Groupoid

This article is about groupoids in category theory. For the algebraic structure with a single binary operation, see magma (algebra).

In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

• Group with a partial function replacing the binary operation;
• Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group.

In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed Template:Tmath, Template:Tmath, say. Composition is then a total function: Template:Tmath, so that Template:Tmath.

Special cases include:

• Setoids: sets that come with an equivalence relation,
• G-sets: sets equipped with an action of a group Template:Tmath.

Groupoids are often used to reason about geometrical objects such as manifolds. Template:Harvs introduced groupoids implicitly via Brandt semigroups.

Definitions

Algebraic

A groupoid can be viewed as an algebraic structure consisting of a set with a binary partial function . Precisely, it is a non-empty set $G$ with a unary operation Template:Tmath, and a partial function Template:Tmath. Here Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle *} is not a binary operation because it is not necessarily defined for all pairs of elements of Template:Tmath. The precise conditions under which Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle *} is defined are not articulated here and vary by situation.

The operations Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ast } and ⁻¹ have the following axiomatic properties: For all Template:Tmath, Template:Tmath, and $c$ in Template:Tmath,

1. Associativity: If $a*b$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b*c} are defined, then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (a*b)*c} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a*(b*c)} are defined and are equal. Conversely, if one of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (a*b)*c} or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a*(b*c)} is defined, then they are both defined (and they are equal to each other), and $a*b$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b*c} are also defined.
2. Inverse: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a^{-1}*a} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a*{a^{-1}}} are always defined.
3. Identity: If $a*b$ is defined, then Template:Tmath, and Template:Tmath. (The previous two axioms already show that these expressions are defined and unambiguous.)

Two easy and convenient properties follow from these axioms:

• Template:Tmath,
• If $a*b$ is defined, then Template:Tmath.

Category-theoretic

A groupoid is a small category in which every morphism is an isomorphism, i.e., invertible. More explicitly, a groupoid $G$ is a set $G_{0}$ of objects with

• for each pair of objects x and y, a (possibly empty) set G(x,y) of morphisms (or arrows) from x to y; we write f : x → y to indicate that f is an element of G(x,y);
• for every object x, a designated element Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {id} _{x}} of G(x, x);
• for each triple of objects x, y, and z, a function Template:Tmath;
• for each pair of objects x, y, a function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {inv} :G(x,y)\rightarrow G(y,x):f\mapsto f^{-1}} satisfying, for any f : x → y, g : y → z, and h : z → w:
  • Template:Tmath and Template:Tmath;
  • Template:Tmath;
  • Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ff^{-1}=\mathrm {id} _{y}} and Template:Tmath.

If f is an element of G(x,y), then x is called the source of f, written s(f), and y is called the target of f, written t(f).

A groupoid G is sometimes denoted as Template:Tmath, where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G_{1}} is the set of all morphisms, and the two arrows Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G_{1}\to G_{0}} represent the source and the target.

More generally, one can consider a groupoid object in an arbitrary category admitting finite fiber products.

Comparing the definitions

The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {comp} } and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {inv} } become partial operations on G, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {inv} } will in fact be defined everywhere. We define ∗ to be Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathrm {comp} } and ⁻¹ to be Template:Tmath, which gives a groupoid in the algebraic sense. Explicit reference to G₀ (and hence to Template:Tmath) can be dropped.

Conversely, given a groupoid G in the algebraic sense, define an equivalence relation $\sim$ on its elements by $a\sim b$ iff a ∗ a⁻¹ = b ∗ b⁻¹. Let G₀ be the set of equivalence classes of Template:Tmath, i.e. Template:Tmath. Denote a ∗ a⁻¹ by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1_{x}} if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a\in G} with Template:Tmath.

Now define $G(x,y)$ as the set of all elements f such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1_{x}*f*1_{y}} exists. Given Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\in G(x,y)} and Template:Tmath, their composite is defined as Template:Tmath. To see that this is well defined, observe that since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (1_{x}*f)*1_{y}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1_{y}*(g*1_{z})} exist, so does Template:Tmath. The identity morphism on x is then Template:Tmath, and the category-theoretic inverse of f is f⁻¹.

Sets in the definitions above may be replaced with classes, as is generally the case in category theory.

Vertex groups and orbits

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G(x,x), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

The orbit of a groupoid G at a point $x\in X$ is given by the set Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s(t^{-1}(x))\subseteq X} containing every point that can be joined to x by a morphism in G. If two points $x$ and $y$ are in the same orbits, their vertex groups Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(x)} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(y)} are isomorphic: if $f$ is any morphism from $x$ to Template:Tmath, then the isomorphism is given by the mapping Template:Tmath.

Orbits form a partition of the set X, and a groupoid is called transitive if it has only one orbit (equivalently, if it is connected as a category). In that case, all the vertex groups are isomorphic (on the other hand, this is not a sufficient condition for transitivity; see the section below for counterexamples).

Subgroupoids and morphisms

A subgroupoid of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G\rightrightarrows X} is a subcategory Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle H\rightrightarrows Y} that is itself a groupoid. It is called wide or full if it is wide or full as a subcategory, i.e., respectively, if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X=Y} or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle G(x,y)=H(x,y)} for every Template:Tmath.

A groupoid morphism is simply a functor between two (category-theoretic) groupoids.

Particular kinds of morphisms of groupoids are of interest. A morphism $p:E\to B$ of groupoids is called a fibration if for each object $x$ of $E$ and each morphism $b$ of $B$ starting at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p(x)} there is a morphism $e$ of $E$ starting at $x$ such that Template:Tmath. A fibration is called a covering morphism or covering of groupoids if further such an $e$ is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.

It is also true that the category of covering morphisms of a given groupoid $B$ is equivalent to the category of actions of the groupoid $B$ on sets.

Examples

Fundamental groupoid

Main article: Fundamental groupoid

Given a topological space Template:Tmath, let $G_{0}$ be the set Template:Tmath. The morphisms from the point $p$ to the point $q$ are equivalence classes of continuous paths from $p$ to Template:Tmath, with two paths being equivalent if they are homotopic. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of Template:Tmath, denoted $\pi _{1}(X)$ (or sometimes, Template:Tmath). The usual fundamental group Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \pi _{1}(X,x)} is then the vertex group for the point Template:Tmath.

The orbits of the fundamental groupoid $\pi _{1}(X)$ are the path-connected components of Template:Tmath. Accordingly, the fundamental groupoid of a path-connected space is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are equivalent as categories (see the section below for the general theory).

An important extension of this idea is to consider the fundamental groupoid Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \pi _{1}(X,A)} where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A\subset X} is a chosen set of "base points". Here Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \pi _{1}(X,A)} is a (full) subgroupoid of Template:Tmath, where one considers only paths whose endpoints belong to Template:Tmath. The set $A$ may be chosen according to the geometry of the situation at hand.

Equivalence relation

If $X$ is a setoid, i.e. a set with an equivalence relation Template:Tmath, then a groupoid "representing" this equivalence relation can be formed as follows:

• The objects of the groupoid are the elements of Template:Tmath;
• For any two elements $x$ and $y$ in Template:Tmath, there is a single morphism from $x$ to $y$ (denote by Template:Tmath) if and only if Template:Tmath;
• The composition of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (z,y)} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (y,x)} is Template:Tmath.

The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples:

• If every element of $X$ is in relation with every other element of Template:Tmath, we obtain the pair groupoid of Template:Tmath, which has the entire $X\times X$ as set of arrows, and which is transitive.
• If every element of $X$ is only in relation with itself, one obtains the unit groupoid, which has $X$ as set of arrows, Template:Tmath, and which is completely intransitive (every singleton Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \{x\}} is an orbit).

Examples

• If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f:X_{0}\to Y} is a smooth surjective submersion of smooth manifolds, then $X_{0}\times _{Y}X_{0}\subset X_{0}\times X_{0}$ is an equivalence relation since $Y$ has a topology isomorphic to the quotient topology of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X_{0}} under the surjective map of topological spaces. If we write, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X_{1}=X_{0}\times _{Y}X_{0}} then we get a groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1 \rightrightarrows X_0,} which is sometimes called the banal groupoid of a surjective submersion of smooth manifolds.
• If we relax the reflexivity requirement and consider partial equivalence relations, then it becomes possible to consider semidecidable notions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, called PER models. Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to the effective topos introduced by Martin Hyland.

Čech groupoid

See also: Simplicial manifold and Nerve of a covering

A Čech groupoid^p. 5 is a special kind of groupoid associated to an equivalence relation given by an open cover Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {U}}=\{U_{i}\}_{i\in I}} of some manifold Template:Tmath. Its objects are given by the disjoint union Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\mathcal {G}}_{0}=\coprod U_{i},} and its arrows are the intersections Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_1 = \coprod U_{ij} .}

The source and target maps are then given by the induced maps

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} s = \phi_j: U_{ij} \to U_j\\ t = \phi_i: U_{ij} \to U_i \end{align}}

and the inclusion map

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon: U_i \to U_{ii}}

giving the structure of a groupoid. In fact, this can be further extended by setting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_n = \mathcal{G}_1\times_{\mathcal{G}_0} \cdots \times_{\mathcal{G}_0}\mathcal{G}_1}

as the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -iterated fiber product where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_n} represents Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} -tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} U_{ijk} & \to & U_{ij} \\ \downarrow & & \downarrow \\ U_{ik} & \to & U_{i} \end{matrix}}

is a cartesian diagram where the maps to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i} are the target maps. This construction can be seen as a model for some ∞-groupoids. Also, another artifact of this construction is k-cocycles

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\sigma] \in \check{H}^k(\mathcal{U},\underline{A})}

for some constant sheaf of abelian groups can be represented as a function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma:\coprod U_{i_1\cdots i_k} \to A}

giving an explicit representation of cohomology classes.

Group action

Main article: action groupoid

If the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} acts on the set Template:Tmath, then we can form the action groupoid (or transformation groupoid) representing this group action as follows:

• The objects are the elements of Template:Tmath;
• For any two elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} in Template:Tmath, the morphisms from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} correspond to the elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} such that Template:Tmath;
• Composition of morphisms interprets the binary operation of Template:Tmath.

More explicitly, the action groupoid is a small category with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ob}(C)=X} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{hom}(C)=G\times X} and with source and target maps Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(g,x) = x} and Template:Tmath. It is often denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \ltimes X} (or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\rtimes G} for a right action). Multiplication (or composition) in the groupoid is then Template:Tmath, which is defined provided Template:Tmath.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} in Template:Tmath, the vertex group consists of those Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (g,x)} with Template:Tmath, which is just the isotropy subgroup at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are the orbit of the group action, and the groupoid is transitive if and only if the group action is transitive.

Another way to describe Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} -sets is the functor category Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Gr}} is the groupoid (category) with one element and isomorphic to the group Template:Tmath. Indeed, every functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} of this category defines a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X=F(\mathrm{Gr})} and for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} (i.e. for every morphism in Template:Tmath) induces a bijection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_g}  : Template:Tmath. The categorical structure of the functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} assures us that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} defines a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} -action on the set Template:Tmath. The (unique) representable functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F : \mathrm{Gr} \to \mathrm{Set}} is the Cayley representation of Template:Tmath. In fact, this functor is isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Hom}(\mathrm{Gr},-)} and so sends Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{ob}(\mathrm{Gr})} to the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Hom}(\mathrm{Gr},\mathrm{Gr})} which is by definition the "set" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and the morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Gr}} (i.e. the element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} of Template:Tmath) to the permutation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_g} of the set Template:Tmath. We deduce from the Yoneda embedding that the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is isomorphic to the group Template:Tmath, a subgroup of the group of permutations of Template:Tmath.

Finite set

Consider the group action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}/2} on the finite set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X = \{-2, -1, 0, 1, 2\}} where 1 acts by taking each number to its negative, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 \mapsto 2} and Template:Tmath. The quotient groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [X/G]} is the set of equivalence classes from this group action Template:Tmath, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0]} has a group action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}/2} on it.

Quotient variety

Any finite group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G } that maps to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle GL(n) } gives a group action on the affine space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{A}^n } (since this is the group of automorphisms). Then, a quotient groupoid can be of the form Template:Tmath, which has one point with stabilizer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G } at the origin. Examples like these form the basis for the theory of orbifolds. Another commonly studied family of orbifolds are weighted projective spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{P}(n_1,\ldots, n_k)} and subspaces of them, such as Calabi–Yau orbifolds.

Inertia groupoid

Main article: Inertia groupoid

The inertia groupoid of a groupoid is roughly a groupoid of loops in the given groupoid.

Fiber product of groupoids

Given a diagram of groupoids with groupoid morphisms

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}&&X\\&&\downarrow \\Y&\rightarrow &Z\end{aligned}}}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f:X\to Z} and Template:Tmath, we can form the groupoid Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle X\times _{Z}Y} whose objects are triples Template:Tmath, where Template:Tmath, Template:Tmath, and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi :f(x)\to g(y)} in Template:Tmath. Morphisms can be defined as a pair of morphisms Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\alpha ,\beta )} where $\alpha :x\to x'$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \beta :y\to y'} such that for triples Template:Tmath, there is a commutative diagram in $Z$ of Template:Tmath, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(\beta ):g(y)\to g(y')} and the Template:Tmath.

Homological algebra

A two term complex

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1 ~\overset{d}{\rightarrow}~ C_0}

of objects in a concrete Abelian category can be used to form a groupoid. It has as objects the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_0} and as arrows the set Template:Tmath; the source morphism is just the projection onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_0} while the target morphism is the addition of projection onto Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_1} composed with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} and projection onto Template:Tmath. That is, given Template:Tmath, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t(c_1 + c_0) = d(c_1) + c_0.}

Of course, if the abelian category is the category of coherent sheaves on a scheme, then this construction can be used to form a presheaf of groupoids.

Puzzles

While puzzles such as the Rubik's Cube can be modeled using group theory (see Rubik's Cube group), certain puzzles are better modeled as groupoids.

The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed). This groupoid acts on configurations.

Mathieu groupoid

The Mathieu groupoid is a groupoid introduced by John Horton Conway acting on 13 points such that the elements fixing a point form a copy of the Mathieu group M₁₂.

Relation to groups

Template:Group-like structures If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of group theory generalize to groupoids, with the notion of functor replacing that of group homomorphism.

Every transitive/connected groupoid - that is, as explained above, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above) Template:Tmath. By transitivity, there will only be one orbit under the action.

Note that the isomorphism just mentioned is not unique, and there is no natural choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object Template:Tmath, a group isomorphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x_0)} to Template:Tmath, and for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} other than Template:Tmath, a morphism in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} to Template:Tmath.

If a groupoid is not transitive, then it is isomorphic to a disjoint union of groupoids of the above type, also called its connected components (possibly with different groups Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} for each connected component).

In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a multiset of unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets Template:Tmath, but only the groups Template:Tmath. For example,

• The fundamental groupoid of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is equivalent to the collection of the fundamental groups of each path-connected component of Template:Tmath, but an isomorphism requires specifying the set of points in each component;
• The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} with the equivalence relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim} is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but an isomorphism requires specifying what each equivalence class is;
• The set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} equipped with an action of the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is equivalent (as a groupoid) to one copy of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} for each orbit of the action, but an isomorphism requires specifying what set each orbit is.

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the entire groupoid. Otherwise, one must choose a way to view each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(x)} in terms of a single group, and this choice can be arbitrary. In the example from topology, one would have to make a coherent choice of paths (or equivalence classes of paths) from each point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} to each point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} in the same path-connected component.

As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector spaces with one endomorphism is nontrivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} of a group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} yields an action of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} on the set of cosets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} and hence a covering morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} from, say, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} to Template:Tmath, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is a groupoid with vertex groups isomorphic to Template:Tmath. In this way, presentations of the group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} can be "lifted" to presentations of the groupoid Template:Tmath, and this is a useful way of obtaining information about presentations of the subgroup Template:Tmath. For further information, see the books by Higgins and by Brown in the References.

Category of groupoids

The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, and is denoted by Grpd.

The category Grpd is, like the category of small categories, Cartesian closed: for any groupoids Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H,K} we can construct a groupoid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{GPD}(H,K)} whose objects are the morphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \to K } and whose arrows are the natural equivalences of morphisms. Thus if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H,K } are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,H,K } there is a natural bijection

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Grpd}(G \times H, K) \cong \operatorname{Grpd}(G, \operatorname{GPD}(H,K)).}

This result is of interest even if all the groupoids Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G,H,K } are just groups.

Another important property of Grpd is that it is both complete and cocomplete.

Relation to Cat

The inclusion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i : \mathbf{Grpd} \to \mathbf{Cat}} has both a left and a right adjoint:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hom_{\mathbf{Grpd}}(C[C^{-1}], G) \cong \hom_{\mathbf{Cat}}(C, i(G)) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hom_{\mathbf{Cat}}(i(G), C) \cong \hom_{\mathbf{Grpd}}(G, \mathrm{Core}(C)) }

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C[C^{-1}]} denotes the localization of a category that inverts every morphism, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Core}(C)} denotes the subcategory of all isomorphisms.

Relation to sSet

The nerve functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N : \mathbf{Grpd} \to \mathbf{sSet}} embeds Grpd as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always a Kan complex.

The nerve has a left adjoint

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hom_{\mathbf{Grpd}}(\pi_1(X), G) \cong \hom_{\mathbf{sSet}}(X, N(G)) }

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi_1(X)} denotes the fundamental groupoid of the simplicial set Template:Tmath.

Groupoids in Grpd

Main article: Double groupoid

There is an additional structure which can be derived from groupoids internal to the category of groupoids, double-groupoids. Because Grpd is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{G}_1,\mathcal{G}_0} with functors

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s,t: \mathcal{G}_1 \to \mathcal{G}_0}

and an embedding given by an identity functor

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i:\mathcal{G}_0 \to\mathcal{G}_1}

One way to think about these 2-groupoids is they contain objects, morphisms, and squares which can compose together vertically and horizontally. For example, given squares

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \bullet & \to & \bullet \\ \downarrow & & \downarrow \\ \bullet & \xrightarrow{a} & \bullet \end{matrix} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \bullet & \xrightarrow{a} & \bullet \\ \downarrow & & \downarrow \\ \bullet & \to & \bullet \end{matrix}}

with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} the same morphism, they can be vertically conjoined giving a diagram

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \bullet & \to & \bullet \\ \downarrow & & \downarrow \\ \bullet & \xrightarrow{a} & \bullet \\ \downarrow & & \downarrow \\ \bullet & \to & \bullet \end{matrix}}

which can be converted into another square by composing the vertical arrows. There is a similar composition law for horizontal attachments of squares.

Groupoids with geometric structures

When studying geometrical objects, the arising groupoids often carry a topology, turning them into topological groupoids, or even some differentiable structure, turning them into Lie groupoids. These last objects can be also studied in terms of their associated Lie algebroids, in analogy to the relation between Lie groups and Lie algebras.

Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, in Poisson geometry one has the notion of a symplectic groupoid, which is a Lie groupoid endowed with a compatible symplectic form. Similarly, one can have groupoids with a compatible Riemannian metric, or complex structure, etc.

See also

• ∞-groupoid
• 2-group
• Homotopy type theory
• Inverse category
• Groupoid algebra (not to be confused with algebraic groupoid)
• R-algebroid

References

• Brandt, H (1927), "Über eine Verallgemeinerung des Gruppenbegriffes", Mathematische Annalen, 96 (1): 360–366, doi:10.1007/BF01209171, S2CID 119597988
• Brown, Ronald, 1987, "From groups to groupoids: a brief survey", Bull. London Math. Soc. 19: 113–34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references.
• —, 2006. Topology and groupoids. Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
• —, Higher dimensional group theory. Explains how the groupoid concept has led to higher-dimensional homotopy groupoids, having applications in homotopy theory and in group cohomology. Many references.
• Dicks, Warren; Ventura, Enric (1996), The group fixed by a family of injective endomorphisms of a free group, Mathematical Surveys and Monographs, 195, AMS Bookstore, ISBN 978-0-8218-0564-0
• Dokuchaev, M.; Exel, R.; Piccione, P. (2000). "Partial Representations and Partial Group Algebras". Journal of Algebra. Elsevier. 226: 505–532. arXiv:math/9903129. doi:10.1006/jabr.1999.8204. ISSN 0021-8693. S2CID 14622598.
• F. Borceux, G. Janelidze, 2001, Galois theories. Cambridge Univ. Press. Shows how generalisations of Galois theory lead to Galois groupoids.
• Cannas da Silva, A., and A. Weinstein, Geometric Models for Noncommutative Algebras. Especially Part VI.
• Golubitsky, M., Ian Stewart, 2006, "Nonlinear dynamics of networks: the groupoid formalism", Bull. Amer. Math. Soc. 43: 305–64
• Template:Springer
• Higgins, P. J., "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) 13 (1976) 145–149.
• Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an orbit space", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115–122.
• Higgins, P. J., 1971. Categories and groupoids. Van Nostrand Notes in Mathematics. Republished in Reprints in Theory and Applications of Categories, No. 7 (2005) pp. 1–195; freely downloadable. Substantial introduction to category theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in topology, e.g. fundamental groupoid.
• Mackenzie, K. C. H., 2005. General theory of Lie groupoids and Lie algebroids. Cambridge Univ. Press.
• Weinstein, Alan, "Groupoids: unifying internal and external symmetry – A tour through some examples". Also available in Postscript, Notices of the AMS, July 1996, pp. 744–752.
• Weinstein, Alan, "The Geometry of Momentum" (2002)
• R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". In Algebraic and geometric combinatorics, volume 423 of Contemp. Math., 305–324. Amer. Math. Soc., Providence, RI (2006)
• Template:Nlab
• Template:Nlab