Semigroupoid
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| Total | Associative | Identity | [[Quasigroup|Page Template:Tooltip/styles.css has no content.Template:Encodefirst]] | |
|---|---|---|---|---|
| Partial magma | Unneeded | Unneeded | Unneeded | 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 |
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. 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.
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
- Yoneda lemma does not hold in general for semicategories.
References
Page Template:Reflist/styles.css has no content.
- ^ Page Module:Citation/CS1/styles.css has no content.Tilson, Bret (1987). "Categories as algebra: an essential ingredient in the theory of monoids". J. Pure Appl. Algebra. 48 (1–2): 83–198. doi:10.1016/0022-4049(87)90108-3., Appendix B
- ^ Page Module:Citation/CS1/styles.css has no content.Rhodes, John; Steinberg, Ben (2009), The q-Theory of Finite Semigroups, Springer, p. 26, ISBN 9780387097817
- ^ See e.g. Page Module:Citation/CS1/styles.css has no content.Gomes, Gracinda M. S. (2002), Semigroups, Algorithms, Automata and Languages, World Scientific, p. 41, ISBN 9789812776884, which requires the objects of a semigroupoid to form a set.
- Page Module:Citation/CS1/styles.css has no content.Mitchell, Barry (1972). "The Dominion of Isbell". Transactions of the American Mathematical Society. 167: 319–331. doi:10.1090/S0002-9947-1972-0294441-0. JSTOR 1996142.
- Page Module:Citation/CS1/styles.css has no content.Moens, M.; Berni-Canani, U.; Borceux, F. (2002). "On regular presheaves and regular semi-categories" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques.
- Page Module:Citation/CS1/styles.css has no content.Stubbe, Isar (2005). "Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques. 46 (2): 99–121.
External links
- Page Module:Citation/CS1/styles.css has no content."Yoneda lemma 6. The Yoneda lemma in semicategories". ncatlab.org.
- Page Module:Citation/CS1/styles.css has no content.The Univalent Foundations Program (2013). "Homotopy Type Theory: Univalent Foundations of Mathematics". Homotopy Type Theory.
