# Groupoid

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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 in several equivalent ways. A groupoid can be seen as a:

Special cases include:

Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt introduced groupoids implicitly via Brandt semigroups in 1926.[1]

## Contents

### Algebraic

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