Groupoid

A groupoid is a set ${\displaystyle G}$ with a unary operation ${\displaystyle {}^{-1}:G\to G,}$ and a partial function ${\displaystyle *:G\times G\rightharpoonup G}$. Here * is not a binary operation because it is not necessarily defined for all pairs of elements of ${\displaystyle G}$. The precise conditions under which ${\displaystyle *}$ is defined are not articulated here and vary by situation.

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

1. Associativity: If ${\displaystyle a*b}$ and ${\displaystyle b*c}$ are defined, then ${\displaystyle (a*b)*c}$ and ${\displaystyle a*(b*c)}$ are defined and are equal. Conversely, if one of ${\displaystyle (a*b)*c}$ and ${\displaystyle a*(b*c)}$ is defined, then so are both ${\displaystyle a*b}$ and ${\displaystyle b*c}$ as well as ${\displaystyle (a*b)*c}$ = ${\displaystyle a*(b*c)}$.
2. Inverse: ${\displaystyle a^{-1}*a}$ and ${\displaystyle a*{a^{-1}}}$ are always defined.
3. Identity: If ${\displaystyle a*b}$ is defined, then ${\displaystyle a*b*{b^{-1}}=a}$, and ${\displaystyle {a^{-1}}*a*b=b}$. (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 ${\displaystyle a*b}$ is defined, then ${\displaystyle (a*b)^{-1}=b^{-1}*a^{-1}}$.[3]

A groupoid is a small category in which every morphism is an isomorphism, i.e. invertible.[1] More precisely, a groupoid G is:

• A set G₀ of objects;
• For each pair of objects x and y in G₀, there exists 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 ${\displaystyle \mathrm {id} _{x}}$ of G(x,x);
• For each triple of objects x, y, and z, a function ${\displaystyle \mathrm {comp} _{x,y,z}:G(y,z)\times G(x,y)\rightarrow G(x,z):(g,f)\mapsto gf}$;
• For each pair of objects x, y a function ${\displaystyle \mathrm {inv} :G(x,y)\rightarrow G(y,x):f\mapsto f^{-1}}$;

satisfying, for any f : x → y, g : y → z, and h : z → w:

• ${\displaystyle f\ \mathrm {id} _{x}=f}$ and ${\displaystyle \mathrm {id} _{y}\ f=f}$;
• ${\displaystyle (hg)f=h(gf)}$;
• ${\displaystyle ff^{-1}=\mathrm {id} _{y}}$ and ${\displaystyle f^{-1}f=\mathrm {id} _{x}}$.

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).

More generally, one can consider a groupoid 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, 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 ${\displaystyle \mathrm {comp} }$ and ${\displaystyle \mathrm {inv} }$ become partial operations on G, and ${\displaystyle \mathrm {inv} }$ will in fact be defined everywhere. We define ∗ to be ${\displaystyle \mathrm {comp} }$ and ⁻¹ to be ${\displaystyle \mathrm {inv} }$, which gives a groupoid in the algebraic sense. Explicit reference to G₀ (and hence to ${\displaystyle \mathrm {id} }$) can be dropped.

Conversely, given a groupoid G in the algebraic sense, define an equivalence relation ${\displaystyle \sim }$ on its elements by ${\displaystyle a\sim b}$ iff a ∗ a⁻¹ = b ∗ b⁻¹. Let G₀ be the set of equivalence classes of ${\displaystyle \sim }$, i.e. ${\displaystyle G_{0}:=G/\!\!\sim }$. Denote a ∗ a⁻¹ by ${\displaystyle 1_{x}}$ if ${\displaystyle a\in x}$ with ${\displaystyle x\in G_{0}}$.

Now define ${\displaystyle G(x,y)}$ as the set of all elements f such that ${\displaystyle 1_{x}*f*1_{y}}$ exists. Given ${\displaystyle f\in G(x,y)}$ and ${\displaystyle g\in G(y,z),}$ their composite is defined as ${\displaystyle gf:=f*g\in G(x,z)}$. To see that this is well defined, observe that since ${\displaystyle (1_{x}*f)*1_{y}}$ and ${\displaystyle 1_{y}*(g*1_{z})}$ exist, so does ${\displaystyle (1_{x}*f*1_{y})*(g*1_{z})=f*g}$. The identity morphism on x is then ${\displaystyle 1_{x}}$, 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.

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.

A subgroupoid is a subcategory that is itself a groupoid. A groupoid morphism is simply a functor between two (category-theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd.

It is useful that this category is, like the category of small categories, Cartesian closed. That is, we can construct for any groupoids ${\displaystyle H,K}$ a groupoid ${\displaystyle \operatorname {GPD} (H,K)}$ 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

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

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

Particular kinds of morphisms of groupoids are of interest. A morphism ${\displaystyle p:E\to B}$ of groupoids is called a fibration 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 a covering morphism or covering of groupoids if further such an ${\displaystyle e}$ is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering 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 a topological space ${\displaystyle X}$, let ${\displaystyle G_{0}}$ be the set ${\displaystyle X}$. The morphisms from the point ${\displaystyle p}$ to the point ${\displaystyle q}$ are equivalence classes of continuous paths from ${\displaystyle p}$ to ${\displaystyle q}$, 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 ${\displaystyle X}$, denoted ${\displaystyle \pi _{1}(X)}$ (or sometimes, ${\displaystyle \Pi _{1}(X)}$).[5] The usual fundamental group ${\displaystyle \pi _{1}(X,x)}$ is then the vertex group for the point ${\displaystyle x}$.

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, one considers only paths whose endpoints belong to ${\displaystyle A}$. ${\displaystyle \pi _{1}(X,A)}$ is a sub-groupoid of ${\displaystyle \pi _{1}(X)}$. The set ${\displaystyle A}$ may be chosen according to the geometry of the situation at hand.

If ${\displaystyle X}$ is a set with an equivalence relation denoted by infix ${\displaystyle \sim }$, 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 ${\displaystyle y}$ if and only if ${\displaystyle x\sim y}$.

If the group ${\displaystyle G}$ acts on the set ${\displaystyle X}$, then we can form the action groupoid (or transformation groupoid) representing this group action as follows:

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

More explicitly, the action groupoid is a small category with ${\displaystyle \mathrm {ob} (C)=X}$ and ${\displaystyle \mathrm {hom} (C)=G\times X}$ with source and target maps ${\displaystyle s(g,x)=x}$ and ${\displaystyle t(g,x)=gx}$. It is often denoted ${\displaystyle G\ltimes X}$ (or ${\displaystyle X\rtimes G}$). 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 the isotropy subgroup at ${\displaystyle x}$ for the given action (which is why vertex groups are also called isotropy groups).

Another way to describe ${\displaystyle G}$-sets is the functor category ${\displaystyle [\mathrm {Gr} ,\mathrm {Set} ]}$, where ${\displaystyle \mathrm {Gr} }$ is the groupoid (category) with one element and isomorphic to the group ${\displaystyle G}$. Indeed, every functor ${\displaystyle F}$ of this category defines a set ${\displaystyle X=F(\mathrm {Gr} )}$ and for every ${\displaystyle g}$ in ${\displaystyle G}$ (i.e. for every morphism in ${\displaystyle \mathrm {Gr} }$) induces a bijection ${\displaystyle F_{g}}$ : ${\displaystyle X\to X}$. 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 ${\displaystyle F}$ : ${\displaystyle \mathrm {Gr} }$→${\displaystyle \mathrm {Set} }$ is the Cayley representation of ${\displaystyle G}$. In fact, this functor is isomorphic to ${\displaystyle \mathrm {Hom} (\mathrm {Gr} ,-)}$ and so sends ${\displaystyle \mathrm {ob} (\mathrm {Gr} )}$ to the set ${\displaystyle \mathrm {Hom} (\mathrm {Gr} ,\mathrm {Gr} )}$ which is by definition the "set" ${\displaystyle G}$ and the morphism ${\displaystyle g}$ of ${\displaystyle \mathrm {Gr} }$ (i.e. the element ${\displaystyle g}$ of ${\displaystyle G}$) to the permutation ${\displaystyle F_{g}}$ of the set ${\displaystyle G}$. We deduce from the Yoneda embedding that the group ${\displaystyle G}$ is isomorphic to the group ${\displaystyle \{F_{g}\mid g\in G\}}$, a subgroup of the group of permutations of ${\displaystyle G}$.

Consider the finite set ${\displaystyle X=\{-2,-1,0,1,2\}}$, we can form the group action ${\displaystyle \mathbb {Z} /2}$ acting on ${\displaystyle X}$ 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.

On ${\displaystyle \mathbb {A} ^{n}}$, any finite group ${\displaystyle G}$ which maps to ${\displaystyle GL(n)}$ give a group action on ${\displaystyle \mathbb {A} ^{n}}$ (since this is the group of automorphisms). Then, a quotient groupoid can be forms ${\displaystyle [\mathbb {A} ^{n}/G]}$, which has one point with stabilizer ${\displaystyle G}$ at the origin.

Given a diagram of groupoids with groupoid morphisms

${\displaystyle {\begin{aligned}&&X\\&&\downarrow \\Y&\rightarrow &Z\end{aligned}}}$

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 ${\displaystyle (x,\phi ,y)}$, where ${\displaystyle x\in {\text{Ob}}(X)}$, ${\displaystyle y\in {\text{Ob}}(Y)}$, and ${\displaystyle \phi :f(x)\to g(y)}$ in ${\displaystyle Z}$. Morphisms can be defined as a pair of morphisms ${\displaystyle (\alpha ,\beta )}$ where ${\displaystyle \alpha :x\to x'}$ and ${\displaystyle \beta :y\to y'}$ such that for triples ${\displaystyle (x,\phi ,y),(x',\phi ',y')}$ , there is a commutative diagram in ${\displaystyle Z}$ of ${\displaystyle f(\alpha ):f(x)\to f(x')}$, ${\displaystyle g(\beta ):g(y)\to g(y')}$ and the ${\displaystyle \phi ,\phi '}$.[6]

A two term complex

${\displaystyle C_{1}{\overset {d}{\rightarrow }}C_{0}}$

of objects in a concrete Abelian category can be used to form a groupoid. It has as objects the set ${\displaystyle C_{0}}$ and arrows ${\displaystyle C_{1}\oplus C_{0}}$ where 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 of coherent sheaves on a scheme, then this construction can be used to form a presheaf of groupoids.

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.[7]

The transformations of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed).[8][9][10] This groupoid acts on configurations.

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.

The inclusion ${\displaystyle i:\mathbf {Grpd} \to \mathbf {Cat} }$ has both a left and a right adjoint:

${\displaystyle \hom _{\mathbf {Grpd} }(C[C^{-1}],G)\cong \hom _{\mathbf {Cat} }(C,i(G))}$

${\displaystyle \hom _{\mathbf {Cat} }(i(G),C)\cong \hom _{\mathbf {Grpd} }(G,\mathrm {Core} (C))}$

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

The nerve functor ${\displaystyle N:\mathbf {Grpd} \to \mathbf {sSet} }$ embeds Grpd as a full subcategory of the category of simplicial sets. The nerve of a groupoid is always Kan complex.

The nerve has a left adjoint

${\displaystyle \hom _{\mathbf {Grpd} }(\pi _{1}(X),G)\cong \hom _{\mathbf {sSet} }(X,N(G))}$

Here, ${\displaystyle \pi _{1}(X)}$ denotes the fundamental groupoid of the simplicial set X.