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Monads with arities and their associated theories

By Clemens Berger, Paul-andré Melliès and Mark Weber


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 application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids. Introduction. In his seminal work [20] Lawvere constructed for every variety of algebras, defined by finitary operations and relations on sets, an algebraic theory whose n-ary operations are the elements of the free algebra on n elements. He showed that the variety of algebras is equivalent to the category of models of the associated algebrai

Year: 2013
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