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 g : A → B {\displaystyle g:A\rightarrow B} , h : B → C {\displaystyle h:B\rightarrow C} , say. Composition is then a total function: ∘ : ( B → C ) → ( A → B ) → ( A → C ) {\displaystyle \circ :(B\rightarrow C)\rightarrow (A\rightarrow B)\rightarrow (A\rightarrow C)} , so that h ∘ g : A → C {\displaystyle h\circ g:A\rightarrow C} . Special cases include:
Setoid: a set that comes with an equivalence relation, G-set: a set equipped with an action of a group G {\displaystyle G} . Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) 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 {\displaystyle G} with a unary operation
− 1 : G → G {\displaystyle {}^{-1}:G\to G} , and a partial function ∗ : G × G ⇀ G {\displaystyle *:G\times G\rightharpoonup G} . Here ∗ {\displaystyle *} is not a binary operation because it is not necessarily defined for all pairs of elements of G {\displaystyle G} . The precise conditions under which ∗ {\displaystyle *} is defined are not articulated here and vary by situation. The operations ∗ {\displaystyle \ast } and
− 1 {\displaystyle {}^{-1}} have the following axiomatic properties: For all a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} in G {\displaystyle G} ,
Associativity: If a ∗ b {\displaystyle a*b} and b ∗ c {\displaystyle b*c} are defined, then ( a ∗ b ) ∗ c {\displaystyle (a*b)*c} and a ∗ ( b ∗ c ) {\displaystyle a*(b*c)} are defined and are equal. Conversely, if one of ( a ∗ b ) ∗ c {\displaystyle (a*b)*c} or a ∗ ( b ∗ c ) {\displaystyle a*(b*c)} is defined, then they are both defined (and they are equal to each other), and a ∗ b {\displaystyle a*b} and b ∗ c {\displaystyle b*c} are also defined. Inverse: a − 1 ∗ a {\displaystyle a^{-1}*a} and a ∗ a − 1 {\displaystyle a*{a^{-1}}} are always defined. Identity: If a ∗ b {\displaystyle a*b} is defined, then a ∗ b ∗ b − 1 = a {\displaystyle a*b*{b^{-1}}=a} , and a − 1 ∗ a ∗ b = b {\displaystyle {a^{-1}}*a*b=b} . (The previous two axioms already show that these expressions are defined and unambiguous.) Two convenient properties follow from these axioms:
( a − 1 ) − 1 = a {\displaystyle (a^{-1})^{-1}=a} , If a ∗ b {\displaystyle a*b} is defined, then ( a ∗ b ) − 1 = b − 1 ∗ a − 1 {\displaystyle (a*b)^{-1}=b^{-1}*a^{-1}} .
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