoids (i.e.: [[internal category|internal]] to [[Set]]) these constructions are canonical. But under mild conditions or else when equipped some suitable extra [[stuff, structure, property|structure]], it generalizes to [[internal categories]]/[[internal groupoids]] in [[geometry|geometric]] contexts, notably in [[topology]] ([[topological groupoids]]), [[differential geometry]] ([[Lie groupoids]]), and [[algebraic geometry]]. In such geometric situations a groupoid convolution algebra equipped with its canonical [[coalgebra]] structure over the functions on its canonical [[atlas]] is also called a \emph{[[Hopf algebroid]]} and may be used to \emph{characterize} the geometric groupoid. Therefore to some extent one may think of the relation between groupoids/categories and their groupoid/category algebras as an incarnation of the [[Isbell duality|general duality]] between [[geometry]] and [[algebra]]. Since category/groupoid algebras are generically non-commutative, this relation identifies groupoids/categories as certain spaces in [[noncommutative geometry]]. From this point of view groupoid convolution algebras have been highlighted and developed notably in (\hyperlink{Connes094}{Connes 94}). Due to this relation the groupoid convolution product is also referred to as a [[star product]] and denoted ``$\star$''. Groupoid [[C\emph{-algebras]] form a rich sub-class of all [[C}-algebras]], including [[crossed product C\emph{-algebras]], [[Cuntz algebras]].} Groupoid convolution algebras may also be understood as generalizations of [[matrix algebras]], to which they reduce for the case of the [[pair groupoid]]. In (\hyperlink{Connes094}{Connes 94, 1.1}) it was famously argued that when [[Werner Heisenberg]] (re-)discovered (infinite-dimensional) [[matrix algebras]] as [[algebras of observables]] in [[quantum mechanics]], conceptually he rather considered groupoid convolution algebras. This perspective has since been fully developed: in (\hyperlink{EH}{EH 06}) [[strict deformation quantization]] is given fairly generally by twisted groupoid convolution algebras. See at \emph{[[geometric quantization of symplectic groupoids]]} for more on this. For the [[discrete groupoid|discrete]] but [[higher geometry]] of [[infinity-Dijkgraaf-Witten theory]] quantization by higher groupoid convolution algebras is indicated in (\hyperlink{FHLT}{FHLT 09}). \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{for_bare_categories_discrete_geometry}{}\subsubsection*{{For bare categories (discrete geometry)}}\label{for_bare_categories_discrete_geometry} \begin{defn} \label{CategoryAlgebra}\hypertarget{CategoryAlgebra}{} Let $\mathcal{C}$ be a [[small category]] and let $R$ be a [[ring]]. The \textbf{category algebra} or \textbf{[[convolution algebra]]} $R[\mathcal{C}]$ of $\mathcal{C}$ over $R$ is the $R$-[[associative algebra|algebra]] \begin{itemize}% \item whose underlying $R$-[[module]] is the [[free module]] $R[\mathcal{C}_1]$ over the set of [[morphisms]] of $\mathcal{C}$; \item whose product operation is defined on [[basis]]-elements $f,g \in \mathcal{C}_1 \hookrightarrow R[\mathcal{C}]$ to be their [[composition]] if they are composable and zero otherwise: \begin{displaymath} f \cdot g := \left\lbrace \itexarray{ g \circ f & if\;composable \\ 0 & otherwise } \right. \,. \end{displaymath} \end{itemize} \end{defn} \begin{remark} \label{AsConvolutionAlgebra}\hypertarget{AsConvolutionAlgebra}{} We may identify elements in $R[\mathcal{C}_1]$ with [[functions]] $\mathcal{C}_1 \to R$ with the property that they are non-vanishing only on finitely many elements. Under this identification for $\phi_1, \phi_2$ two such functions, their product in $R[\mathcal{C}]$ is given by the formula \begin{displaymath} \phi_1 \cdot \phi_2 \; \colon \; f \mapsto \sum_{f_2 \circ f_1 = f} \phi_2(f_2) \cdot \phi_1(f_1) \,, \end{displaymath} where $f,f_1,f_2 \in \mathcal{C}_1$. In particular if $\mathcal{C}$ is a [[groupoid]] so that every [[morphisms]] $f$ has an [[inverse]] $f^{-1}$ then this is equivalently \begin{displaymath} \phi_1 \cdot \phi_2 \; \colon \; f \mapsto = \sum_{g \in \mathcal{C}_1} \phi_2(f \circ g^{-1}) \cdot \phi_1(g) \,. \end{displaymath} This expresses [[convolution]] of functions. \end{remark} \begin{defn} \label{InvolutionByPullbackAlongInversion}\hypertarget{InvolutionByPullbackAlongInversion}{} If the small category $\mathcal{C}_\bullet$ is a [[groupoid]], hence equipped with an [[inverse|inversion]] map \begin{displaymath} inv \colon \mathcal{C}_1 \to \mathcal{C}_1 \end{displaymath} then [[pullback of functions]] along this map makes is an [[involution]] of the convolution algebra of $\mathcal{C}$ and hence makes it into a [[star-algebra]]. More generally for $\mathcal{C}$ equipped with the structure of a [[dagger-category]], pullback along the dagger-functor \begin{displaymath} \dagger \colon \mathcal{C}_1 \to \mathcal{C}_1 \end{displaymath} makes the convolution algebra a star-algebra. \end{defn} \begin{defn} \label{catalgebraIsWeakHopfalgebra}\hypertarget{catalgebraIsWeakHopfalgebra}{} If the category has finitely many objects and finitely many morphisms, then the category algebra is also a coalgebra via the unique comultiplication where every morphism in $C_1\subset R[C_1]$ is group like; the structure of the coalgebra and that of an algebra are compatible in the sense that they form a [[weak Hopf algebra]]. \end{defn} \hypertarget{ForLieGroupoids}{}\subsubsection*{{For Lie groupoids}}\label{ForLieGroupoids} We discuss groupoid convolution [[C\emph{-algebras]] for [[Lie groupoids]]/[[differentiable stacks]] (def. \ref{ContinuousCategoryAlgebraWithHalfDensities}, prop. \ref{GroupoidConvolutionIs2Functor} below).} \begin{remark} \label{ForLieGroupoids}\hypertarget{ForLieGroupoids}{} If $\mathcal{C}$ is a groupoid with extra [[geometry|geometric]] structure, then there are natural variants of the above definition. Notably if $\mathcal{C}$ is a [[Lie groupoid]] then there is a variant where the functions in remark \ref{AsConvolutionAlgebra} are taken to be [[smooth functions]] and where the convolution [[sum]] is replaced by an [[integration]]. In order for this to make sense one needs to consider in fact functions with values in half-[[densities]] over the [[manifold]] $\mathcal{C}_1$. More generally, for a [[bundle gerbe]] over a Lie groupoid $\mathcal{C}$, hence a multiplicative [[line bundle]] over $\mathcal{C}_1$, one can consider a convolution product on [[sections]] of this line bundle tensored with half-densities. \end{remark} See the \hyperlink{ReferencesForSmoothGeometry}{References -- For continuous/smooth geometry}. \begin{defn} \label{ContinuousCategoryAlgebraWithHalfDensities}\hypertarget{ContinuousCategoryAlgebraWithHalfDensities}{} \textbf{([[Lie groupoid convolution algebra]])} Let $\mathcal{G}_\bullet$ be a [[Lie groupoid]]. Write $C^\infty_c(\mathcal{G}_1, \sqrt{\Omega})$ for the space of smooth [[half-densities]] in $T_{d s = 0}\Gamma_1 \oplus T_{d t = 0}\Gamma_1$ of [[compact support]] on its [[manifold]] $\mathcal{G}_1$ of [[morphisms]]. Let the [[convolution product]] \begin{displaymath} \star \;\colon\; C_c(\mathcal{G}_1, \sqrt{\Omega}) \times C_c(\mathcal{G}_1, \sqrt{\Omega}) \to C_c(\mathcal{G}_1, \sqrt{\Omega}) \end{displaymath} be given on elements $f,g \in C_c(\mathcal{G}_1, \sqrt{\Omega})$ over any element $\gamma \in \mathcal{G}_1$ by the [[integral]] \begin{displaymath} (f \star g) \colon \gamma \mapsto \int_{\gamma_2\circ \gamma_1 = \gamma} f(\gamma_1) \cdot f(\gamma_2) \,. \end{displaymath} (Here we regard the integrand naturally as taking values in actual [[densities]] tensored with the pullback of $\sqrt{\Omega}$ along the composition map. This defines the [[integration]] of density-factor which then takes values in $\sqrt{\Omega}$.) The algebra $(C_c^\infty(\mathcal{G}_1, \sqrt{\Omega}), \star)$ is the \textbf{groupoid convolution algebra} of smooth compactly supported functions. As in remark \ref{AsConvolutionAlgebra}, this is naturally a [[star-algebra]] with [[involution]] $inv^\ast$. \end{defn} This construction originates around (\hyperlink{Connes82}{Connes 82}). \begin{prop} \label{}\hypertarget{}{} For $\mathcal{G}_\bullet$ a [[Lie groupoid]] and for $x \in \mathcal{G}_0$ any point in the manifold of [[objects]] there is an involutive [[representation]] $\pi_x$ of the convolution algebra $(C_c^\infty(\mathcal{G}_1, \sqrt{\Omega}), \star, inv^\ast)$ of def. \ref{ContinuousCategoryAlgebraWithHalfDensities} on the [[canonical Hilbert space of half-densities]] $L^2(s^{-1}(x))$ of the source fiber of $x$ given on any $\xi \in L^2(s^{-1}(x))$ by \begin{displaymath} \pi_x(f) \xi \colon \gamma \mapsto \int_{\gamma_1 \in t^{-1}(\gamma)} f(\gamma_1) \xi(\gamma_1^{-1}\gamma) \,. \end{displaymath} This defines a [[norm]] $|{\Vert \Vert}$ on the [[vector space]] $C_c^\infty(\mathcal{G}_1, \Omega)$ given by the [[supremum]] of the norms in $L^2(s^{-1}(x))$ over all points $x$: \begin{displaymath} {\Vert f\Vert} \coloneqq Sup_{x \in \mathcal{G}_0} {\Vert \pi_x(f)\Vert} \,. \end{displaymath} The [[Cauchy completion]] of the [[star algebra]] $(C_c^\infty(\mathcal{G}_1, \sqrt{\Omega}), \star, inv^\ast)$ with respect to this [[norm]] is a [[C-star-algebra]], the \textbf{convolution $C^\ast$-algebra} of the Lie groupoid $\mathcal{G}_\bullet$. \end{prop} This is recalled at (\hyperlink{Connes94}{Connes 94, prop. 3 on p. 106}). \begin{remark} \label{ExtensionToBibundlesAndBimodules}\hypertarget{ExtensionToBibundlesAndBimodules}{} With suitable definitions, this construction constitutes something at least close to a [[2-functor]] from [[differentiable stacks]] to [[C-star-algebras]] and [[Hilbert bimodules]] between them: In (\hyperlink{Mrcun99}{Mrun 99}) the convolution algebra construction for [[étale Lie groupoids]] is extended to groupoid [[bibundles]] and shown to produce a [[functor]] to [[C-star-algebras]] with (isomorphism classes of) [[bimodules]] between them. In (\hyperlink{KalisnikMrcun}{Kali\v{s}nik-Mrun 07}) it is shown that if one remembers the additional [[Hopf algebroid]] structure on the convolution $C^\ast$-alegras (the algebraic analog of remebering the [[atlas]] of the [[differentiable stack]] of a Lie groupoid) then this construction becomes a [[full subcategory]] inclusion of \'e{}tale Lie groupoids into their convolution Hopf algebroids. In (\hyperlink{MuhleReaultWilliams87}{Muhly-Renault-Williams 87}, \hyperlink{Landsman00}{Landsman 00}) the generalization of the construction of a $C^\ast$-bimodule from a groupoid [[bibundle]] to general [[Lie groupoids]] is discussed (not necessarily [[étale Lie groupoid|étale]]), but only equivalence-bibundles are considered and are shown to yield [[Morita equivalence]] bimodules (no discussion of composition and functoriality here). \end{remark} A precise form of this statement is the following \hyperlink{Nuiten13}{Nuiten 13, theorem 3.3.1} \begin{prop} \label{GroupoidConvolutionIs2Functor}\hypertarget{GroupoidConvolutionIs2Functor}{} \textbf{([[groupoid convolution algebra]] is [[(2,1)-functor]] from [[differentiable stacks]] to [[C\emph{-algebras]])*}} The above construction of groupoid convolution algebras (def. \ref{ContinuousCategoryAlgebraWithHalfDensities}) extends to a [[(2,1)-functor]] \begin{displaymath} C^\ast(-) \;\colon\; DiffStack^{prop} \overset{\phantom{AAA}}{\longrightarrow} C^\ast Alg^{op}_{bim} \end{displaymath} between \begin{enumerate}% \item the [[(2,1)-category]] $DiffStack_{prop}$ of [[differentiable stacks]] (i.e. [[Lie groupoids]] regarded as [[smooth stacks]]) with proper morphisms between them (\hyperlink{Nuiten13}{Nuiten 13, def. 2.2.35}) \item the [[opposite 2-category|opposite]] [[(2,1)-category]] of [[C\emph{-algebras]] with [[Hilbert bimodules]] between them and [[intertwiners]] between those.} \end{enumerate} \end{prop} (\hyperlink{Nuiten13}{Nuiten 13, theorem 3.3.1}) This way, much of [[noncommutative geometry]] is exhibited as actually being the [[higher geometry]] of [[differentiable stacks]] inside all [[smooth stacks]]. In particular, [[groupoid K-theory]] is realized as the [[KK-theory]] of convolution C\emph{-algebras.} \hypertarget{equivalent_characterizations}{}\subsection*{{Equivalent characterizations}}\label{equivalent_characterizations} We discuss equivalent characterizations of category algebras/groupoid algebras that are useful in certain context \begin{itemize}% \item \hyperlink{WeakColimit}{As a weak colimit over a constant 2Vect-valued functor} \item \hyperlink{InTermsOfCompositionOfSpans}{In terms of composition of spans} \end{itemize} \hypertarget{WeakColimit}{}\subsubsection*{{As a weak colimit over a constant $2Vect$-valued functor}}\label{WeakColimit} Apparently for $\mathcal{C}$ a [[groupoid]] the category algebra of $C$ is the [[weak limit|weak colimit]] over $\mathcal{C}$ of the functor $\mathcal{C} \to Vect\text{-}Mod$ constant on the ground field algebra. This statement is for instance in (\hyperlink{FHLT}{FHLT, section 8.4}). The 2-cell in the universal co-cone corresponding to the morphism $f \in C$ is the $k\text{-}k[C]$-bimodule homomorphism $f \cdot (-) : A \to A$ that multiplies by $f \in k[C]$ from the left. \begin{displaymath} \itexarray{ x &&\stackrel{f}{\to}&& y \\ k &&\stackrel{k}{\to}&& k \\ & {k[C]}_{\mathllap{}}\searrow &\swArrow_{f \cdot (-)}& \swarrow_{\mathrlap{k[C]}} \\ && k[C] } \end{displaymath} This description should be compared with the analogous description of the [[action groupoid]] by a weak colimit. One sees that the groupoid algebra is a linear incarnation of the action groupoid in some sense. \hypertarget{InTermsOfCompositionOfSpans}{}\subsubsection*{{In terms of composition of spans}}\label{InTermsOfCompositionOfSpans} The category algebra of a category $C$ is a special case of a general construction of [[spans]] (see also at \emph{[[bi-brane]]}). In order not to get distracted by inessential technicalities, consider the case of a finite [[category]] $C$, i.e. an [[internal category]] in [[FinSet]]. This is a [[span]] \begin{displaymath} \itexarray{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&&& C_0 } \end{displaymath} equipped with a composition operation: a morphism of spans from the composite span \begin{displaymath} \itexarray{ &&&& C_1 \times_{t,s} C_1 \\ &&& \swarrow && \searrow \\ && C_1 &&&& C_1 \\ & {}^{s}\swarrow && \searrow^{t} && {}^{s}\swarrow && \searrow^{t} \\ C_0 &&&& C_0 &&&& C_0 } \end{displaymath} to the original one, i.e. a morphism \begin{displaymath} comp : C_1 \times_{t,s} C_1 \to C_1 \end{displaymath} which respects source and target morphisms. Given this, consider the trivial vector bundle on the set of objects $C_0$. This is nothing but an assignment \begin{displaymath} I : C_0 \to Vect_k \end{displaymath} of the [[ground field]] $k$ to each element of $C_0$. There are two different ways to pull this vector bundle on objects back to a vector bundle on morphisms, once along the source, once along the target map. Then notice that the set of natural transformations between these two vector bundles \begin{displaymath} Hom_{[Sets,Vect_k]}(s^* I , t^* I) \end{displaymath} whose elements are 2-arrows of the form \begin{displaymath} \itexarray{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{f}{\Rightarrow}&& C_0 \\ & {}_I \searrow && \swarrow_I \\ && Vect_k } \end{displaymath} are canonically in bijection with $k$-calued functions on $C_1$, hence with the vector space spanned by $C_1$, hence with the vector space underlying the category algebra \begin{displaymath} Hom_{[Sets,Vect_k]}(s^* I , t^* I) \simeq k[C] \,. \end{displaymath} The algebra structure on $k[C]$ is similarly encoded in the diagrammatics: given two elements \begin{displaymath} \itexarray{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{f}{\Rightarrow}&& C_0 \\ & {}_I \searrow && \swarrow_I \\ && Vect_k } \;\;\;\; and \;\;\;\; \itexarray{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{g}{\Rightarrow}&& C_0 \\ & {}_I \searrow && \swarrow_I \\ && Vect_k } \end{displaymath} their pre-composite is the diagram \begin{displaymath} \itexarray{ &&&& C_1 \times_{t,s} C_1 \\ &&& \swarrow && \searrow \\ && C_1 &&&& C_1 \\ & {}^{s}\swarrow && \searrow^{t} && {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{f}{\Rightarrow}&& C_0 &&\stackrel{g}{\Rightarrow}&& C_0 \\ & \searrow &&& \downarrow &&& \swarrow \\ && \to &&Vect_k&& \leftarrow } \,. \end{displaymath} This is a composite transformation between three trivial vector bundles on the set $C_1 \times_{t,s} C_1$ of composable morphisms in $C$. As such, it is a function, which on the element consisting of the composable pair $\stackrel{r}{\to}\stackrel{s}{\to}$ takes the value $f(r)\cdot g(s)$. In order to get back a transformation between vector bundles on $C_1$, hence a transformation between vector bundles on $C_1$, we \emph{push forward} along the composition map $comp: C_1 \times_{t,s} C_1 \to C_1$. This just means that we add up the values on the fibers of this map. The result is the [[convolution product]] \begin{displaymath} (f\star g) : t \mapsto \sum_{s\circ r = t} f(r)\cdot g(s) \,. \end{displaymath} This is indeed the product in the category algebra. Looking at category algebras realizes them as a puny special case of a bigger story which involves [[bi-brane]]s as morphisms between $n$-bundles/$(n-1)$-gerbes which live on spaces connected by correspondence spaces. This is related to a bunch of things, such as T-duality, Fourier-Mukai transformations and other issues of quantization. A description of this perspective is in \begin{itemize}% \item [[schreiber:Nonabelian cocycles and their quantum symmetries]]. \end{itemize} This is related to observations such as described here: \begin{itemize}% \item [[John Baez]], \href{http://golem.ph.utexas.edu/category/2007/03/quantization_and_cohomology_we_16.html}{\emph{Quantization and Cohomology (Week 17)}} \item Urs Schreiber, \href{http://golem.ph.utexas.edu/category/2007/02/qft_of_charged_nparticle_tdual.html}{\emph{QFT of Charged n-Particle: T-Duality}} \end{itemize} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{basic_examples}{}\subsubsection*{{Basic examples}}\label{basic_examples} \begin{example} \label{}\hypertarget{}{} The convolution algebra of a [[set]]/[[manifold]] $X$ regarded as a [[discrete groupoid]]/[[Lie groupoid]] with only [[identity]] [[morphisms]] is the ordinary [[function algebra]] of $X$. \end{example} \begin{example} \label{}\hypertarget{}{} For $X$ a [[set]] the convolution algebra of the [[pair groupoid]] $Pair(X)_\bullet$ is the [[matrix algebra]] of ${\vert X\vert} \times {\vert X\vert}$ matrices. \end{example} \begin{example} \label{}\hypertarget{}{} For $X$ a [[smooth manifold]] and $Pair(X)_\bullet$ its [[pair groupoid]] regarded as a [[Lie groupoid]] its smooth convolution algebra is the algebra of [[smoothing kernels]] on $X$. \end{example} \begin{example} \label{}\hypertarget{}{} $\mathcal{C} = \mathbf{B}G$ is the [[delooping]] [[groupoid]] of a [[discrete group]] $G$ (the groupoid with a single object and $G$ as its set of morphisms), then def. \ref{CategoryAlgebra} reduces to that of the [[group algebra]] of $G$: \begin{displaymath} R[\mathbf{B}G] \simeq R[G] \,. \end{displaymath} \end{example} \hypertarget{incidence_algebras_poset_convolution_algebras}{}\subsubsection*{{Incidence algebras (poset convolution algebras)}}\label{incidence_algebras_poset_convolution_algebras} See at [[Möbius inversion\#IncidenceAlgebrasZetaFn]]. \hypertarget{HigherGroupoidConvolutionAlgebras}{}\subsubsection*{{Higher groupoid convolution algebras and n-vector spaces/n-modules}}\label{HigherGroupoidConvolutionAlgebras} \begin{quote}% under construction \end{quote} We discuss here a natural generalization of the notion of [[groupoid convolution algebras]] to [[higher algebra|higher algebras]] for [[n-groupoid|higher groupoids]]. There may be several sensible such generalizations. The one discussed now follows the principle of iterated [[internalization]] and naturally matches to the concept of [[n-modules]] ([[n-vector spaces]]) as they appear in [[extended prequantum field theory]]. In order to disentangle conceptual from technical aspects, we first discuss [[discrete ∞-groupoid|geometrically discrete higher groupoids]]. The results of this discussion then in particular help to suggest what the right definition of ``higher Lie groupoid'' in the context of higher convolution algebras should be in the first place. The considerations are based on the following \begin{remark} \label{}\hypertarget{}{} By the discussion at \emph{[[2-module]]} we may think of the [[2-category]] $k Alg_b$ of $k$-[[associative algebras]] and [[bimodules]] between them as a model for the 2-category [[2Mod]] of $k$-[[2-modules]] that admit a 2-[[basis]] ([[2-vector spaces]]). Hence the groupoid convolution algebra constructiuon is a 2-functor \begin{displaymath} C \;\colon\; Grpd \to 2 Mod \,. \end{displaymath} There is then the following systematic refinement of this to higher groupoids and higher algebra: by the discussion at \emph{[[n-module]]}, 3-modules are algebra objects in [[2Mod]] and maps between them are [[bimodule]] objects in there. An algebra object in $k Alg_b$ is equivalently a [[sesquialgebra]], an algebra equipped with a second algebra structure up to coherent homotopy, that is exhibited by structure bimodules. Special cases of this are [[bialgebras]], for which these structure bimodules come from actual algebra homomorphisms. Examples of these in turn are [[Hopf algebras]]. These we naturally re-discover as special higher groupoid convolution higher algebras in example \ref{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup} below. \end{remark} This iterated internalization on the codomain of the groupoid convolution algebra functor has a natural analog on its domain: a 2-groupoid we may present by a [[double groupoid]], namely a [[groupoid object in an (∞,1)-category]] in [[Grpd]] which is 3-[[coskeleton|coskeletal]] as a [[simplicial object]] in [[Grpd]]. \begin{remark} \label{}\hypertarget{}{} Given a [[groupoid object in an (∞,1)-category|groupoid object]] $\mathcal{G}_\bullet$ in the [[(2,1)-topos]] [[Grpd]] hence a [[double groupoid]], applying the groupoid convolution algebra $(2,1)$-functor $C$ to the corresponding [[simplicial object]] $\mathcal{G}_\bullet \in Grpd^{\Delta^{op}}$ yields: \begin{itemize}% \item groupoid convolution algebras $C(\mathcal{G}_0)$ and $C(\mathcal{G}_1)$, \item a $C(\mathcal{G}_1) \otimes_{C(\mathcal{G}_{0,1})} C(\mathcal{G}_1)-C(\mathcal{G}_{0})$-bimodule, assigned to the [[composition]] functor $\partial_1 \colon \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1$. \end{itemize} Under the 2-functoriality of $C$, the [[Segal conditions]] satisfied by $\mathcal{G}_\bullet$ make this bimodule exhibi a [[sesquialgebra]] structure over $C(\mathcal{G}_{0,1})$. This sesquialgebra we call the the \textbf{double groupoid convolution 2-algebra} of $\mathcal{G}_\bullet$. (Here we make invariant sense of the [[tensor product]] by evaluating on a [[Reedy model structure|Reedy fibrant]] representative.) \end{remark} \begin{example} \label{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup}\hypertarget{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup}{} Let $G$ be a [[finite group]]. Write $\mathbf{B}G$ for its [[delooping]] [[groupoid]] (the connected groupoid with $\pi_1 = G$). Since this is just a [[1-groupoid]], there are two natural ways to present $\mathbf{B}G$ as a [[double groupoid]]: \begin{enumerate}% \item $\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G\stackrel{\to}{\stackrel{\to}{\to}} \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G$; \item $\underset{\longrightarrow}{\lim}(\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *) \simeq \mathbf{B}G$. \end{enumerate} (The first is ``vertically constant'', the second is ``horizontally constant''). Applying the [[groupoid convolution algebra]] functor to the first presentation yields the groupoid convolution algebra $C(\mathbf{B}G)$ equipped with a trivial multiplication bimodule, hence just the group convolution algebra $C(\mathbf{B}G) \simeq C_{conv}(G)$. Applying however the groupoid convolution algebra functor to the second presentation yields the \emph{commutative} algebra of functions $C(G)$ equipped with the multiplication bimodule which is $C(G \times G)$ regarded as a $(C(G\times G), C(G))$-bimdodule, where the right action is induced by pullback along the group product map $G \times G \to G$. This bimodule is in the image of the functor $Alg \to Alg_b$ that sends algebra homomorphisms to their induced bimodules, by sending $f \colon A \to B$ to $A$ regarded as an $(A,B)$-bimodule with the canonical left action on itself and the right action induced by $f$. Namely this bimdoule correspondonds to the map \begin{displaymath} \Delta \colon C(G) \to C(G \times G) \simeq C(G) \otimes C(G) \end{displaymath} given on $\phi \in C(G)$ and $g_1, g_2 \in G$ by \begin{displaymath} \Delta \phi \colon (g_1, g_2) \mapsto \phi(g_1 \cdot g_2) \,. \end{displaymath} This is that standard [[coalgebra|coproduct]] on the standard dual [[Hopf algebra]] associated with $G$. In summary this means that (for $G$ a finite group): \begin{enumerate}% \item If we regard $\mathbf{B}G$ as presented as a double groupoid constant on $\mathbf{B}G$, then the corresponding groupoid convolution [[sesquialgebra]] (basis for a [[n-module|3-module]]) is the convolution algebra of $G$; \item If instead we regard $\mathbf{B}G$ as presented as the double groupoid which is degreewise constant as a groupoid, then the corresponding groupoid convolution sesquialgebra is the standard (``dual'') [[Hopf algebra]] structure on the commutative pointwise product algebra of functions on $G$. \end{enumerate} \end{example} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{RelationToTwistedKTheory}{}\subsubsection*{{Relation to (twisted) K-theory}}\label{RelationToTwistedKTheory} The [[operator K-theory]] of the convolution $C^\ast$-algebra of a [[topological groupoid]] $\mathcal{X}_\bullet$ may be thought of as the [[topological K-theory]] of the corresponding [[topological stack]]. More generally, for $\mathcal{X} \to \mathbf{B}^2 U(1)$ a [[principal 2-bundle]] ([[bundle gerbe]]) on the groupoid/stack, the [[operator K-theory]] of the corresponding twisted convolution algebra is the [[twisted K-theory]] of the stack. (\hyperlink{TXLG}{Tu, Xu, Laurent-Gengoux 04}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[groupoid quantale]] - an analogue of groupoid ring/algebra where free abelian groups/vector spaces are replaced by free [[sup-lattice]]s \item [[noncommutative topology]], [[noncommutative geometry]] [[KK-theory]] \item [[bibundle]], [[bimodule]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{for_partial_orders}{}\subsubsection*{{For partial orders}}\label{for_partial_orders} \begin{itemize}% \item [[Gian-Carlo Rota]], \emph{On the foundations of combinatorial theory I: theory of M\"o{}bius functions} , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340--368. \end{itemize} \hypertarget{for_discrete_geometry}{}\subsubsection*{{For discrete geometry}}\label{for_discrete_geometry} The [[homotopy colimit]]-interpretation of category algebras over discrete categories is discussed in \begin{itemize}% \item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], \emph{[[Topological Quantum Field Theories from Compact Lie Groups]]} (\href{http://arxiv.org/abs/0905.0731}{arXiv}) \end{itemize} Groupoid algebras of geometrically discrete groupoids twisted by [[principal 2-bundles]]/[[bundle gerbes]]/[[central extension of groupoids|groupoid central extension]] is reviwed in \begin{itemize}% \item Eitan Angel, \emph{A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras}, PhD thesis (2010) Cyclic cocycles on twisted convolution algebras, (\href{http://arxiv.org/abs/1103.0578}{arXiv.1103.0578}) \end{itemize} \hypertarget{ReferencesForSmoothGeometry}{}\subsubsection*{{For continuous/smooth geometry}}\label{ReferencesForSmoothGeometry} \hypertarget{convolution_algebras}{}\paragraph*{{Convolution $C^\ast$-algebras}}\label{convolution_algebras} The study of convolution [[C-star algebras]] of [[Lie groupoids]] goes back to \begin{itemize}% \item [[Jean Renault]], \emph{A groupoid approach to $C^\ast$ algebras}, Springer Lecture Notes in Mathematics, 793, Springer-Verlag, New York, 1980. \end{itemize} Where the [[integration]] is performed against a fixed [[Haar measure]]. Surveys are for instance in \begin{itemize}% \item [[Nigel Higson]], \emph{Groupoids, $C^\ast$-algebras and Index theory} (\href{http://folk.uio.no/rognes/higson/zurich.pdf}{pdf}) \item PlanetMath, \emph{\href{http://planetmath.org/groupoidcconvolutionalgebras}{groupoid $C^\ast$-convolution algebras}}. \end{itemize} The construction via [[sections]] of [[bundles]] of [[half-densities]] (avoiding a choice of Haar measure) is due to \begin{itemize}% \item [[Alain Connes]], \emph{A survey of foliations and operator algebras}, Proc. Sympos. Pure Math., AMS Providence, 32 (1982), 521--628 \end{itemize} A review is on page 106 of \begin{itemize}% \item [[Alain Connes]], \emph{[[Noncommutative Geometry]]}, Academic Press, San Diego, CA, (1994) \end{itemize} See also \begin{itemize}% \item [[Joost Nuiten]], \emph{[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]}, master thesis, August 2013 \end{itemize} More along these lines is in \begin{itemize}% \item [[Paul Muhly]], Dana P. Williams, \emph{Continuous trace groupoid $C^\ast$-algebras II} Math. Scand. 70 (1992), no. 1, 127--145; MR 93i:46117). (\href{http://www.math.dartmouth.edu/~dana/dpwpdfs/muhwil-ms90.pdf}{pdf}) \item [[Paul Muhly]], [[Jean Renault]], Dana P. Williams, \emph{Continuous trace groupoid $C^\ast$-algebras III} , Transactions of the AMS, vol 348, Number 9 (1996) (\href{http://www.jstor.org/stable/2155247}{jstor}) \item Mdlina Roxana Buneci, \emph{Groupoid Representations,} Ed. Mirton: Timishoara (2003). \item Mdlina Roxana Buneci, \emph{Groupoid $C^\ast$-Algebras}, Surveys in Mathematics and its Applications, Volume 1: 71--98. (\href{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf}{pdf}) \item Mdlina Roxana Buneci, \emph{Convolution algebras for topological groupoids with locally compact fibers} (2011) (\href{http://journals.bg.agh.edu.pl/OPUSCULA/31-2/31-2-02.pdf}{pdf}) \end{itemize} A review in the context of [[geometric quantization]] is in section 4.3 of \begin{itemize}% \item [[Rogier Bos]], \emph{Groupoids in geometric quantization} PhD Thesis (2007) \href{http://www.math.ist.utl.pt/~rbos/ProefschriftA4.pdf}{pdf} \end{itemize} Specifically the convolution $C^\ast$-algebras of [[bundle gerbes]] regarded as [[centrally extended groupoids]] (algebras whose [[modules]] (see \hyperlink{ReferencesModulesOverConvolutionAlgebra}{below}) are [[gerbe modules]]/[[twisted bundle]]) is discussed in section 5 of \begin{itemize}% \item [[Alan Carey]], Stuart Johnson, [[Michael Murray]], \emph{Holonomy on D-Branes}, J. Geom. Phys. 52 (2004) 186-216 (\href{http://arxiv.org/abs/hep-th/0204199}{arXiv:hep-th/0204199}) \end{itemize} A discussion of convolution algebras of [[symplectic groupoids]] (in the context of [[geometric quantization of symplectic groupoids]]) is in \begin{itemize}% \item [[Eli Hawkins]], \emph{A groupoid approach to quantization} (\href{http://arxiv.org/abs/math.SG/0612363}{arXiv:math.SG/0612363}) \end{itemize} Functoriality of the construction of $C^\ast$-convolution algebras (its extension to groupoid-[[bibundles]]) is discussed in \begin{itemize}% \item [[Paul Muhly]], [[Jean Renault]], D. Williams, \emph{Equivalence and isomorphism for groupoid $C^\ast$-algebras}, J. Operator Theory 17 (1987), no. 1, 3--22. \end{itemize} \begin{itemize}% \item [[Janez Mr?un]], \emph{Functoriality of the bimodule associated to a Hilsum-Skandalis map}. K-Theory 18 (1999) 235--253. \end{itemize} \begin{itemize}% \item [[Klaas Landsman]], \emph{The Muhly-Renault-Williams theorem for Lie groupoids and its classical counterpart}, Lett. Math. Phys. 54 (2000), no. 1, 43--59. (\href{http://arxiv.org/abs/math-ph/0008005}{arXiv:math-ph/0008005}) \end{itemize} \begin{itemize}% \item [[Klaas Landsman]], \emph{Operator Algebras and Poisson Manifolds Associated to Groupoids}, Commun. Math. Phys. 222, 97 -- 116 (2001) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.2716}{web}) \end{itemize} \hypertarget{ReferencesConvolutionHopfAlgebroids}{}\paragraph*{{Convolution Hopf algebroids}}\label{ReferencesConvolutionHopfAlgebroids} A characterization of the convolution algebras of [[étale groupoids]] with their [[Hopf algebroid]] structure is in \begin{itemize}% \item [[Janez Mr?un]], \emph{On spectral representation of coalgebras and Hopf algebroids} (\href{http://arxiv.org/abs/math/0208199}{arXiv:math/0208199}) \item [[Jure Kali?nik]], [[Janez Mr?un]], \emph{Equivalence between the Morita categories of etale Lie groupoids and of locally grouplike Hopf algebroids} (\href{http://arxiv.org/abs/math/0703374}{arXiv:math/0703374}) \end{itemize} \hypertarget{ReferencesModulesOverConvolutionAlgebra}{}\paragraph*{{Modules over Lie groupoid convolution algebras and K-theory}}\label{ReferencesModulesOverConvolutionAlgebra} Discussion of [[modules]] over Lie groupoid convolution algebras is in the following articles. In (\hyperlink{Renault80}{Renault80}) [[measurable space|measurable]] representations of topological groupoids are related to modules over their $L^1$ convolution [[star algebra]] [[Banach algebras]] hence over their envoloping $C^\ast$-algebras. In (\hyperlink{Bos}{Bos, chapter 7}) is discussion refining this to continuous representations and representation of a convolution $C^\ast$-algebra, also in section 4 of: \begin{itemize}% \item [[Rogier Bos]], \emph{Continuous representations of groupoids} (\href{http://arxiv.org/abs/math/0612639}{arXiv:math/0612639}) \end{itemize} Representation of convolution algebras of [[étale groupoids]] is in \begin{itemize}% \item [[Jure Kali?nik]], \emph{Groupoid representations and modules over the convolution algebras} (\href{http://arxiv.org/abs/0806.1832}{arXiv:0806.1832}) \end{itemize} The [[operator K-theory]] of groupoid convolution algebras (the [[topological K-theory]] of the corresponding [[differentiable stacks]]) is discussed in \begin{itemize}% \item [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], \emph{Twisted K-theory of differentiable stacks}, Annales scientifiques de l'\'E{}cole Normale Sup\'e{}rieure (2004) Volume: 37, Issue: 6, page 841-910 (\href{http://arxiv.org/abs/math/0306138}{arXiv:math/0306138}) \end{itemize} Construction of cocycles in [[KK-theory]] and [[spectral triples]] from groupoid convolution is in \begin{itemize}% \item Bram Mesland, \emph{Groupoid cocycles and K-theory} (\href{http://arxiv.org/abs/1005.3677}{arXiv:1005.3677}) \end{itemize} [[!redirects category algebras]] [[!redirects groupoid algebra]] [[!redirects groupoid algebras]] [[!redirects incidence algebra]] [[!redirects incidence algebras]] [[!redirects groupoid convolution algebra]] [[!redirects groupoid convolution algebras]] [[!redirects convolution algebra of a Lie groupoid]] [[!redirects convolution algebras of a Lie groupoid]] [[!redirects convolution algebra of Lie groupoids]] [[!redirects convolution algebras of Lie groupoids]] [[!redirects Lie groupoid convolution algebra]] [[!redirects Lie groupoid convolution algebras]] [[!redirects category convolution algebra]] [[!redirects category convolution algebras]] [[!redirects twisted groupoid algebra]] [[!redirects twisted groupoid algebras]] [[!redirects twisted groupoid convolution algebra]] [[!redirects twisted groupoid convolution algebras]] [[!redirects groupoid convolution C\emph{-algebra]] [[!redirects groupoid convolution C}-algebras]] [[!redirects groupoid convolution C-star-algebra]] [[!redirects groupoid convolution C-star-algebras]] [[!redirects groupoid convolution product]] [[!redirects groupoid convolution products]] \end{document}