Inmathematics, especially incategory theory andhomotopy theory, agroupoid (less oftenBrandt groupoid orvirtual group) generalises the notion ofgroup in several equivalent ways. A groupoid can be seen as a:

• Group with apartial function replacing thebinary operation;
• Category in which everymorphism is invertible. A category of this sort can be viewed as augmented with aunary operation on the morphisms, calledinverse by analogy withgroup theory.^[1] A groupoid where there is only one object is a usual group.

In the presence ofdependent typing, a category in general can be viewed as a typedmonoid, 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${\displaystyle g:A\to B}$⁠,⁠${\displaystyle h:B\to C}$⁠, say. Composition is then a total function:⁠${\displaystyle \circ :(B\to C)\to (A\to B)\to (A\to C)}$⁠, so that⁠${\displaystyle h\circ g:A\to C}$⁠.

Special cases include:

• Setoids:sets that come with anequivalence relation,
• G-sets: sets equipped with anaction of a group⁠${\displaystyle G}$⁠.

Groupoids are often used to reason aboutgeometrical objects such asmanifolds.Heinrich Brandt (1927) introduced groupoids implicitly viaBrandt semigroups.^[2]

A groupoid can be viewed as an algebraic structure consisting of a set with a binarypartial function^{[citation needed]}.Precisely, it is a non-empty set${\displaystyle G}$ with aunary operation⁠${\displaystyle {}^{-1}:G\to G}$⁠, and apartial function⁠${\displaystyle \ast :G\times G\rightharpoonup G}$⁠. Here${\displaystyle \ast }$ is not abinary operation because it is not necessarily defined for all pairs of elements of⁠${\displaystyle G}$⁠. The precise conditions under which${\displaystyle \ast }$ is defined are not articulated here and vary by situation.

The operations${\displaystyle \ast }$ and⁻¹ have the following axiomatic properties: For all⁠${\displaystyle a}$⁠,⁠${\displaystyle b}$⁠, and${\displaystyle c}$ in⁠${\displaystyle G}$⁠,

1. Associativity: If${\displaystyle a\ast b}$ and${\displaystyle b\ast c}$ are defined, then${\displaystyle (a\ast b)\ast c}$ and${\displaystyle a\ast (b\ast c)}$ are defined and are equal. Conversely, if one of${\displaystyle (a\ast b)\ast c}$ or${\displaystyle a\ast (b\ast c)}$ is defined, then they are both defined (and they are equal to each other), and${\displaystyle a\ast b}$ and${\displaystyle b\ast c}$ are also defined.
2. Inverse:${\displaystyle a^{-1}\ast a}$ and${\displaystyle a\ast a^{-1}}$ are always defined.
3. Identity: If${\displaystyle a\ast b}$ is defined, then⁠${\displaystyle a\ast b\ast b^{-1}=a}$⁠, and⁠${\displaystyle a^{-1}\ast a\ast b=b}$⁠. (The previous two axioms already show that these expressions are defined and unambiguous.)

Two easy and convenient properties follow from these axioms:

• ⁠${\displaystyle (a^{-1}{)}^{-1}=a}$⁠,
• If${\displaystyle a\ast b}$ is defined, then⁠${\displaystyle (a\ast b{)}^{-1}=b^{-1}\ast a^{-1}}$⁠.^[3]

A groupoid is asmall category in which everymorphism is anisomorphism, i.e., invertible.^[1] More explicitly, a groupoid${\displaystyle G}$ is a set${\displaystyle G_0}$ ofobjects with

• for each pair of objectsx andy, a (possibly empty) setG(x,y) ofmorphisms (orarrows) fromx toy; we writef :x →y to indicate thatf is an element ofG(x,y);
• for every objectx, a designated element${\displaystyle {\mathrm{i}\mathrm{d}}_x}$ ofG(x,x);
• for each triple of objectsx,y, andz, afunction⁠${\displaystyle {\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}_{x,y,z}:G(y,z)\times G(x,y)\to G(x,z):(g,f)\mapsto gf}$⁠;
• for each pair of objectsx,y, a function${\displaystyle \mathrm{i}\mathrm{n}\mathrm{v}:G(x,y)\to G(y,x):f\mapsto f^{-1}}$ satisfying, for anyf :x →y,g :y →z, andh :z →w:
  • ⁠${\displaystyle f{\mathrm{i}\mathrm{d}}_x=f}$⁠ and⁠${\displaystyle {\mathrm{i}\mathrm{d}}_{y}f=f}$⁠;
  • ⁠${\displaystyle (hg)f=h(gf)}$⁠;
  • ${\displaystyle f{f}^{-1}={\mathrm{i}\mathrm{d}}_y}$ and⁠${\displaystyle f^{-1}f={\mathrm{i}\mathrm{d}}_x}$⁠.

Iff is an element ofG(x,y), thenx is called thesource off, writtens(f), andy is called thetarget off, writtent(f).

A groupoidG is sometimes denoted as⁠${\displaystyle G_1\rightrightarrows G_0}$⁠, where${\displaystyle G_1}$ is the set of all morphisms, and the two arrows${\displaystyle G_1\to G_0}$ represent the source and the target.

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

The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, letG be thedisjoint union of all of the setsG(x,y) (i.e. the sets of morphisms fromx toy). Then${\displaystyle \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ and${\displaystyle \mathrm{i}\mathrm{n}\mathrm{v}}$ become partial operations onG, and${\displaystyle \mathrm{i}\mathrm{n}\mathrm{v}}$ will in fact be defined everywhere. We define ∗ to be${\displaystyle \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ and⁻¹ to be⁠${\displaystyle \mathrm{i}\mathrm{n}\mathrm{v}}$⁠, which gives a groupoid in the algebraic sense. Explicit reference toG₀ (and hence to⁠${\displaystyle \mathrm{i}\mathrm{d}}$⁠) can be dropped.

Conversely, given a groupoidG in the algebraic sense, define an equivalence relation${\displaystyle \sim }$ on its elements by${\displaystyle a\sim b}$ iffa ∗a⁻¹ =b ∗b⁻¹. LetG₀ be the set of equivalence classes of⁠${\displaystyle \sim }$⁠, i.e.⁠${\displaystyle G_0:=G/\sim }$⁠. Denotea ∗a⁻¹ by${\displaystyle 1_x}$ if${\displaystyle a\in G}$ with⁠${\displaystyle x\in G_0}$⁠.

Now define${\displaystyle G(x,y)}$ as the set of all elementsf such that${\displaystyle 1_x\ast f\ast 1_y}$ exists. Given${\displaystyle f\in G(x,y)}$ and⁠${\displaystyle g\in G(y,z)}$⁠, their composite is defined as⁠${\displaystyle gf:=f\ast g\in G(x,z)}$⁠. To see that this is well defined, observe that since${\displaystyle (1_x\ast f)\ast 1_y}$ and${\displaystyle 1_y\ast (g\ast 1_z)}$ exist, so does⁠${\displaystyle (1_x\ast f\ast 1_y)\ast (g\ast 1_z)=f\ast g}$⁠. The identity morphism onx is then⁠${\displaystyle 1_x}$⁠, and the category-theoretic inverse off isf⁻¹.

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

Given a groupoidG, thevertex groups orisotropy groups orobject groups inG are the subsets of the formG(x,x), wherex is any object ofG. 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.

Theorbit of a groupoidG at a point${\displaystyle x\in X}$ is given by the set${\displaystyle s(t^{-1}(x))\subseteq X}$ containing every point that can be joined to x by a morphism in G. If two points${\displaystyle x}$ and${\displaystyle y}$ are in the same orbits, their vertex groups${\displaystyle G(x)}$ and${\displaystyle G(y)}$ areisomorphic: if${\displaystyle f}$ is any morphism from${\displaystyle x}$ to⁠${\displaystyle y}$⁠, then the isomorphism is given by the mapping⁠${\displaystyle g\to fg{f}^{-1}}$⁠.

Orbits form a partition of the set X, and a groupoid is calledtransitive if it has only one orbit (equivalently, if it isconnected 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 sectionbelow for counterexamples).

Asubgroupoid of${\displaystyle G\rightrightarrows X}$ is asubcategory${\displaystyle H\rightrightarrows Y}$ that is itself a groupoid. It is calledwide orfull if it iswide orfull as a subcategory, i.e., respectively, if${\displaystyle X=Y}$ or${\displaystyle G(x,y)=H(x,y)}$ for every⁠${\displaystyle x,y\in Y}$⁠.

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

Particular kinds of morphisms of groupoids are of interest. A morphism${\displaystyle p:E\to B}$ of groupoids is called afibration if for each object${\displaystyle x}$ of${\displaystyle E}$ and each morphism${\displaystyle b}$ of${\displaystyle B}$ starting at${\displaystyle p(x)}$ there is a morphism${\displaystyle e}$ of${\displaystyle E}$ starting at${\displaystyle x}$ such that⁠${\displaystyle p(e)=b}$⁠. A fibration is called acovering morphism orcovering of groupoids if further such an${\displaystyle e}$ is unique. The covering morphisms of groupoids are especially useful because they can be used to modelcovering maps of spaces.^[4]

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

Given atopological space⁠${\displaystyle X}$⁠, let${\displaystyle G_0}$ be the set⁠${\displaystyle X}$⁠. The morphisms from the point${\displaystyle p}$ to the point${\displaystyle q}$ areequivalence classes ofcontinuouspaths from${\displaystyle p}$ to⁠${\displaystyle q}$⁠, with two paths being equivalent if they arehomotopic.Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition isassociative. This groupoid is called thefundamental groupoid of⁠${\displaystyle X}$⁠, denoted${\displaystyle {\pi }_1(X)}$ (or sometimes,⁠${\displaystyle {\mathrm{\Pi }}_1(X)}$⁠).^[5] The usual fundamental group${\displaystyle {\pi }_1(X,x)}$ is then the vertex group for the point⁠${\displaystyle x}$⁠.

The orbits of the fundamental groupoid${\displaystyle {\pi }_1(X)}$ are the path-connected components of⁠${\displaystyle X}$⁠. Accordingly, the fundamental groupoid of apath-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 areequivalent as categories (see the sectionbelow for the general theory).

An important extension of this idea is to consider the fundamental groupoid${\displaystyle {\pi }_1(X,A)}$ where${\displaystyle A\subset X}$ is a chosen set of "base points". Here${\displaystyle {\pi }_1(X,A)}$ is a (full) subgroupoid of⁠${\displaystyle {\pi }_1(X)}$⁠, where one considers only paths whose endpoints belong to⁠${\displaystyle A}$⁠. The set${\displaystyle A}$ may be chosen according to the geometry of the situation at hand.

If${\displaystyle X}$ is asetoid, i.e. a set with anequivalence relation⁠${\displaystyle \sim }$⁠, then a groupoid "representing" this equivalence relation can be formed as follows:

• The objects of the groupoid are the elements of⁠${\displaystyle X}$⁠;
• For any two elements${\displaystyle x}$ and${\displaystyle y}$ in⁠${\displaystyle X}$⁠, there is a single morphism from${\displaystyle x}$ to${\displaystyle y}$ (denote by⁠${\displaystyle (y,x)}$⁠) if and only if⁠${\displaystyle x\sim y}$⁠;
• The composition of${\displaystyle (z,y)}$ and${\displaystyle (y,x)}$ is⁠${\displaystyle (z,x)}$⁠.

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${\displaystyle X}$ is in relation with every other element of⁠${\displaystyle X}$⁠, we obtain thepair groupoid of⁠${\displaystyle X}$⁠, which has the entire${\displaystyle X\times X}$ as set of arrows, and which is transitive.
• If every element of${\displaystyle X}$ is only in relation with itself, one obtains theunit groupoid, which has${\displaystyle X}$ as set of arrows,⁠${\displaystyle s=t={\mathrm{i}\mathrm{d}}_X}$⁠, and which is completely intransitive (every singleton${\displaystyle \{x\}}$ is an orbit).

• If${\displaystyle f:X_0\to Y}$ is a smoothsurjectivesubmersion ofsmooth manifolds, then${\displaystyle X_0{\times }_Y{X}_0\subset X_0\times X_0}$ is an equivalence relation^[6] since${\displaystyle Y}$ has a topology isomorphic to thequotient topology of${\displaystyle X_0}$ under the surjective map of topological spaces. If we write,${\displaystyle X_1=X_0{\times }_Y{X}_0}$ then we get a groupoid$${\displaystyle X_1\rightrightarrows X_0,}$$ which is sometimes called thebanal groupoid of a surjective submersion of smooth manifolds.
• If we relax the reflexivity requirement and considerpartial equivalence relations, then it becomes possible to considersemidecidable notions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, calledPER models. Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to theeffective topos introduced byMartin Hyland.

A Čech groupoid^[6]p. 5 is a special kind of groupoid associated to an equivalence relation given by an open cover${\displaystyle \mathcal{U}=\{U_i{\}}_{i\in I}}$ of some manifold⁠${\displaystyle X}$⁠. Its objects are given by the disjoint union$${\displaystyle {\mathcal{G}}_0=\coprod U_i,}$$and its arrows are the intersections$${\displaystyle {\mathcal{G}}_1=\coprod U_{ij}.}$$

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

${\displaystyle \begin{array}{r}s={\varphi }_j:U_{ij}\to U_j\\ t={\varphi }_i:U_{ij}\to U_i\end{array}}$

and the inclusion map

${\displaystyle \epsilon :U_i\to U_{ii}}$

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

${\displaystyle {\mathcal{G}}_n={\mathcal{G}}_1{\times }_{{\mathcal{G}}_0}\cdots {\times }_{{\mathcal{G}}_0}{\mathcal{G}}_1}$

as the${\displaystyle n}$-iterated fiber product where the${\displaystyle {\mathcal{G}}_n}$ represents${\displaystyle n}$-tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since

is a cartesian diagram where the maps to${\displaystyle U_i}$ are the target maps. This construction can be seen as a model for some∞-groupoids. Also, another artifact of this construction isk-cocycles

${\displaystyle [\sigma ]\in {\check{H}}^k(\mathcal{U},\underset{\_}{A})}$

for some constantsheaf of abelian groups can be represented as a function

${\displaystyle \sigma :\coprod U_{i_1\cdots i_k}\to A}$

giving an explicit representation of cohomology classes.

If thegroup${\displaystyle G}$ acts on the set⁠${\displaystyle X}$⁠, then we can form theaction groupoid (ortransformation groupoid) representing thisgroup action as follows:

• The objects are the elements of⁠${\displaystyle X}$⁠;
• For any two elements${\displaystyle x}$ and${\displaystyle y}$ in⁠${\displaystyle X}$⁠, themorphisms from${\displaystyle x}$ to${\displaystyle y}$ correspond to the elements${\displaystyle g}$ of${\displaystyle G}$ such that⁠${\displaystyle gx=y}$⁠;
• Composition of morphisms interprets thebinary operation of⁠${\displaystyle G}$⁠.

More explicitly, theaction groupoid is a small category with${\displaystyle \mathrm{o}\mathrm{b}(C)=X}$ and${\displaystyle \mathrm{h}\mathrm{o}\mathrm{m}(C)=G\times X}$ and with source and target maps${\displaystyle s(g,x)=x}$ and⁠${\displaystyle t(g,x)=gx}$⁠. It is often denoted${\displaystyle G⋉X}$ (or${\displaystyle X⋊G}$ for a right action). Multiplication (or composition) in the groupoid is then⁠${\displaystyle (h,y)(g,x)=(hg,x)}$⁠, which is defined provided⁠${\displaystyle y=gx}$⁠.

For${\displaystyle x}$ in⁠${\displaystyle X}$⁠, the vertex group consists of those${\displaystyle (g,x)}$ with⁠${\displaystyle gx=x}$⁠, which is just theisotropy subgroup at${\displaystyle x}$ for the given action (which is why vertex groups are also called isotropy groups). Similarly, the orbits of the action groupoid are theorbit of the group action, and the groupoid is transitive if and only if the group action istransitive.

Another way to describe${\displaystyle G}$-sets is thefunctor category⁠${\displaystyle [\mathrm{G}\mathrm{r},\mathrm{S}\mathrm{e}\mathrm{t}]}$⁠, where${\displaystyle \mathrm{G}\mathrm{r}}$ is the groupoid (category) with one element andisomorphic to the group⁠${\displaystyle G}$⁠. Indeed, every functor${\displaystyle F}$ of this category defines a set${\displaystyle X=F(\mathrm{G}\mathrm{r})}$ and for every${\displaystyle g}$ in${\displaystyle G}$ (i.e. for every morphism in⁠${\displaystyle \mathrm{G}\mathrm{r}}$⁠) induces abijection${\displaystyle F_g}$ :⁠${\displaystyle X\to X}$⁠. The categorical structure of the functor${\displaystyle F}$ assures us that${\displaystyle F}$ defines a${\displaystyle G}$-action on the set⁠${\displaystyle G}$⁠. The (unique)representable functor${\displaystyle F:\mathrm{G}\mathrm{r}\to \mathrm{S}\mathrm{e}\mathrm{t}}$ is theCayley representation of⁠${\displaystyle G}$⁠. In fact, this functor is isomorphic to${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{G}\mathrm{r},-)}$ and so sends${\displaystyle \mathrm{o}\mathrm{b}(\mathrm{G}\mathrm{r})}$ to the set${\displaystyle \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{G}\mathrm{r},\mathrm{G}\mathrm{r})}$ which is by definition the "set"${\displaystyle G}$ and the morphism${\displaystyle g}$ of${\displaystyle \mathrm{G}\mathrm{r}}$ (i.e. the element${\displaystyle g}$ of⁠${\displaystyle G}$⁠) to the permutation${\displaystyle F_g}$ of the set⁠${\displaystyle G}$⁠. We deduce from theYoneda embedding that the group${\displaystyle G}$ is isomorphic to the group⁠${\displaystyle \{F_g\mid g\in G\}}$⁠, asubgroup of the group ofpermutations of⁠${\displaystyle G}$⁠.

Consider the group action of${\displaystyle \mathbb{Z}/2}$ on the finite set${\displaystyle X=\{-2,-1,0,1,2\}}$ where 1 acts by taking each number to its negative, so${\displaystyle -2\mapsto 2}$ and⁠${\displaystyle 1\mapsto -1}$⁠. The quotient groupoid${\displaystyle [X/G]}$ is the set of equivalence classes from this group action⁠${\displaystyle \{[0],[1],[2]\}}$⁠, and${\displaystyle [0]}$ has a group action of${\displaystyle \mathbb{Z}/2}$ on it.

Any finite group${\displaystyle G}$ that maps to${\displaystyle GL(n)}$ gives a group action on theaffine space${\displaystyle {\mathbb{A}}^n}$ (since this is the group of automorphisms). Then, a quotient groupoid can be of the form⁠${\displaystyle [{\mathbb{A}}^n/G]}$⁠, which has one point with stabilizer${\displaystyle G}$ at the origin. Examples like these form the basis for the theory oforbifolds. Another commonly studied family of orbifolds areweighted projective spaces${\displaystyle \mathbb{P}(n_1,\dots ,n_k)}$ and subspaces of them, such asCalabi–Yau orbifolds.

Given a diagram of groupoids with groupoid morphisms

where${\displaystyle f:X\to Z}$ and⁠${\displaystyle g:Y\to Z}$⁠, we can form the groupoid${\displaystyle X{\times }_{Z}Y}$ whose objects are triples⁠${\displaystyle (x,\varphi ,y)}$⁠, where⁠${\displaystyle x\in \text{Ob}(X)}$⁠,⁠${\displaystyle y\in \text{Ob}(Y)}$⁠, and${\displaystyle \varphi :f(x)\to g(y)}$ in⁠${\displaystyle Z}$⁠. Morphisms can be defined as a pair of morphisms${\displaystyle (\alpha ,\beta )}$ where${\displaystyle \alpha :x\to x^{\prime }}$ and${\displaystyle \beta :y\to y^{\prime }}$ such that for triples⁠${\displaystyle (x,\varphi ,y),(x^{\prime },{\varphi }^{\prime },y^{\prime })}$⁠, there is a commutative diagram in${\displaystyle Z}$ of⁠${\displaystyle f(\alpha ):f(x)\to f(x^{\prime })}$⁠,${\displaystyle g(\beta ):g(y)\to g(y^{\prime })}$ and the⁠${\displaystyle \varphi ,{\varphi }^{\prime }}$⁠.^[7]

A two term complex

${\displaystyle C_1\stackrel{d}{\to }{C}_0}$

of objects in aconcreteAbelian category can be used to form a groupoid. It has as objects the set${\displaystyle C_0}$ and as arrows the set⁠${\displaystyle C_1\oplus C_0}$⁠; the source morphism is just the projection onto${\displaystyle C_0}$ while the target morphism is the addition of projection onto${\displaystyle C_1}$ composed with${\displaystyle d}$ and projection onto⁠${\displaystyle C_0}$⁠. That is, given⁠${\displaystyle c_1+c_0\in C_1\oplus C_0}$⁠, we have

${\displaystyle t(c_1+c_0)=d(c_1)+c_0.}$

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

While puzzles such as theRubik's Cube can be modeled using group theory (seeRubik's Cube group), certain puzzles are better modeled as groupoids.^[8]

The transformations of thefifteen puzzle form a groupoid (not a group, as not all moves can be composed).^[9][10][11] Thisgroupoid acts on configurations.

TheMathieu groupoid is a groupoid introduced byJohn Horton Conway acting on 13 points such that the elements fixing a point form a copy of theMathieu group M₁₂.

If a groupoid has only one object, then the set of its morphisms forms agroup. Using the algebraic definition, such a groupoid is literally just a group.^[12] Many concepts ofgroup theory generalize to groupoids, with the notion offunctor replacing that ofgroup 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)⁠${\displaystyle (G,X)}$⁠. By transitivity, there will only be oneorbit under the action.

Note that the isomorphism just mentioned is not unique, and there is nonatural choice. Choosing such an isomorphism for a transitive groupoid essentially amounts to picking one object⁠${\displaystyle x_0}$⁠, agroup isomorphism${\displaystyle h}$ from${\displaystyle G(x_0)}$ to⁠${\displaystyle G}$⁠, and for each${\displaystyle x}$ other than⁠${\displaystyle x_0}$⁠, a morphism in${\displaystyle G}$ from${\displaystyle x_0}$ to⁠${\displaystyle x}$⁠.

If a groupoid is not transitive, then it is isomorphic to adisjoint union of groupoids of the above type, also called itsconnected components (possibly with different groups${\displaystyle G}$ and sets${\displaystyle X}$ for each connected component).

In category-theoretic terms, each connected component of a groupoid isequivalent (but notisomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to amultiset of unrelated groups. In other words, for equivalence instead of isomorphism, one does not need to specify the sets⁠${\displaystyle X}$⁠, but only the groups⁠${\displaystyle G}$⁠. For example,

• The fundamental groupoid of${\displaystyle X}$ is equivalent to the collection of thefundamental groups of eachpath-connected component of⁠${\displaystyle X}$⁠, but an isomorphism requires specifying the set of points in each component;
• The set${\displaystyle X}$ with the equivalence relation${\displaystyle \sim }$ is equivalent (as a groupoid) to one copy of thetrivial group for eachequivalence class, but an isomorphism requires specifying what each equivalence class is;
• The set${\displaystyle X}$ equipped with anaction of the group${\displaystyle G}$ is equivalent (as a groupoid) to one copy of${\displaystyle G}$ for eachorbit of the action, but anisomorphism 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 notnatural. 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${\displaystyle G(x)}$ in terms of a single group, and this choice can be arbitrary. In the example fromtopology, one would have to make a coherent choice of paths (or equivalence classes of paths) from each point${\displaystyle p}$ to each point${\displaystyle q}$ in the same path-connected component.

As a more illuminating example, the classification of groupoids with oneendomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification ofvector 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, andquotient morphisms. Thus a subgroup${\displaystyle H}$ of a group${\displaystyle G}$ yields an action of${\displaystyle G}$ on the set ofcosets of${\displaystyle H}$ in${\displaystyle G}$ and hence a covering morphism${\displaystyle p}$ from, say,${\displaystyle K}$ to⁠${\displaystyle G}$⁠, where${\displaystyle K}$ is a groupoid withvertex groups isomorphic to⁠${\displaystyle H}$⁠. In this way, presentations of the group${\displaystyle G}$ can be "lifted" to presentations of the groupoid⁠${\displaystyle K}$⁠, and this is a useful way of obtaining information about presentations of the subgroup⁠${\displaystyle H}$⁠. For further information, see the books by Higgins and by Brown in the References.

The category whose objects are groupoids and whose morphisms are groupoid morphisms is called thegroupoid category, or thecategory of groupoids, and is denoted byGrpd.

The categoryGrpd is, like the category of small categories,Cartesian closed: for any groupoids${\displaystyle H,K}$ we can construct a groupoid${\displaystyle \mathrm{GPD}(H,K)}$ whose objects are the morphisms${\displaystyle H\to K}$ and whose arrows are the natural equivalences of morphisms. Thus if${\displaystyle H,K}$ are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids${\displaystyle G,H,K}$ there is a natural bijection

${\displaystyle \mathrm{Grpd}(G\times H,K)\cong \mathrm{Grpd}(G,\mathrm{GPD}(H,K)).}$

This result is of interest even if all the groupoids${\displaystyle G,H,K}$ are just groups.

Another important property ofGrpd is that it is bothcomplete andcocomplete.

The inclusion${\displaystyle i:\mathbf{G}\mathbf{r}\mathbf{p}\mathbf{d}\to \mathbf{C}\mathbf{a}\mathbf{t}}$ has both a left and a rightadjoint:

${\displaystyle {\mathrm{hom}}_{\mathbf{G}\mathbf{r}\mathbf{p}\mathbf{d}}(C[C^{-1}],G)\cong {\mathrm{hom}}_{\mathbf{C}\mathbf{a}\mathbf{t}}(C,i(G))}$

${\displaystyle {\mathrm{hom}}_{\mathbf{C}\mathbf{a}\mathbf{t}}(i(G),C)\cong {\mathrm{hom}}_{\mathbf{G}\mathbf{r}\mathbf{p}\mathbf{d}}(G,\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e}(C))}$

Here,${\displaystyle C[C^{-1}]}$ denotes thelocalization of a category that inverts every morphism, and${\displaystyle \mathrm{C}\mathrm{o}\mathrm{r}\mathrm{e}(C)}$ denotes the subcategory of all isomorphisms.

Thenerve functor${\displaystyle N:\mathbf{G}\mathbf{r}\mathbf{p}\mathbf{d}\to \mathbf{s}\mathbf{S}\mathbf{e}\mathbf{t}}$ embedsGrpd as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always aKan complex.

The nerve has a left adjoint

${\displaystyle {\mathrm{hom}}_{\mathbf{G}\mathbf{r}\mathbf{p}\mathbf{d}}({\pi }_1(X),G)\cong {\mathrm{hom}}_{\mathbf{s}\mathbf{S}\mathbf{e}\mathbf{t}}(X,N(G))}$

Here,${\displaystyle {\pi }_1(X)}$ denotes the fundamental groupoid of the simplicial set⁠${\displaystyle X}$⁠.

There is an additional structure which can be derived from groupoids internal to the category of groupoids,double-groupoids.^[13][14] BecauseGrpd is a 2-category, these objects form a 2-category instead of a 1-category since there is extra structure. Essentially, these are groupoids${\displaystyle {\mathcal{G}}_1,{\mathcal{G}}_0}$ with functors

${\displaystyle s,t:{\mathcal{G}}_1\to {\mathcal{G}}_0}$

and an embedding given by an identity functor

${\displaystyle i:{\mathcal{G}}_0\to {\mathcal{G}}_1}$

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

and

with${\displaystyle a}$ the same morphism, they can be vertically conjoined giving a diagram

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

When studying geometrical objects, the arising groupoids often carry atopology, turning them intotopological groupoids, or even somedifferentiable structure, turning them intoLie groupoids. These last objects can be also studied in terms of their associatedLie algebroids, in analogy to the relation betweenLie groups andLie algebras.

Groupoids arising from geometry often possess further structures which interact with the groupoid multiplication. For instance, inPoisson geometry one has the notion of asymplectic groupoid, which is a Lie groupoid endowed with a compatiblesymplectic form. Similarly, one can have groupoids with a compatibleRiemannian metric, orcomplex structure, etc.

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

• 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 inhomotopy theory and in groupcohomology. Many references.
• Dicks, Warren; Ventura, Enric (1996),The group fixed by a family of injective endomorphisms of a free group, Mathematical Surveys and Monographs, vol. 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.226. Elsevier: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 ofGalois theory lead toGalois groupoids.
• Cannas da Silva, A., andA. 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
• "Groupoid",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Higgins, P. J., "The fundamental groupoid of agraph 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 anorbit 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 inReprints in Theory and Applications of Categories, No. 7 (2005) pp. 1–195;freely downloadable. Substantial introduction tocategory theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation ofGrushko'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 inPostscript, 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". InAlgebraic and geometric combinatorics, volume 423 ofContemp. Math., 305–324. Amer. Math. Soc., Providence, RI (2006)
• fundamental groupoid at thenLab
• core at thenLab

• Algebraic structures
• Category theory
• Homotopy theory

• Webarchive template wayback links
• Articles with short description
• Short description matches Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from July 2024