symmetric monoidal (∞,1)-category of spectra
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 category convolution algebra or just category algebra for short.
Often this is considered specifically for groupoids and hence accordingly called groupoid convolution algebra or just groupoid algebra for short. (For one-object delooping groupoids of groups, groupoid algebras reduce to group algebras.) The 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 twisted groupoid convolution algebra, twisted by the characteristic class of the line 2-bundle (TXLG).
For “bare” categories/groupoids (i.e.: internal to Set) these constructions are canonical. But under mild conditions or else when equipped some suitable extra structure, it generalizes to internal categories/internal groupoids in 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 Hopf algebroid and may be used to 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 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 (Connes 94). Due to this relation the groupoid convolution product is also referred to as a star product and denoted “$\star$”. Groupoid C*-algebras form a rich sub-class of all C*-algebras, including crossed product C*-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 (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 (EH 06) strict deformation quantization is given fairly generally by twisted groupoid convolution algebras. See at geometric quantization of symplectic groupoids for more on this. For the discrete but higher geometry of infinity-Dijkgraaf-Witten theory quantization by higher groupoid convolution algebras is indicated in (FHLT 09).
Let $\mathcal{C}$ be a small category and let $R$ be a ring.
The category algebra or convolution algebra $R[\mathcal{C}]$ of $\mathcal{C}$ over $R$ is the $R$-algebra
whose underlying $R$-module is the free module $R[\mathcal{C}_1]$ over the set of morphisms of $\mathcal{C}$;
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:
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
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
This expresses convolution of functions.
If the small category $\mathcal{C}_\bullet$ is a groupoid, hence equipped with an inversion map
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-fuctor
makes the convolution algebra a star-algebra.
If $\mathcal{C}$ is a groupoid with extra 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 1 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.
See the References – For continuous/smooth geometry.
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
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
(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 groupoid convolution algebra of smooth compactly supported functions. As in remark 1, this is naturally a star-algebra with involution $inv^\ast$.
This construction originates around (Connes 82).
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. 3 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
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 $x$:
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 convolution $C^\ast$-algebra of the Lie groupoid $\mathcal{G}_\bullet$.
This is recalled at (Connes 94, prop. 3 on p. 106).
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 (Mrčun 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 (Kališnik-Mrčun 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 étale Lie groupoids into their convolution Hopf algebroids.
In (Muhly-Renault-Williams 87, 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), but only equivalence-bibundles are considered and are shown to yield Morita equivalence bimodules (no discussion of composition and functoriality here).
We discuss equivalent characterizations of category algebras/groupoid algebras that are useful in certain context
Apparently for $\mathcal{C}$ a groupoid the category algebra of $C$ is the 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 (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.
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.
The category algebra of a category $C$ is a special case of a general construction of spans (see also at 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
equipped with a composition operation: a morphism of spans from the composite span
to the original one, i.e. a morphism
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
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
whose elements are 2-arrows of the form
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
The algebra structure on $k[C]$ is similarly encoded in the diagrammatics: given two elements
their pre-composite is the diagram
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 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
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-branes 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
This is related to observations such as described here:
Urs Schreiber, QFT of Charged n-Particle: T-Duality
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 $X$.
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.
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 $X$.
$\mathcal{C} = \mathbf{B}G$ is the delooping groupoid of a discrete group $G$ (the groupoid with a single object and $G$ as its set of morphisms), then def. 1 reduces to that of the group algebra of $G$:
under construction
We discuss here a natural generalization of the notion of groupoid convolution algebras to higher algebras for 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 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
By the discussion at 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
There is then the following systematic refinement of this to higher groupoids and higher algebra: by the discussion at 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 4 below.
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-coskeletal as a simplicial object in Grpd.
Given a 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:
groupoid convolution algebras $C(\mathcal{G}_0)$ and $C(\mathcal{G}_1)$,
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 $\partial_1 \colon \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1$.
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 double groupoid convolution 2-algebra of $\mathcal{G}_\bullet$.
(Here we make invariant sense of the tensor product by evaluating on a Reedy fibrant representative.)
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:
$\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G\stackrel{\to}{\stackrel{\to}{\to}} \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G$;
$\underset{\longrightarrow}{\lim}(\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *) \simeq \mathbf{B}G$.
(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 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
given on $\phi \in C(G)$ and $g_1, g_2 \in G$ by
This is that standard coproduct on the standard dual Hopf algebra associated with $G$.
In summary this means that (for $G$ a finite group):
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 3-module) is the convolution algebra of $G$;
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 $G$.
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.
The homotopy colimit-interpretation of category algebras over discrete categories is discussed in
Groupoid algebras of geometrically discrete groupoids twisted by principal 2-bundles/bundle gerbes/groupoid central extension is reviwed in
Eitan Angel, A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras, PhD thesis (2010)
Cyclic cocycles on twisted convolution algebras, (arXiv.1103.0578)
The study of convolution C-star algebras of Lie groupoids goes back to
Where the integration is performed against a fixed Haar measure. Surveys are for instance in
Nigel Higson, Groupoids, $C^\ast$-algebras and Index theory (pdf)
PlanetMath, groupoid $C^\ast$-convolution algebras.
The construction via sections of bundles of half-densities (avoiding a choice of Haar measure) is due to
A review is on page 106 of
More along these lines is in
Paul Muhly, Dana P. Williams, Continuous trace groupoid $C^\ast$-algebras II Math. Scand. 70 (1992), no. 1, 127–145; MR 93i:46117). (pdf)
Paul Muhly, Jean Renault, Dana P. Williams, Continuous trace groupoid $C^\ast$-algebras III , Transactions of the AMS, vol 348, Number 9 (1996) (jstor)
Mădălina Roxana Buneci, Groupoid Representations, Ed. Mirton: Timishoara (2003).
Mădălina Roxana Buneci, Groupoid $C^\ast$-Algebras, Surveys in Mathematics and its Applications, Volume 1: 71–98. (pdf)
Mădălina Roxana Buneci, Convolution algebras for topological groupoids with locally compact fibers (2011) (pdf)
A review in the context of geometric quantization is in section 4.3 of
Specifically the convolution $C^\ast$-algebras of bundle gerbes regarded as centrally extended groupoids (algebras whose modules (see below) are gerbe modules/twisted bundle) is discussed in section 5 of
A discussion of convolution algebras of symplectic groupoids (in the context of geometric quantization of symplectic groupoids) is in
Functoriality of the construction of $C^\ast$-convolution algebras (its extension to groupoid-bibundles) is discussed in
A characterization of the convolution algebras of étale groupoids with their Hopf algebroid structure is in
Janez Mrčun, On spectral representation of coalgebras and Hopf algebroids (arXiv:math/0208199)
Jure Kališnik, Janez Mrčun, Equivalence between the Morita categories of etale Lie groupoids and of locally grouplike Hopf algebroids (arXiv:math/0703374)
Discussion of modules over Lie groupoid convolution algebras is in the following articles.
In (Renault80) 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 (Bos, chapter 7) is discussion refining this to continuous representations and representation of a convolution $C^\ast$-algebra, also in section 4 of:
Representation of convolution algebras of étale groupoids is in
The operator K-theory of groupoid convolution algebras (the topological K-theory of the corresponding differentiable stacks) is discussed in
Construction of cocycles in KK-theory and spectral triples from groupoid convolution is in