Editing Groupoid (section)

{{Short description|Category where every morphism is invertible; generalization of a group}}
{{About|groupoids in category theory|the algebraic structure with a single binary operation|magma (algebra)}}
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In [[mathematics]], especially in [[category theory]] and [[homotopy theory]], a '''groupoid''' (less often '''Brandt groupoid''' or '''virtual group''') generalises the notion of [[group (mathematics)|group]] in several equivalent ways. A groupoid can be seen as a:
* ''[[group (mathematics)|Group]]'' with a [[partial function]] replacing the [[binary operation]];
* ''[[category theory|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]].<ref name="dicks-ventura-96">{{cite book|author=Dicks & Ventura|year=1996|title=The Group Fixed by a Family of Injective Endomorphisms of a Free Group|url={{Google books|plainurl=y|id=3sWSRRfNFKgC|page=6|text=G has the structure of a graph}}|page=6}}</ref> A groupoid where there is only one object is a usual group.

In the presence of [[Dependent type|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 {{tmath|1= g:A \rightarrow B }},  {{tmath|1= h:B \rightarrow C }}, say. Composition is then a total function: {{tmath|1= \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow (A \rightarrow C) }}, so that {{tmath|1= h \circ g : A \rightarrow C }}.

Special cases include:
* ''[[Setoid]]s'': [[Set (mathematics)|sets]] that come with an [[equivalence relation]],
* ''[[G-set]]s'': sets equipped with an [[Group action (mathematics)|action]] of a group {{tmath|1= G }}.

Groupoids are often used to reason about [[geometrical]] objects such as [[manifold]]s. {{harvs|txt|first=Heinrich |last=Brandt|authorlink=Heinrich Brandt|year=1927}} introduced groupoids implicitly via [[Brandt semigroup]]s.<ref>{{SpringerEOM|title=Brandt semi-group|ISBN=1-4020-0609-8}}</ref>