Semigroupoid
Partial algebra
From Wikipedia, the free encyclopedia
In mathematics, a semigroupoid (also called semicategory, naked category or precategory) is a partial algebra that satisfies the axioms for a small[1][2][3] category, except possibly for the requirement that there be an identity at each object. While this definition is due to Tilson, Exel has introduced a different definition, one in which there is no underlying graph.[4] The term semicategory usually refers to a Tilson's graphed semigroupoid. Semigroupoids generalise semigroups in the same way that small categories generalise monoids and groupoids generalise groups. Semigroupoids have applications in the structural theory of semigroups.
| Total | Associative | Identity | Divisible | |
|---|---|---|---|---|
| Partial magma | Unneeded | Unneeded | Unneeded | Unneeded |
| Multiplicative | Unneeded | Unneeded | Required | Unneeded |
| Semigroupoid | Unneeded | Required | Unneeded | Unneeded |
| Small category | Unneeded | Required | Required | Unneeded |
| Groupoid | Unneeded | Required | Required | Required |
| Magma | Required | Unneeded | Unneeded | Unneeded |
| Quasigroup | Required | Unneeded | Unneeded | Required |
| Unital magma | Required | Unneeded | Required | Unneeded |
| Loop | Required | Unneeded | Required | Required |
| Semigroup | Required | Required | Unneeded | Unneeded |
| Associative quasigroup | Required | Required | Unneeded | Required |
| Monoid | Required | Required | Required | Unneeded |
| Group | Required | Required | Required | Required |
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 morphisms 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.
Examples
- The Yoneda lemma does not hold in general for semicategories.
