Monads with arities and their associated theories

By C. Berger, Paul-André Melliès, Mark Weber. 2011

86 citations recorded by the source corpus.

Abstract 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 an application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a631c27f835ed9ad51fc5e03ccf8f2a2c910f8e1

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness (2015, background): “The notions of generic and free maps are general notions in category theory, introduced by Weber [34, 35], who extracted the notions from earlier work of Joyal [17]; a recommended entry point to the theory is Berger–Melliès–Weber [3].”
Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory (2015, background): “A recommended entry point to the theory is Berger–Melliès–Weber [6].”
Monads need not be endofunctors (2010, background): “ in a good sense, the Kleisli construction for arrows. Spivey [31] has studied a generalization of monads, which diﬀers from ours, but is similar in spirit and related (see Conclusion). Berger et al. [10, 16] have introduced a generalization of ﬁnitary monads, called monads with arities. Monads with”