mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra 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: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

, 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 together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...

s in the same way that small categories generalise

monoid In abstract algebra, 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 . Monoids are semigroups with identity ...

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 fu ...

s generalise groups. Semigroupoids have applications in the structural theory of semigroups. Formally, a ''semigroupoid'' consists of: * a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of things called ''objects''. * for every two objects ''A'' and ''B'' a set Mor(''A'',''B'') of things called ''

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

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 {{categorytheory-stub