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 theo...

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Bibliographic Details
Published in:Journal of pure and applied algebra 2012-08, Vol.216 (8-9), p.2029-2048
Main Authors: Berger, Clemens, Melliès, Paul-André, Weber, Mark
Format: Article
Language:English
Subjects:
Mathematics
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Summary: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.
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2012.02.039