In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group and of category 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.
A groupoid is a set G with a unary oper...
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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 and of category 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.
A groupoid is a set G with a unary operation and a partial function * 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.
and have the following axiomatic properties. Let a, b, and c be elements of G. Then:
In short:
From these axioms, two easy and convenient theorems follow:
A groupoid is a small category in which every morphism is an isomorphism, and hence invertible. More precisely, a groupoid G is:
The objects and morphisms have...
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