Monads with arities and their associated theories
Copyright policy )Monads with arities and their associated theories
After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids.
New introduction; Section 1 shortened and redispatched with Section 2; Subsections on symmetric operads (3.14) and symmetric simplicial sets (4.17) added; Bibliography completed
- Univ Paris
- Université Paris Cité France
- French National Centre for Scientific Research France
- Macquarie University Australia
- University of Paris France
monad/theory equivalences, Algebra and Number Theory, Mathematics - Category Theory, Theories (e.g., algebraic theories), structure, and semantics, [MATH] Mathematics [math], globular operads, Monads with arities, Primary 18C10, 18C15, Secondary 18D50, 18B40, generic factorization system, algebraic theories, homogeneous theory, symmetric operads, Closed categories (closed monoidal and Cartesian closed categories, etc.), FOS: Mathematics, Operads, Category Theory (math.CT), nerve theorem, simplicial nerve, category of theories, Groupoids, semigroupoids, semigroups, groups (viewed as categories), Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
monad/theory equivalences, Algebra and Number Theory, Mathematics - Category Theory, Theories (e.g., algebraic theories), structure, and semantics, [MATH] Mathematics [math], globular operads, Monads with arities, Primary 18C10, 18C15, Secondary 18D50, 18B40, generic factorization system, algebraic theories, homogeneous theory, symmetric operads, Closed categories (closed monoidal and Cartesian closed categories, etc.), FOS: Mathematics, Operads, Category Theory (math.CT), nerve theorem, simplicial nerve, category of theories, Groupoids, semigroupoids, semigroups, groups (viewed as categories), Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
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