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:

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 ⁠${\displaystyle g:A\rightarrow B}$⁠, ⁠${\displaystyle h:B\rightarrow C}$⁠, say. Composition is then a total function: ⁠${\displaystyle \circ :(B\rightarrow C)\rightarrow (A\rightarrow B)\rightarrow (A\rightarrow C)}$⁠, so that ⁠${\displaystyle h\circ g:A\rightarrow C}$⁠.

Special cases include:

Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups.

A groupoid can be viewed as an algebraic structure consisting of a set with a binary partial function.^{[citation needed]} Precisely, it is a non-empty set ${\displaystyle G}$ with a unary operation ⁠${\displaystyle {}^{-1}:G\to G}$⁠, and a partial function ⁠${\displaystyle *:G\times G\rightharpoonup G}$⁠. Here ${\displaystyle *}$ is not a binary operation because it is not necessarily defined for all pairs of elements of ⁠${\displaystyle G}$⁠. The precise conditions under which ${\displaystyle *}$ 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 ⁠${\displaystyle G}$⁠,

Two easy and convenient properties follow from these axioms:

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