Semigroupoid
| Total | Associative | Identity | Divisible | |
|---|---|---|---|---|
| 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. 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.
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
[edit]- Yoneda lemma does not hold in general for semicategories.
References
[edit]- ↑ 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
- ↑ Rhodes, John; Steinberg, Ben (2009), The q-Theory of Finite Semigroups, Springer, p. 26, ISBN 9780387097817
- ↑ See e.g. 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.
- ↑ Exel, R. (May 2011). "Semigroupoid C ⁎ -algebras". Journal of Mathematical Analysis and Applications. 377 (1): 303–318. doi:10.1016/j.jmaa.2010.10.061.
- 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.
- Moens, M.; Berni-Canani, U.; Borceux, F. (2002). "On regular presheaves and regular semi-categories" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques.
- 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
[edit]- "Yoneda lemma 6. The Yoneda lemma in semicategories". ncatlab.org.
- The Univalent Foundations Program (2013). "Homotopy Type Theory: Univalent Foundations of Mathematics". Homotopy Type Theory.
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