In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a

partial algebra In abstract algebra, a partial algebra is a generalization of universal algebra to partial operations. Example(s) * partial groupoid * field — the multiplicative inversion is the only proper partial operation * effect algebra Effect algebras ...

that satisfies the axioms for a smallSee e.g. , which requires the objects of a semigroupoid to form a set.

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

, except possibly for the requirement that there be an identity at each object. Semigroupoids generalise

semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...

s in the same way that small categories generalise

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

s and

groupoid 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: *'' Group'' with a partial fun ...

s generalise

groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

. Semigroupoids have applications in the structural theory of semigroups. Formally, a ''semigroupoid'' consists of: * a set of things called ''objects''. * for every two objects ''A'' and ''B'' a set Mor(''A'',''B'') of things called ''

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...

s from A to B''. If ''f'' is in Mor(''A'',''B''), we write ''f'' : ''A'' → ''B''. * for every three objects ''A'', ''B'' and ''C'' a binary operation Mor(''A'',''B'') × Mor(''B'',''C'') → Mor(''A'',''C'') called ''composition of morphisms''. The composition of ''f'' : ''A'' → ''B'' and ''g'' : ''B'' → ''C'' is written as ''g'' ∘ ''f'' or ''gf''. (Some authors write it as ''fg''.) such that the following axiom holds: * (associativity) if ''f'' : ''A'' → ''B'', ''g'' : ''B'' → ''C'' and ''h'' : ''C'' → ''D'' then ''h'' ∘ (''g'' ∘ ''f'') = (''h'' ∘ ''g'') ∘ ''f''.

References

Algebraic structures Category theory {{algebra-stub