categorification via groupoid schemes

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### Context
#### Categorification
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#### Arithmetic geometry
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#Contents#
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## General case

On this page, we explain a connection between [[decategorification]] and [[degroupoidification]], which appears when one considers [[groupoids internal]] to [[scheme]]s.  One can construct a category of sheaves attached to such a groupoid, and a sequence of finite groups, given by points over a [[finite field]].  This connection has been exploited quite widely within [[arithmetic geometry]], but has not really been stated in the context of categorification.

### Basic data

Let $X\rightrightarrows Y$ be a groupoid internal to the category of schemes ($Y$ is the scheme of [[object]]s and $X$ the scheme of [[morphism]]s) where both schemes are of finite type.

Then we can make two constructions from this scheme:

* The $\mathbb{F}_p$-points $X(\mathbb{F}_p) \rightrightarrows Y(\mathbb{F}_p)$ form a finite groupoid (with no additional structure).

* There is a [[simplicial object|simplicial]] scheme $N(X\rightrightarrows Y)$ given by the [[nerve]] of $X\rightrightarrows Y$ (this sends $\Delta^n$ to $X\times_Y X\times_Y\cdots  \times_Y X$, with the obvious face and degeneracy maps).

  * One special case is when this is an [[action groupoid]], for an algebraic group $G$ (say) acting on $Y$.  The resulting simplicial scheme should be thought of as the Borel space for the action on $Y$. The notation $Y//G$ (more properly $Y \sslash G$, but this doesn\'t seem to render for everybody) for this 'homotopy' or weak quotient is used in other fields, but this conflicts here with [[GIT quotient|GIT quotients]].

  * More generally, but for this notational clash, $Y//X$ could denote the nerve. Another option is $\mathcal{N}Y$, where $Y$ denotes the whole groupoid, rather than just the scheme of arrows.

### Linearization, and the function sheaf correspondence

The latter construction can be "linearized" in an analogous way to [[geometric function theory]], except using the [[constructible derived category|derived category of sheaves with finite rank constructible cohomology]] with coefficients in the $\ell$-[[adic number|adics]] $\mathbb{Q}_\ell$ for some prime $\ell$ of $N(X\rightrightarrows Y)$.  Let $D(N(X\rightrightarrows Y))$ denote this category.  In fact, we will require the graded version of this category, provided by [[mixed sheaf|mixed sheaves]].  We denote this graded category $D_{mix}(N(X\rightrightarrows Y))$

There is a map $\alpha_{q,\ell}$ from $K^0(D_{mix}(N(X\rightrightarrows Y))$ to the set $\mathbb{Q}_\ell[N(X \rightrightarrows Y)(\mathbb{F}_q)]$ of $\mathbb{Q}_\ell$-valued functions on any $\mathbb{F}_q$ point of $N(X\rightrightarrows Y)$ where $\ell$ and $q$ where $\ell$ is a prime and $\ell\nmid q$, given by the supertrace of automorphism of the stalk of the sheaf at that point induced by the action of the Frobenius $q^n$ on $X\times_{\mathrm{Spec}\mathbb{Z}}  \mathrm{Spec}\overline{\mathbb{F}_q}$.

This map has the property that multiplying by the [[motivic integral]] of $\mathbb{A}^1$ (i.e., $m_!m^*$ for the map $m:X\times \mathbb{A}^1\to X$) multiplies the corresponding function by $q$.

Furthermore, no non-zero element of $K^0(D_{mix}(N(X\rightrightarrows Y))$ is killed by this map for all  $q$.

If $N(X_1\rightrightarrows Y_1)\overset{f}\leftarrow N(Z\rightrightarrows W) \overset{g}\rightarrow N(X_2 \rightrightarrows Y_2)$ is a [[span]] of groupoids of schemes, then we have a functor $g_!f^*:D_{mix}(N(X_1\rightrightarrows Y_1))\to D_{mix}(N(X_2 \rightrightarrows Y_2))$, given by pull-back followed by compactly supported pushforward.

Furthermore, there is an induced span of groupoids on this set of $\mathbb{F}_q$-points, which in turn induces a map $(gf)^\dagger$ from $\mathbb{Q}_\ell[N(X_1\rightrightarrows Y_1)(\mathbb{F}_q)]\to \mathbb{Q}_\ell[N(X_2 \rightrightarrows Y_2)(\mathbb{F}_q)]$.

The content of the [[Grothendieck trace formula]] connects these two constructions.

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###### Theorem

The functor  $g_!f^*:D_{mix}(N(X_1\rightrightarrows Y_1))\to D_{mix}(N(X_1\rightrightarrows Y_1))$ induces a map $(gf)_\dagger:K^0(D_{mix}(N(X_1\rightrightarrows Y_1)))\to K^0(D_{mix}(N(X_2 \rightrightarrows Y_2)))$ such that the diagram
$$
  \array{
      K^0(D_{mix}(N(X_1\rightrightarrows Y_1))&\overset{(gf)_\dagger}\to& K^0(D_{mix}(N(X_2 \rightrightarrows Y_2)))   \\
     \downarrow^{\alpha_{q,\ell}} && \downarrow^{\alpha_{q,\ell}}
     \\
    \mathbb{Q}_\ell[(N(X_1\rightrightarrows Y_1)\mathbb{F}_q)] &\overset{(gf)^\dagger}\to& \mathbb{Q}_\ell[N(X_2 \rightrightarrows Y_2)(\mathbb{F}_q)]
  }
$$
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That is, modulo a few details:

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The [[decategorification]] of the mixed derived category and [[degroupoidification]] of the $\mathbb{F}_q$ points coincide.
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We note that in many examples of geometric [[categorification]], the fact that the decategorification is correct is checked by   understanding the degroupoidification  and using this theorem, though it is typically not stated this explicitly.

## Examples
Since all groupoids appearing below are action groupoids, I'll denote them $X/G$ in place of $G\times X\rightrightarrows X$.

### The Hecke algebra
 {#TheHeckeAlgebra}

If one takes the span of groupoids
$$\array{B\backslash G/B &&&&\\
&\nwarrow&&&\\
&& B\backslash G\times_B G/B &\to &B\backslash G/B\\
&\swarrow&&&\\
B\backslash G/B &&&&\\}$$
over the action groupoid for $B\times B$ acting on the left and right on $G$ for a simple algebraic group, then the resulting monoidal category is the [[Hecke category]] of G.  Its decategorification is the [[Hecke algebra]] of G, which is checked by showing the Hecke algebra is groupoidified
by the F_q points of this, a result which was proved [Iwahori in 1964](http://www.ams.org/mathscinet-getitem?mr=165016).

### Geometric Satake
 {#GeometricSatake}

Consider the groupoid given by the [[action]] of $G(\mathbb{Z}[t])\times G(\mathbb{Z}[t])$ (for $\mathbb{Z}[t]$ the [[polynomial ring]]) on the left and right on $G(\mathbb{Z}(t))$ considered as as [[pro-object|pro-]][[formal schemes|ind-schemes]] over $\mathbb{Z}$.  The [[derived category]] of this groupoid scheme is monoidal by the analogous diagram to that [above](#TheHeckeAlgebra).

The groupoidification of the $\mathbb{F}_q$ points was shown by Satake (see at _[[Satake equivalence]]_) to be the [[representation ring]]
of ${^L G}$, the [[Langlands dual group]].  Thus, $D_{mix}(G(\mathbb{Z}[t])\backslash G(\mathbb{Z}(t)/ G(\mathbb{Z}[t])$ also contains a categorification of the representation ring of ${^L G}$.

In fact, Mirković and Vilonen showed that the subcategory of $D_{mix}(G(\mathbb{Z}[t])\backslash G(\mathbb{Z}(t))/ G(\mathbb{Z}[t])$ consisting of [[perverse sheaves]] is equivalent to the category of representations of ${^L G}\times_{\mathrm{Spec}\mathbb{Z}} \mathrm{Spec}\overline{\mathbb{Q}_\ell}$ as an [[algebraic group]].  Moreover, if one replaces $\mathbb{Q}_\ell$ by any other ring both in the [[coefficient]] of the sheaves, and the base of the algebraic group, the result still holds.

### The Hall algebra and Lusztig's categorification

If one takes the groupoid of representations of a Dynkin quiver with fixed dimension vector and basis $E_d$ (with each vector only having components over one dot), with the morphisms given by isomorphisms of representations (note, this is the action groupoid for a product of general linear groups $GL_d$ acting on a finite dimensional vector space), the resulting simplicial scheme is the fine moduli space of representations of that quiver.

There is a span of groupoids
$$
\array{E_{d''}/GL_{d''} &&&&\\
&\nwarrow&&&\\
&& \mathrm{SES}(d,d',d'')&\to &E_d/GL_d\\
&\swarrow&&&\\
E_{d'}/GL_{d'} &&&&\\}
$$
where the span in the middle is the set of short exact sequences of representations with basis of dimension vector $d',d''$ mapping into and out of one of dimension vector $d$, preserving all bases, and the morphism are the obvious possible basis changes in this situation (this the action of groupoid for the parabolic in $GL_d$ preserving this filtration).

It was proved by Ringel that the $\mathbb{F}_q$ points of this variety groupoidify the upper half of the quantum group (of course, not in that language).  The natural resulting categorification was described by Lusztig and used to construct the canonical basis of $U_q(\mathfrak{g})$ where $ \mathfrak{g}$ is the simple Lie algebra associated to this quiver.

This categorification was described independently by Rouquier and Khovanov--Lauda combinatorially.

## References

* Wikipedia, _[Satake isomorphism](http://en.wikipedia.org/wiki/Satake_isomorphism)_


categorified Dold-Kan correspondence

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#### Homological algebra
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#### Higher category theory
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#Contents#
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##Idea

A [[categorification]] of the [[Dold-Kan correspondence]] appears in ([Dyckerhoff17](#Dyckerhoff17)), where [[Ab]], the [[category]] of [[abelian groups]], is replaced by the [[(∞,2)-category]] of [[stable (∞, 1)-categories]].

The [[equivalence]] between [[simplicial abelian groups]] and [[connective chain complex|connective]] [[chain complexes]] from the ordinary [[Dold-Kan correspondence]] becomes an equivalence between 2-simplicial stable (∞, 1)-categories and connective chain complexes of stable (∞, 1)-categories.

This should lead to a categorified version of [[homological algebra]] and of [[cohomology]], for example, through a categorified version of an [[Eilenberg-Mac Lane spectrum]].

## Categorified nerve functor

The [nerve](nerve#nerve_of_chain_complexes) in the ordinary Dold-Kan correspondence sends a connective chain complex in an [[abelian category]] $\mathcal{A}$ to a [[simplicial object]] in $\mathcal{A}$. The _categorified nerve_, $\mathcal{N}$, is playing an analogous role on connective chain complexes of stable (∞, 1)-categories. This nerve unifies various known constructions from [[algebraic K-theory]], in particular, for $\mathcal{B}, \mathcal{B}_0, \mathcal{B}_1$ all stable (∞, 1)-categories:

* For $\mathcal{B}[1]$, the chain complex concentrated in degree 1, $\mathcal{N}(\mathcal{B}[1])$ is a $(\infty,1)$-version of the [[Waldhausen S-construction]].

* For $f: \mathcal{B}_1 \to \mathcal{B}_0$, then applying $\mathcal{N}$ to the corresponding chain complex concentrated in degrees $0$ and $1$ is an $(\infty,1)$-version of the _relative_ Waldhausen S-construction.

* For the chain complex concentrated in degree $2$, $\mathcal{N}(\mathcal{B}[2])$ is an $(\infty,1)$-version of Hesselholt-Madsen's $S^{2,1}_{\bullet}$-construction. (See the reference at [[real algebraic K-theory]].)

* For the chain complex concentrated in degree $k$, $\mathcal{N}(\mathcal{B}[k])$ can be seen as a categorification of the [[Eilenberg-Mac Lane space]] $N(B[k])$.

## Related concepts

* [[Waldhausen S-construction]]
* [[real algebraic K-theory]]

##References

* {#Dyckerhoff17} [[Tobias Dyckerhoff]], _A categorified Dold-Kan correspondence_ ([arXiv:1710.08356](https://arxiv.org/abs/1710.08356))


category

> This page is about the concept in [[mathematics]]. For the concept of the same name in [[philosophy]] see at _[[category (philosophy)]]_.

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### Context
#### Category theory
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# Contents
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## Idea

* A category consists of a collection of things and binary relationships (or transitions) between them, such that these relationships can be combined and include the "identity" relationship "is the same as."

* A category is a [[quiver]] (a [[directed graph]] with multiple edges) with a rule saying how to _compose_ two edges that fit together to get a new edge.  Furthermore, each vertex has an edge starting and ending at that vertex, which acts as an identity for this composition.

* A category is a combinatorial model for a [[directed space]] -- a "directed [[homotopy 1-type]]" in some sense.  It has "points", called _objects_, and also directed "paths", or "processes" connecting these points, called _morphisms_.  There is a rule for how to compose paths; and for each object there is an identity path that starts and ends there.

* More precisely, a category consists of a collection of [[object|objects]] and
a collection of [[morphism|morphisms]].  Every morphism
has a [[source]] object and a [[target]] object.  If $f$ is a morphism with $x$ as its source and $y$ as its target,
we write
$$f : x \to y$$
and we say that $f$ is a morphism from $x$ to $y$.  In a category, we can [[composition|compose]] a morphism $g : x \to y$ and a morphism $f : y \to z$ to get a morphism $f \circ g : x \to z$.  Composition is associative and satisfies the left and right unit laws.

A good example to keep in mind is the category [[Set]], in which the objects are sets and a morphism $f : x \to y$ is a function from the set $x$ to the set $y$.  Here composition is the usual composition of functions.

For more background on and context for categories see

* [[category theory]].

## Definitions

There are two broad ways to write down the definition of category; in the usual [[foundations of mathematics]], these two definitions are equivalent.  It is good to know both, for several reasons:

*  Each introduces its own system of notation, both of which are useful in other parts of category theory, so one should know them.

*  One definition generalises quite nicely to the notion of [[internal category]], while the other generalises quite nicely to the notion of [[enriched category]]; these are both important concepts.

*  When examining alternative foundations, sometimes one definition or the other may be more appropriate; in any case, one will want to examine the question of their equivalence.

The two definitions may be distinguished by whether they use a single collection of all [[morphisms]] or several collections of morphisms, a [[family of sets|family of collections]] indexed by pairs of [[objects]].

### With one collection of morphisms
 {#OneCollectionOfMorphisms}

[[Grothendieck 61, Section 4](#Grothendieck61)]

A __category__ $C$ consists of

*  a [[collection]] (see [Foundational issues](#foundations) and [Size issues](#size)) $C_0$ of __[[objects]]__;

*  a collection $C_1$ of __[[morphisms]]__ (or __arrows__);

*  for every morphism $f$, an object $s(f)$ (called its __[[source]]__ or __domain__), and an object $t(f)$ (called its __[[target]]__ or __codomain__);

*  for every pair of morphisms $f$ and $g$, where $t(f) = s(g)$, a morphism $g \circ f$, called their __[[composite]]__ (also  written $g f$ or sometimes $f;g$— see [[diagrammatic order]]);

*  for every object $x$, a morphism $id_x$ (or $1_x$), called the __[[identity morphism]]__ on $x$;

*  such that the following properties are satisfied:

   *  source and target are respected by composition: $s(g \circ f) = s(f)$ and $t(g\circ f) = t(g)$;

   *  source and target are respected by identities: $s(1_x) = x$ and $t(1_x) = x$;

   *  composition is [[associative]]:
      $(h \circ g)\circ f = h\circ (g \circ f)$ whenever $t(f) = s(g)$ and $t(g) = s(h)$;

   *  composition satisfies the [[unit law|left and right unit laws]]:
      if $s(f) = x$ and $t(f) = y$, then $1_y \circ f = f = f \circ 1_x$.

People also often write $x \in C$ instead of $x \in C_0$ as a short way to indicate that $x$ is an object of $C$.  Also, some people write $Ob(C)$ and $Mor(C)$ instead of $C_0$ and $C_1$.  One usually writes $f\colon x \to y$ if $f \in C_1$ to state that $s(f) = x$ and $t(f) = y$.  Finally, people often write $hom(x,y)$, $hom_C(x,y)$, or $C(x,y)$ for the collection of morphisms $f\colon x \to y$.

If the [[identity]]-assigning map and its axiom is omitted, then one speaks of a _[[semicategory]]_.

### With a family of collections of morphisms
 {#AFamilyOfCollectionsOfMorphisms}

A __category__ $C$ consists of

*  a [[collection]] (see [Foundational issues](#foundations) and [Size issues](#size)) $C_0$ of __[[objects]]__;

*  for each pair $x,y$ of objects, a collection $C_1(x,y)$ of __[[morphisms]] from $x$ to $y$__;

*  for each pair of morphisms $f$ in $C_1(x,y)$ and $g$ in $C_1(y,z)$, a morphism $g \circ f$ in $C_1(x,z)$, called their __[[composite]]__ (also  written $g f$ or sometimes $f;g$— see [[diagrammatic order]]);

*  for each object $x$, a morphism $id_x$ (or $1_x$) in $C_1(x,x)$, called the __[[identity morphism]]__ on $x$;

*  such that the following properties are satisfied:

   *  composition is [[associative]]:
      for each quadruple $w,x,y,z$ of objects, if $f \in C_1(y,z)$, $g \in C_1(x,y)$, and $h \in C_1(w,x)$, then $(f \circ g)\circ h = f\circ (g \circ h)$;

   *  composition satisfies the [[unit law|left and right unit laws]]:
      for each pair $x,y$ of objects, if $f \in C_1(x,y)$, then $1_y \circ f = f = f \circ 1_x$.

People also often write $x \in C$ instead of $x \in C_0$ as a short way to indicate that $x$ is an object of $C$.  Also, some people write $Ob(C)$ instead of $C_0$ and $hom(x,y)$, $hom_C(x,y)$, or $C(x,y)$ instead of $C_1(x,y)$.  One usually writes $f\colon x \to y$ to state that $f \in C_1(x,y)$.  Finally, people often write $C_1$ or $Mor(C)$ for the [[disjoint union]] $\biguplus_{x \in C_0} \biguplus_{y \in C_0} C_1(x,y)$.

### Equivalence between the two definitions

Given a one-collection-of-morphisms category $C_1\rightrightarrows C_0$, we define a family-of-collections-of-morphisms category by taking $C_1(x,y)$ to be the [[preimage]] of $(x,y)$ under the [[function]] $(s,t):C_1 \to C_0\times C_0$.  Conversely, given a family-of-collections-of-morphisms category we define a one-collection-of-morphisms category by taking $C_1$ to be the [[disjoint union]] of the families of morphisms $C_1 = \coprod_{x,y\in C_0} C_1(x,y)$.  With the straightforward definitions of [[functor]] and [[natural transformation]] in both cases, this sets up a strict [[2-equivalence]] of [[2-categories]].

Note, though, that this 2-equivalence is not an *isomorphism* of 2-categories, because the disjoint union operation has to "tag" each morphism with its domain and codomain.  It seems that the strongest thing that can be said is that, in a [[material set theory]], if a family-of-collections-of-morphisms category $C$ has the property that sets $C_1(x,y)$ are all disjoint, then there is a one-collection-of-morphisms category with $C_1 = \bigcup_{x,y\in C_0} C_1(x,y)$ (the *non*-disjoint union) that gives rise to $C$ on the nose.  The notion of [[protocategory]] is a way to formalize a family-of-collections-of-morphisms category together with the information about how its hom-sets "overlap".

In ordinary category theory one rarely specifies which of the two definitions is being used, although often language implicitly suggests that it is the latter, e.g. when defining a category one first specifies the objects and then specifies what "the morphisms from $X$ to $Y$ are" rather than specifying what "a morphism" is and then what the domain and codomain of each morphism are.  Indeed, when defining a category in this way, one rarely worries about whether the hom-sets are disjoint, meaning that it must be the second definition in use.  Even for the prototypical category [[Set]], if constructed in a material-set-theoretic foundation like [[ZFC]], the natural definition "a morphism from $X$ to $Y$ is a function, i.e. a subset of $X\times Y$ that is total and functional", produces hom-sets that are not disjoint, since a total and functional set of ordered pairs can have any codomain that is a superset of its range.  Moreover, categorical constructions do not "naturally" preserve disjointness of homsets, e.g. in the [[category of elements]] $el(P)$ of a functor $P:C\to Set$ a given morphism in $C$ can "be" a morphism between many different elements of $P$, and similarly for a [[slice category]] and so on.

### Foundational issues
{#foundations}

We said a category has a 'collection' of objects and 'collection'(s) of morphisms. However, different [[mathematical foundations]] have different notions of [[equality]]. For foundations whose notion of 'collection' have [[proposition]]-valued equality, such as in [[set theory]], [[class]]-set theory, or [[extensional type theory]], the two definitions above suffice for defining a category. However, in for other foundations, with a weaker notion of equality, such as [[internalization|internally]] in a [[(2,1)-topos]] like [[Grpd]], in [[Thomas Streicher]]'s groupoid model of types, or in [[homotopy type theory]], there are multiple definitions of a category. The naive definition of the category above with a collection of objects and a collection of morphisms results in a [[wild category]]. If the morphism collections are sets, then the resulting structure is a [[precategory]], and if the collection of objects is a set as well, then the resulting structure is a [[strict category]]. The above definitions in set theory foundations are the same as strict categories in the alternative foundations, but in the alternative foundations categories like [[Set]] are not strict categories. Instead, they happen to be a special kind of precategory called [[univalent categories]], where equality of objects is isomorphism of objects.

### Size issues
{#size}

We said a category has a 'collection' of objects and 'collection'(s) of morphisms.  A category is said to be [[small category|small]] if these collections are all [[sets]] — as opposed to [[proper class|proper classes]], for example.  (The alternatives depend on ones [[foundations|foundations for mathematics]].)

Similarly, a category is [[locally small category|locally small]] if $C_1(x,y)$ is a set for every pair of objects $x,y$ in that category.  The most common motivating examples of categories (such as [[Set]]) are all locally small but not small (unless one restricts their objects in some way).

### Alternative definitions

For some purposes it is useful or necessary to vary the way the ordinary definition of category is expressed. See

* [[single-sorted definition of a category]] -- a variant of the first definition, with only _one_ collection at all ($C_1$, no $C_0$). This is sometimes convenient for technical reasons.

* [[type-theoretic definition of category]] -- a variant of the second definition, interpreted explicitly in [[dependent type theory]].

* [[protocategory]] -- a sort of mixture of the two definitions, in which all the hom-sets $C_1(x,y)$ are subsets of a single set $C_1$, but not necessarily disjoint.

### Equivalent definitions
 {#EquivalenceDefinitions}
 {#EquivalentDefinitions}

A [[category]] is equivalently

* a [[monad]] in the [[2-category]] of [[spans]] of [[sets]];

* a [[monoid]] in the [[monoidal category]] of endospans on the set of objects;

* a [[simplicial set]] which satisfies the [[Segal conditions]];

* a [[simplicial set]] which satisfies the [[weak Kan complex]] conditions strictly:

* hence a [[directed homotopy type theory|directed homotopy type]] which is "1-truncated"

### Generalizations

#### Internal categories

The first definition, with a single collection $C_1$ of morphisms, generalises to the notion of [[internal category]].  Essentially, we define a category internal to (some other category) $D$ as above, with 'collection' interpreted as an object of $D$ and 'function' interpreted as a morphism of $D$.  In particular, a category internal to [[Set]] is the same thing as a small category.

#### Enriched categories

The second definition of a category, as a family $C_1(x,y)$ of collections of morphisms, generalises to the notion of [[enriched category]]: we define a category enriched over (some other category) $D$ as above, with the collection of objects still a 'collection' as before, but with objects of $D$ in place of the collections of morphisms and morphisms of $D$ in place of the various functions.  In particular, a category enriched over [[Set]] is the same thing as a locally small category.

#### Indexed categories

The notion of [[indexed category]] captures the idea of woking "over a base" other than [[Set]].

#### Multicategories etc.

There is a generalization of the notion of category where one allows a morphism to go from several objects to a single object. This is called a [[multicategory]] or [[operad]].  If we additionally allow a morphism to go *to* several objects, we obtain a _[[polycategory]]_ or _[[PROP]]_.

#### Higher categories

See [[higher category theory]].

## Examples

There is the beginning of a [[database of categories]] listing well-known categories (with links to articles on these categories, if such articles exist) and some of their properties.

The classic example of a category is [[Set]], the category with [[set]]s as objects and [[function]]s as morphisms, and the usual composition of functions as composition.  Here are some other famous examples, which arise as variations on this theme:

* [[Vect]] - [[vector space]]s as objects, linear maps as morphisms.

* [[Grp]] - [[group]]s as objects, homomorphisms as morphisms.

* [[Top]] - [[topological space]]s as objects, continuous functions as morphisms.

* [[Diff]] - smooth [[manifold]]s as objects, smooth maps as morphisms.

* [[Ring]] - [[ring]]s as objects, ring homomorphisms as morphisms.

Note that in all these cases the morphisms are actually special sorts of functions. these are [[concrete categories]].  That need not be the case in general!

These classic examples are the original motivation for the term "category": all of the above categories encapsulate one "kind of mathematical structure".  These are often called "concrete" categories (that term also has a [[concrete category|technical definition]] that these examples all satisfy).  But just as widespread in applications as these categorization examples of categories are are other categories (often "[[small category|small]]" ones) which, roughly, model something like _states_ and _processes_ of some system.

* **Poset** A [[partial order|poset]] can be thought of as a category with its elements as objects and one morphism in each $hom(x,y)$ if $x$ is less than or equal to $y$, but none otherwise.

* **Group** A [[group]] is just a category where there's one object and all the morphisms have inverses - we call the morphisms "elements" of the group.  This may seem weird, but it's actually a very useful viewpoint. Here's another way to say it: _A group is a [[groupoid]] with a single object_.

* **Monoid** More generally, a [[monoid]] is a category with a single object. In fact, this is one way to motivate the concept of categories: categories are the [[horizontal categorification|many object version]] of monoids.

* **Groupoid** A [[groupoid]] is a category in which all morphisms are [[isomorphism]]s.

* **Quiver** A [[quiver]] may be identified with the [[free category]] on its [[directed graph]].  Given a directed graph $G$ with collection of vertices $G_0$ and collection of edges $G_1$, there is the _[[free functor|free]] category_ $F(G)$ on the graph whose collection of objects coincides with the collection of vertices, and whose collection of morphisms consists of finite sequences of edges in $G_1$ that fit together head-to-tail. The composition operation in this free category is the concatenation of sequences of edges.

* **Universal structure** A category bearing a structure making it [[initial object|initial]] (or 2-initial) in some [[doctrine]]. Examples include the [[permutation category]] as the free symmetric monoidal category generated by a single object, or the [[simplex category]] which is initial among monoidal categories equipped with a monoid.

## Basic notions

A homomorphism between categories is a [[functor]].

This way [[small categories]] themselves form a category, the category [[Cat]] whose objects are small categories and whose morphisms are functors. This naturally enhances to a [[2-category]] whose [[2-morphism]]s are [[natural transformation]]s between functors.

An [[equivalence of categories]] is an equivalence in [[Cat]], hence a pair of functors going back and forth, that are each others inverse up to [[natural isomorphism]].

Weaker than the notion of a pair of functors exhibiting an equivalence is the notion of a pair of [[adjoint functor]]s.

Other standard operations on categories include

* [[opposite category]]

* [[localization]]

* [[geometric realization of categories]]

## Related concepts

* [[enriched category]]

  * [[continuous category]]

* [[EI-category]]

* [[0-category]], [[(0,1)-category]]

* **category**

  [[semicategory]]

* [[2-category]]

* [[3-category]]

* [[n-category]]

* [[(∞,0)-category]]

* [[(n,1)-category]]

* [[(∞,1)-category]]

* [[(∞,2)-category]]

* [[(∞,n)-category]]

* [[(n,r)-category]]

[[!include oidification - table]]

## References

(For more extensive references see at _[[category theory]]_.)

The concept originates in

* {#EilenbergMacLane45} [[Samuel Eilenberg]], [[Saunders MacLane]], _[[General Theory of Natural Equivalences]]_,  Transactions of the American Mathematical Society **58** 2 (1945) 231-294 [[doi:10.1090/S0002-9947-1945-0013131-6](https://doi.org/10.1090/S0002-9947-1945-0013131-6), [jstor:1990284](http://www.jstor.org/stable/1990284)]

The definition "with a single set of morphisms" (i.e. as [[internal categories]] in [[Set]]) appears in:

* {#Grothendieck61} [[Alexander Grothendieck]], Section 4 of: _Techniques de construction et théorèmes d'existence en géométrie algébrique III: préschémas quotients_, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf))

Textbook accounts:

* {#MacLane97} [[Saunders MacLane]], *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5**, Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]]_, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994)

* [[Roy L. Crole]], *Categories for types*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139172707](https://doi.org/10.1017/CBO9781139172707)]

  > ([[categorical semantics]] of [[type theory]])

Expositions:

* A. Martini, H. Ehrig, and D. Nunes, [Elements of basic category theory](http://citeseer.ist.psu.edu/martini96element.html), Technical Report 96-5, Technical University Berlin.

* Jaap van Oosten, [Basic category theory] (http://www.math.uu.nl/people/jvoosten/syllabi/catsmoeder.pdf).

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_. ([arXiv](http://arxiv.org/abs/1912.10642))

* Andrea Schalk and H. Simmons, [An introduction to category theory in four easy movements](http://www.cs.man.ac.uk/~hsimmons/BOOKS/CatTheory.pdf), notes for a course offered as part of the MSc. in Mathematical Logic, Manchester University.

* Daniele Turim, [Category theory lecture notes](http://www.dcs.ed.ac.uk/home/dt/CT/categories.pdf), 1996-2001.  Based on Mac Lane's book (1998).

A textbook that introduces categories by examples arising in [[mathematical physics]] is

* {#Geroch85} [[Robert Geroch]], *Mathematical Physics*, University of Chicago Press (1985) [[ISBN:9780226223063](https://press.uchicago.edu/ucp/books/book/chicago/M/bo4158035.html)]

A textbook with an eye towards the theory of [[categories of sheaves]] and their application in [[homological algebra]] is

* Kashiwara, Shapira, _[[Categories and Sheaves]]_

The definition of categories in the [[foundations]] of [[homotopy type theory]] (see at _[[internal category in homotopy type theory]]_) is discussed in

* {#AhrensKapulkinShulman13} [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], _Univalent categories and the Rezk completion_,  Mathematical Structures in Computer Science 25.5 (2015): 1010-1039 ([arXiv:1303.0584](http://arxiv.org/abs/1303.0584))

* {#CapriottiKraus17} [[Paolo Capriotti]], [[Nicolai Kraus]], _Univalent Higher Categories via Complete Semi-Segal Types_ ([arXiv:1707.03693](https://arxiv.org/abs/1707.03693))

* [[Jason Gross]], [[Adam Chlipala]], [[David Spivak]], _Experience Implementing a Performant Category-Theory Library in Coq_ ([arXiv:1401.7694](http://arxiv.org/abs/1401.7694))

For more references see [[category theory]].

[[!redirects category]]
[[!redirects categories]]
[[!redirects Category]]
[[!redirects Categories]]

[[!redirects monoidoid]]
[[!redirects monoidoids]]


category (philosophy)

> This entry is about the concept in [[philosophy]]. For the concept of the same name in [[mathematics]] see at _[[category]]_.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A philosophical system of categories looks to provide a list of the basic kinds of entity. Often such systems derive from the linguistic form of the judgements we make about the world.

[[Per Martin-Löf]] ([1993](#Martin-Löf93)) sees a strong affinity with his idea of [[types]]:

> Given how basic the very idea of type is, it is of course unthinkable that there shouldn't be a word already in the tradition for what we here call types. You will all know that the traditional word for what I here call type is category. It was introduced by Aristotle and heavily used by Kant. I will show that the traditional use of the word coincides with the way I am using it here. What I call here the doctrine of types, the idea that an object is always an object of a certain type, really goes back to Aristotle.

Notice that from the perspective of [[homotopy type theory]] then [[types]] are [[infinity-groupoids]], hence kinds of ([[higher category theory|higher]]) [[categories]] in the sense of [Eilenberg-MacLane 45](category#EilenbergMacLane45). For more on this see [below](#LawvereCoRfletiveCategories) and at _[[Science of Logic]]_.

### Aristotle's categories

In his work _[Categories](#AristCat)_, [[Aristotle]] divides entities in the world into the most general kinds according to ten _categories_:

1. Substance
1. Quantity
1. Quality
1. Relation
1. Place
1. Date
1. Posture
1. State
1. Action
1. Passion (being acted upon)

### Kant's categories of pure understanding

> In the Transcendental Analytic, the most crucial as well as the most difficult part of [Kant 1781 "The critique of pure reason"](#Kant1781), he maintained that physics is a priori and synthetic because in its ordering of experience it uses concepts of a special sort. These concepts—"categories," he called them—are not so much read out of experience as read into it and, hence, are a priori, or pure, as opposed to empirical. But they differ from empirical concepts in something more than their origin: their whole role in knowledge is different. For, whereas empirical concepts serve to correlate particular experiences and so to bring out in a detailed way how experience is ordered, the categories have the function of prescribing the general form that this detailed order must take. They belong, as it were, to the very framework of knowledge. But although they are indispensable for objective knowledge, the sole knowledge that the categories can yield is of objects of possible experience; they yield valid and real knowledge only when they are ordering what is given through sense in space and time. (from Encyclopedia Britannica)

According to [[Kant]] the _categories of pure understanding_  are (forming what is known as "the table of categories"):

1. [[quantity]]

   * unity

   * plurality

   * totality

1. [[quality]]

   * reality

   * negation

   * limitation

1. [[relation]]

   * inherence and subsistence ([[substance]] and accident)

   * causality and dependence (cause and effect)

   * community (reciprocity)

1. [[modality]]

   * [[possibility]]

   * actuality (*Wirklichkeit*)

   * [[necessity]]

### Hegel's categories of being
 {#CategoryOfBeing}

According to [[Hegel]] there is refinement of the table, both by adding more categories (and maybe by omitting some?) and by claiming to relate them more systematically by a system of [[unity of opposites]] and of [[Aufhebung|sublation]] of these oppositions. See at _[[Science of Logic]]_ for more on this. Hegel calls the categories the _determinations of being_.

According to [WdL §864](Science+of+Logic#864):

> Die Kategorie ist ihrer Etymologie und der Definition des Aristoteles nach, dasjenige, was von dem Seyenden gesagt, behauptet wird.

> According to its etymology and Aristotle's definition, category is what is predicated or asserted of the existent.

### Lawvere's (co-)reflective categories
 {#LawvereCoRfletiveCategories}

According to [[Lawvere]] the system of categories of Hegel [above](#CategoryOfBeing) are usefully formalized in [[categorical logic]] ("[[objective logic]]") as systems of (co-)[[reflective subcategories]] (in the sense of [[category]] in [[category theory]]) of some ambient [[topos]]. See also at _[[Science of Logic]]_ for more on this.

## Related entries

* [[category of being]]

## References

* {#Kant1781} [[Kant]], _[[Critique of Pure Reason]]_, 1781

* {#AristCat} [[Aristotle]], _Categories_, [online](http://classics.mit.edu/Aristotle/categories.html)

* Amie Thomasson, _[SEP: Categories](http://plato.stanford.edu/entries/categories/)_, Stanford Encyclopedia of Philosophy, 2013.

* Paul Studtmann, _[SEP: Aristotle's Categories](https://plato.stanford.edu/entries/aristotle-categories/)_, Stanford Encyclopedia of Philosophy, 2013.

* {#Martin-Löf93} [[Per Martin-Löf]], _Philosophical aspects of intuitionistic type theory_, unpublished lectures given at the Faculteit der Wijsbegeerte, Rijksuniversiteit, Leiden, Sept-Dec 1993, (Cited in P. Boldini, _Des catégories aux types : un itinéraire en mathématiques appliquées_, [pdf](http://pboldini.free.fr/Papers/mathmu.pdf))

* German Wikipedia, _[Kategorie (Philosophie)](http://de.wikipedia.org/wiki/Kategorie_(Philosophie)_

* Wikipedia, _Categories (Aristotle)_

* Wikipedia, _Category (Kant)_C


category algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#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 _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 [[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 _twisted groupoid convolution algebra_, twisted by the [[characteristic class]] of the [[line 2-bundle]] ([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 _[[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 [[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 ([Connes 94](#Connes094)). 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](#Connes094)) 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](#EH)) [[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 groupoid|discrete]] but [[higher geometry]] of [[infinity-Dijkgraaf-Witten theory]] quantization by higher groupoid convolution algebras is indicated in ([FHLT 09](#FHLT)).

## Definition

### For bare categories (discrete geometry)

+-- {: .num_defn #CategoryAlgebra}
###### Definition

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 $R$-[[associative algebra|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:

  $$
    f \cdot g := \left\lbrace
      \array{
        g \circ f & if\;composable
        \\
        0 & otherwise
      }
    \right.
   \,.
  $$

=--

+-- {: .num_remark #AsConvolutionAlgebra}
###### Remark

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

$$
  \phi_1 \cdot \phi_2
  \;
   \colon
  \;
  f \mapsto
  \sum_{f_2 \circ f_1 = f}
  \phi_2(f_2)
  \cdot
  \phi_1(f_1)
  \,,
$$

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

$$
  \phi_1 \cdot \phi_2
  \;
   \colon
  \;
  f \mapsto
  =
  \sum_{g \in \mathcal{C}_1}
  \phi_2(f \circ g^{-1})
  \cdot
  \phi_1(g)
  \,.
$$

This expresses [[convolution]] of functions.

=--

+-- {: .num_defn #InvolutionByPullbackAlongInversion}
###### Remark

If the small category $\mathcal{C}_\bullet$ is a [[groupoid]], hence equipped with an [[inverse|inversion]] map

$$
  inv \colon \mathcal{C}_1 \to \mathcal{C}_1
$$

then [[pullback of functions]] along this map 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

$$
  \dagger \colon \mathcal{C}_1 \to \mathcal{C}_1
$$

makes the convolution algebra a star-algebra.

=--
+-- {: .num_defn #catalgebraIsWeakHopfalgebra}
###### Remark

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]].

=--

### For Lie groupoids
 {#ForLieGroupoids}

We discuss groupoid convolution [[C*-algebras]] for [[Lie groupoids]]/[[differentiable stacks]] (def. \ref{ContinuousCategoryAlgebraWithHalfDensities}, prop. \ref{GroupoidConvolutionIs2Functor} below).

+-- {: .num_remark #ForLieGroupoids}
###### Remark

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.

=--

See the [References -- For continuous/smooth geometry](#ReferencesForSmoothGeometry).

+-- {: .num_defn #ContinuousCategoryAlgebraWithHalfDensities}
###### Definition
**([[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]]

$$
  \star \;\colon\; C_c(\mathcal{G}_1, \sqrt{\Omega}) \times C_c(\mathcal{G}_1, \sqrt{\Omega}) \to C_c(\mathcal{G}_1, \sqrt{\Omega})
$$

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]]

$$
  (f \star g) \colon \gamma
  \mapsto
  \int_{\gamma_2\circ \gamma_1 = \gamma}
  f(\gamma_1) \cdot f(\gamma_2)
  \,.
$$

(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 \ref{AsConvolutionAlgebra}, this is naturally a [[star-algebra]] with [[involution]] $inv^\ast$.

=--

This construction originates around ([Connes 82](#Connes82)).

+-- {: .num_prop}
###### Proposition/Definition

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

$$
\pi_x(f) \xi \colon \gamma
 \mapsto
  \int_{\gamma_1 \in t^{-1}(\gamma)}
  f(\gamma_1) \xi(\gamma_1^{-1}\gamma)
  \,.
$$

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 $x$:

$$
  {\Vert f\Vert}
  \coloneqq
  Sup_{x \in \mathcal{G}_0} {\Vert \pi_x(f)\Vert}
  \,.
$$

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](#Connes94)).

+-- {: .num_remark #ExtensionToBibundlesAndBimodules}
###### Remark

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](#Mrcun99)) 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](#KalisnikMrcun)) 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](#MuhleReaultWilliams87), [Landsman 00](#Landsman00)) 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).

=--

A precise form of this statement is the following [Nuiten 13, theorem 3.3.1](#Nuiten13)

+-- {: .num_prop #GroupoidConvolutionIs2Functor}
###### Proposition
**([[groupoid convolution algebra]] is [[(2,1)-functor]] from [[differentiable stacks]] to [[C*-algebras]])**

The above construction of groupoid convolution algebras (def. \ref{ContinuousCategoryAlgebraWithHalfDensities}) extends to a [[(2,1)-functor]]

$$
  C^\ast(-)
  \;\colon\;
  DiffStack^{prop}
    \overset{\phantom{AAA}}{\longrightarrow}
  C^\ast Alg^{op}_{bim}
$$

between

1. the [[(2,1)-category]] $DiffStack_{prop}$ of [[differentiable stacks]] (i.e. [[Lie groupoids]] regarded as [[smooth stacks]]) with proper morphisms between them ([Nuiten 13, def. 2.2.35](#Nuiten13))

1. the [[opposite 2-category|opposite]] [[(2,1)-category]] of [[C*-algebras]] with [[Hilbert bimodules]] between them and [[intertwiners]] between those.

=--

([Nuiten 13, theorem 3.3.1](#Nuiten13))

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*-algebras.

## Equivalent characterizations

We discuss equivalent characterizations of category algebras/groupoid algebras that are useful in certain context

* [As a weak colimit over a constant 2Vect-valued functor
](#WeakColimit)

* [In terms of composition of spans](#InTermsOfCompositionOfSpans)

### As a weak colimit over a constant $2Vect$-valued functor
 {#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 ([FHLT, section 8.4](#FHLT)).

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.

$$
  \array{
     x &&\stackrel{f}{\to}&& y
     \\
     k &&\stackrel{k}{\to}&& k
     \\
     & {k[C]}_{\mathllap{}}\searrow
      &\swArrow_{f \cdot (-)}&
     \swarrow_{\mathrlap{k[C]}}
     \\
     && k[C]
  }
$$

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.

### In terms of composition of spans
 {#InTermsOfCompositionOfSpans}

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]]
$$
  \array{
     && C_1
      \\
      & {}^{s}\swarrow
      && \searrow^{t}
     \\
     C_0
     &&&&
     C_0
  }
$$
equipped with a composition operation: a morphism
of spans from the composite span
$$
  \array{
     &&&&
     C_1 \times_{t,s} C_1
     \\
     &&& \swarrow && \searrow
     \\
     && C_1
     &&&& C_1
      \\
      & {}^{s}\swarrow
      && \searrow^{t}
      && {}^{s}\swarrow
      && \searrow^{t}
     \\
     C_0
     &&&&
     C_0
     &&&&
     C_0
  }
$$
to the original one, i.e. a morphism
$$
  comp : C_1 \times_{t,s} C_1 \to C_1
$$
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
$$
  I : C_0 \to Vect_k
$$
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
$$
  Hom_{[Sets,Vect_k]}(s^* I , t^* I)
$$
whose elements are 2-arrows of the form
$$
  \array{
     && C_1
      \\
      & {}^{s}\swarrow
      && \searrow^{t}
     \\
     C_0
     &&\stackrel{f}{\Rightarrow}&&
     C_0
     \\
     & {}_I \searrow && \swarrow_I
     \\
     &&
     Vect_k
  }
$$
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
$$
  Hom_{[Sets,Vect_k]}(s^* I , t^* I)
  \simeq
  k[C]
  \,.
$$

The algebra structure on $k[C]$ is similarly encoded in the diagrammatics: given two elements
$$
  \array{
     && C_1
      \\
      & {}^{s}\swarrow
      && \searrow^{t}
     \\
     C_0
     &&\stackrel{f}{\Rightarrow}&&
     C_0
     \\
     & {}_I \searrow && \swarrow_I
     \\
     &&
     Vect_k
  }

  \;\;\;\;
  and
  \;\;\;\;

  \array{
     && C_1
      \\
      & {}^{s}\swarrow
      && \searrow^{t}
     \\
     C_0
     &&\stackrel{g}{\Rightarrow}&&
     C_0
     \\
     & {}_I \searrow && \swarrow_I
     \\
     &&
     Vect_k
  }
$$
their pre-composite is the diagram
$$
  \array{
     &&&&
     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
  }
  \,.
$$

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]]
$$
  (f\star g) : t \mapsto \sum_{s\circ r = t}
  f(r)\cdot g(s)
  \,.
$$
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

* [[schreiber:Nonabelian cocycles and their quantum symmetries]].

This is related to observations such as described here:

* [[John Baez]], [_Quantization and Cohomology (Week 17)_](http://golem.ph.utexas.edu/category/2007/03/quantization_and_cohomology_we_16.html)

* Urs Schreiber, [_QFT of Charged n-Particle: T-Duality_](http://golem.ph.utexas.edu/category/2007/02/qft_of_charged_nparticle_tdual.html)

## Examples

### Basic examples

+-- {: .num_example}
###### Example

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 $X$.

=--

+-- {: .num_example}
###### Example

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.

=--

+-- {: .num_example}
###### Example

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 $X$.

=--

+-- {: .num_example}
###### Example

$\mathcal{C} = \mathbf{B}G$ 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 $G$:
$$
  R[\mathbf{B}G] \simeq R[G]
  \,.
$$

=--

### Incidence algebras (poset convolution algebras)

See at [[Möbius inversion#IncidenceAlgebrasZetaFn]].

### Higher groupoid convolution algebras and n-vector spaces/n-modules
 {#HigherGroupoidConvolutionAlgebras}

> under construction

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

+-- {: .num_remark}
###### Remark

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

$$
  C \;\colon\; Grpd \to 2 Mod
  \,.
$$

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 \ref{DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup} 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-[[coskeleton|coskeletal]] as a [[simplicial object]] in [[Grpd]].

+-- {: .num_remark}
###### Remark

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:

* 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 exhibit 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 model structure|Reedy fibrant]] representative.)

=--

+-- {: .num_example #DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup}
###### Example

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]]:

1. $\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G\stackrel{\to}{\stackrel{\to}{\to}} \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G$;

1. $\underset{\longrightarrow}{\lim}(\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *) \simeq \mathbf{B}G$.

(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

$$
  \Delta \colon C(G) \to C(G \times G) \simeq C(G) \otimes C(G)
$$

given on $\phi \in C(G)$ and $g_1, g_2 \in G$ by

$$
  \Delta \phi \colon (g_1, g_2) \mapsto \phi(g_1 \cdot g_2)
  \,.
$$

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):

1. 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 $G$;

1. 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 $G$.

=--

## Properties

### Relation to (twisted) K-theory
 {#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.

([Tu, Xu, Laurent-Gengoux 04](#TXLG))

## Related concepts

* [[groupoid quantale]] - an analogue of groupoid ring/algebra where free abelian groups/vector spaces are replaced by free [[sup-lattice]]s

* [[noncommutative topology]], [[noncommutative geometry]]

  [[KK-theory]]

* [[bibundle]], [[bimodule]]

## References

### For partial orders

* [[Gian-Carlo Rota]], _On the foundations of combinatorial theory I: theory of Möbius functions_ , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.
 {#Rota64}

### For discrete geometry

The [[homotopy colimit]]-interpretation of category algebras over discrete categories is discussed in

* [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]], _[[Topological Quantum Field Theories from Compact Lie Groups]]_ ([arXiv](http://arxiv.org/abs/0905.0731))
 {#FHLT}

Groupoid algebras of geometrically discrete groupoids twisted by [[principal 2-bundles]]/[[bundle gerbes]]/[[central extension of groupoids|groupoid central extension]] is reviewed 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](http://arxiv.org/abs/1103.0578))

### For continuous/smooth geometry
 {#ReferencesForSmoothGeometry}

#### Convolution $C^\ast$-algebras

The study of convolution [[C-star algebras]] of [[Lie groupoids]] goes back to

* {#Renault80} [[Jean Renault]], _A groupoid approach to $C^\ast$ algebras_, Springer Lecture Notes in Mathematics **793**, Springer (1980)

Where the [[integration]] is performed against a fixed [[Haar measure]].

Surveys:

* [[Nigel Higson]], _Groupoids, $C^\ast$-algebras and Index theory_ ([pdf](http://folk.uio.no/rognes/higson/zurich.pdf))

* PlanetMath, _[groupoid $C^\ast$-convolution algebras](http://planetmath.org/groupoidcconvolutionalgebras)_.

The construction via [[sections]] of [[bundles]] of [[half-densities]] (avoiding a choice of Haar measure) is due to

* {#Connes82} [[Alain Connes]], _A survey of foliations and operator algebras_, Proc. Sympos. Pure Math., AMS Providence, 32 (1982), 521–628

Review:

* {#Connes94} [[Alain Connes]], p. 106: _[[Noncommutative Geometry]]_, Academic Press, San Diego, CA, (1994)

Groupoid algebras as [[strict deformation quantization]] of [[Lie-Poisson structures]] given by [[Lie algebroids]]:

* [[Nicolaas P. Landsman]], B. Ramazan, *Quantization of Poisson algebras associated to Lie algebroids*, in: *Groupoids in Analysis, Geometry, and Physics*, Contemporary Mathematics **282** (2001) [[arXiv:math-ph/0001005](https://arxiv.org/abs/math-ph/0001005), [ams:conm/282](http://www.ams.org/books/conm/282)]

  > (specifically [[Weyl algebras]] induced from [[tangent Lie algebroids]] in Ex. 11.3)

Specifically of [[symplectic groupoids]] in the context of [[geometric quantization of symplectic groupoids]]

* {#EH} [[Eli Hawkins]], _A groupoid approach to quantization_ ([arXiv:math.SG/0612363](http://arxiv.org/abs/math.SG/0612363))

See also:

* {#Nuiten13} [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, master thesis, August 2013

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](http://www.math.dartmouth.edu/~dana/dpwpdfs/muhwil-ms90.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]( http://www.jstor.org/stable/2155247))

* 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](http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf))

* Mădălina Roxana Buneci, _Convolution algebras for topological groupoids with locally compact fibers_ (2011) ([pdf](http://journals.bg.agh.edu.pl/OPUSCULA/31-2/31-2-02.pdf))

A review in the context of [[geometric quantization]] is in section 4.3 of

* [[Rogier Bos]], _Groupoids in geometric quantization_ PhD Thesis (2007) [pdf](http://www.math.ist.utl.pt/~rbos/ProefschriftA4.pdf)
 {#Bos}

Specifically the convolution $C^\ast$-algebras of [[bundle gerbes]] regarded as [[centrally extended groupoids]] (algebras whose [[modules]] (see [below](#ReferencesModulesOverConvolutionAlgebra)) are [[gerbe modules]]/[[twisted bundle]]) is discussed in section 5 of

* [[Alan Carey]], Stuart Johnson, [[Michael Murray]], _Holonomy on D-Branes_, J. Geom. Phys. 52 (2004) 186-216 ([arXiv:hep-th/0204199](http://arxiv.org/abs/hep-th/0204199))

Functoriality of the construction of $C^\ast$-convolution algebras (its extension to groupoid-[[bibundles]]) is discussed in

* [[Paul Muhly]], [[Jean Renault]], D. Williams, _Equivalence and isomorphism for groupoid $C^\ast$-algebras_, J. Operator Theory 17 (1987), no. 1, 3–22.
 {#MuhleReaultWilliams87}

* {#Mrcun99} [[Janez Mrčun]], _Functoriality of the bimodule associated to a Hilsum-Skandalis map_. K-Theory 18 (1999) 235–253.

* [[Klaas Landsman]], _The Muhly-Renault-Williams theorem for Lie groupoids and its classical counterpart_, Lett. Math. Phys. 54 (2000), no. 1, 43–59. ([arXiv:math-ph/0008005](http://arxiv.org/abs/math-ph/0008005))
 {#Landsman00}

* [[Klaas Landsman]], _Operator Algebras and Poisson Manifolds Associated to Groupoids_, Commun. Math. Phys. 222, 97 – 116 (2001) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.2716))

#### Convolution Hopf algebroids
 {#ReferencesConvolutionHopfAlgebroids}

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](http://arxiv.org/abs/math/0208199))

* [[Jure Kališnik]], [[Janez Mrčun]], _Equivalence between the Morita categories of etale Lie groupoids and of locally grouplike Hopf algebroids_ ([arXiv:math/0703374](http://arxiv.org/abs/math/0703374))
 {#KalisnikMrcun}

#### Modules over Lie groupoid convolution algebras and K-theory
 {#ReferencesModulesOverConvolutionAlgebra}

Discussion of [[modules]] over Lie groupoid convolution algebras is in the following articles.

In ([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 ([Bos, chapter 7](#Bos)) is discussion refining this to continuous representations and representation of a convolution $C^\ast$-algebra, also in section 4 of:

* [[Rogier Bos]], _Continuous representations of groupoids_ ([arXiv:math/0612639](http://arxiv.org/abs/math/0612639))

Representation of convolution algebras of [[étale groupoids]] is in

* [[Jure Kališnik]], _Groupoid representations and modules over the convolution algebras_ ([arXiv:0806.1832](http://arxiv.org/abs/0806.1832))

The [[operator K-theory]] of groupoid convolution algebras (the [[topological K-theory]] of the corresponding [[differentiable stacks]]) is discussed in

* [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], _Twisted K-theory of differentiable stacks_, Annales scientifiques de l'École Normale Supérieure (2004) Volume: 37, Issue: 6, page 841-910 ([arXiv:math/0306138](http://arxiv.org/abs/math/0306138))
 {#TXLG}

Construction of cocycles in [[KK-theory]] and [[spectral triples]] from groupoid convolution is in

* Bram Mesland, _Groupoid cocycles and K-theory_ ([arXiv:1005.3677](http://arxiv.org/abs/1005.3677))
 {#Meland}

[[!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*-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]]


category enriched in a bicategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _category enriched in a bicategory_ is the many-object-generalization of the notion of an [[enriched category]] enriched in a [[monoidal category]] via regarding a [[monoidal category]] as a [[bicategory]] with a single object.

Originally Bénabou called these *polyads*.

## Definition

Let $B$ be a [[bicategory]], and write $\otimes$ for horizontal (1-cell) composition (written in Leibniz order). A [[category enriched in a bicategory|category enriched in the bicategory]] $B$ consists of a [[set]] $X$ together with

* A function $p: X \to B_0$,
* A function $\hom: X \times X \to B_1$, satisfying the typing constraint $\hom(x, y): p(x) \to p(y)$,
* A function $\circ: X \times X \times X \to B_2$, satisfying the constraint $\circ_{x, y, z}: \hom(y, z) \otimes \hom(x, y) \to \hom(x, z)$,
* A function $j: X \to B_2$, satisfying the constraint $j_x: 1_{p(x)} \to \hom(x, x)$,

such that the associativity and unitality diagrams, as written above, commute. Viewing a monoidal category $M$ as a 1-object bicategory $\Sigma M$, the notion of enrichment in $M$ coincides with the notion of enrichment in the bicategory $\Sigma M$.

Equivalently this is simply a [[lax functor]] from the [[codiscrete category]] on $X$ into $B$. In particular if $X$ is the singleton set then this is the same as a [[monad]].

If $X$, $Y$ are sets which come equipped with enrichments in $B$, then a $B$-functor consists of a function $f: X \to Y$ such that $p_Y \circ f = p_X$, together with a function $f_1: X \times X \to B_2$, satisfying the constraint $f_1(x, y): \hom_X(x, y) \to \hom_Y(f(x), f(y))$, and satisfying equations expressing coherence with the composition and unit data $\circ$, $j$ of $X$ and $Y$. (Diagram to be inserted, perhaps.)

## Related concepts

* [2-category equipped with proarrows -- Categories enriched in an equipment](2-category+equipped+with+proarrows#CategoriesEnrichedInAnEquipment)
* [[locally cocomplete bicategory]]

## References

Discussion of [[Kleisli objects]] (collages) for [[monads]] generalized to categories enriched in bicategories is in section 15.9 of

* [[Richard Garner]], [[Michael Shulman]], _Enriched categories as a free cocompletion_ ([arXiv:1301.3191](http://arxiv.org/abs/1301.3191))
 {#GarnerShulman13}

[[!redirects categories enriched in a bicategory]]
[[!redirects category enriched over a bicategory]]
[[!redirects categories enriched over a bicategory]]

[[!redirects category enriched in a 2-category]]
[[!redirects categories enriched in a 2-category]]
[[!redirects category enriched over a 2-category]]
[[!redirects categories enriched over a 2-category]]

[[!redirects bicategory-enriched category]]
[[!redirects bicategory-enriched categories]]

[[!redirects polyad]]
[[!redirects polyads]]


category mistake

[[!redirects Category mistake]]

For the moment see at

* Wikipedia, *[Category mistake](https://en.wikipedia.org/wiki/Category_mistake)*

* Stanford Encyclopedia of Philosophy, *[Category Mistakes](https://plato.stanford.edu/entries/category-mistakes/)*