{{Short description|Partial algebra}} {{Group-like structures}} In mathematics, a '''semigroupoid''' (also called '''semicategory''', '''naked category''' or '''precategory''') is a partial algebra that satisfies the axioms for a small<ref>{{cite journal|last=Tilson|first=Bret|title=Categories as algebra: an essential ingredient in the theory of monoids|journal=J. Pure Appl. Algebra|volume=48|year=1987|issue=1-2|pages=83–198|doi=10.1016/0022-4049(87)90108-3|doi-access=free}}, Appendix B</ref><ref>{{citation|title=The q-Theory of Finite Semigroups|first1=John|last1=Rhodes|first2=Ben|last2=Steinberg|publisher=Springer|year=2009|isbn=9780387097817|page=26}}</ref><ref>See e.g. {{citation|title=Semigroups, Algorithms, Automata and Languages|first=Gracinda M. S.|last=Gomes|publisher=World Scientific|year=2002|isbn=9789812776884|page=41|url=https://books.google.com/books?id=IL58mAsfXOgC&pg=PA41}}, which requires the objects of a semigroupoid to form a set.</ref> 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== {{reflist}} *{{cite journal |doi=10.1090/S0002-9947-1972-0294441-0|jstor=1996142 |title=The Dominion of Isbell |last1=Mitchell |first1=Barry |journal=Transactions of the American Mathematical Society |year=1972 |volume=167 |pages=319–331 |doi-access=free }} *{{cite journal |last1=Moens |first1=M. |last2=Berni-Canani |first2=U. |last3=Borceux |first3=F. |title=On regular presheaves and regular semi-categories |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=2002 |url=https://www.numdam.org/article/CTGDC_2002__43_3_163_0.pdf}} *{{cite journal |last1=Stubbe |first1=Isar |title=Categorical structures enriched in a quantaloid : regular presheaves, regular semicategories |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=2005 |volume=46 |issue=2 |pages=99-121 |url=http://archive.numdam.org/article/CTGDC_2005__46_2_99_0.pdf}}
==External links== *{{cite web|title=Yoneda lemma 6. The Yoneda lemma in semicategories |url=https://ncatlab.org/nlab/show/Yoneda+lemma#the_yoneda_lemma_in_semicategories |website=ncatlab.org}} *{{cite web |author=The Univalent Foundations Program |title=Homotopy Type Theory: Univalent Foundations of Mathematics |url=https://homotopytypetheory.org/book/ |website=Homotopy Type Theory |language=en |date=2013}} Category:Algebraic structures Category:Category theory
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