groupoid

Noun

Meaning

(algebra) A magma: a set with a total binary operation.

(algebra and category theory) A set with a partial binary operation that is associative and has identities and inverses.

Examples

A groupoid is a category in which every morphism is an isomorphism. Source: Internet

A groupoid can be seen as a: * Group with a partial function replacing the binary operation ; * Category in which every morphism is invertible. Source: Internet

Examples Linear algebra Given a field K, the corresponding general linear groupoid GL * (K) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition. Source: Internet

Further the stabilizers of the action are the vertex groups, and the orbits of the action are the components, of the action groupoid. Source: Internet

Category of groupoids A subgroupoid is a subcategory that is itself a groupoid. Source: Internet

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Source: Internet

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