Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general class of monads called monads with arities, so that not only algebraic theories can be computed from a proper set of arities, but also more general structures like n-categories, the computing process being realized using Kan extensions. This Master thesis compiles the required material in order to understand this question of arities and reconstruction of monads, and tries to give some examples of relevant interest from both semantics and higher category theory. A discussion on the promising field of operads is then provided as appendix.
In logics or semantics, it is common to think of a theory as generated by elements together with relations between them. Every operation of the theory is then obtained by composition of the generators in a way that respects their arities: for example, the usual addition + as arity 2 since it takes two inputs. In [10], Lawvere gives a functorial presentation of this, where there are no distinguished generators anymore, but only n-ary operations whose compositions respect the relations of the theory. The set of all arities is then N: for example, in the case of arithmetics, the composite of n additions gives a (n + 1)-ary operation, and this for every natural integer n. These theories are in turn equivalent to finitary monads, that is, monads whose values on finite ordinals determine values on all sets, the computation being realized by a filtered colimit. But monads have an interest outside of semantics: for example, they allow to build mathematical structures such as categories, which are built out of graphs. A natural question is then to determine whether the case of finitary monads can be generalized: is there a finite set of arities, to be thought of as some set of elementary pieces, which can be glued by a proper process to realize all categories ? It happens to be the case: weighted colimits of filiform graphs give birth to all categories, as to be shown in this paper. More generally, a notion of monads with arities is provided (originally introduced by Weber in [27]), together with a notion of Lawvere theories with arities, extending the usual correspondence. This paper is structured as follows: we start at Section 2 by recalling usual Lawvere theories, finitary monads, and the traditional correspondence between them, after what we investigate the structure of categories to determine their set of arities, and then we introduce Kan extensions, which give a proper way of describing the computation of a monad with arities on every value, from its values on arities. An introduction to Yoneda structures is then given. At Section 3, we give the general axiomatization of monads with arities, together with their corresponding Lawvere theories, and investigate the arities of the usual free category and free 2-category monads. At Section 4, we give several examples of monads encountered in the practice of semantics, together with their corresponding theories; a special treatment of the state monad is made, following [20]. The appendix enlarges the discussion to the promising field that is the one of operads: after a short introduction to them, we explain following [12] that, in spite of their similarities with Lawvere theories, operads are not equivalent to them; we then have a look at how the generalization of arities is treated in the operadic case.
Acknowledgements I would like to thank Paul-André Melliès for his advices and the quality of the discussions we have had, as well as for the freedom I enjoyed while working on this master thesis, Jonas Frey for answering my questions with a lot of patience, the people of the working group “Catégories supérieures, polygraphes et homotopie” of the Laboratoire Programmes, Preuves, Systèmes for the interesting talks on the notions of operads and arities they gave, and to the contributors to the nLab for their amazing and very useful work. Special thanks go to Antoine Delignat-Lavaud for his support in improving the language quality of this document. I also would like to apologize by advance to all the people whose paternity in ideas detailed here was not mentioned due to my lack of knowledge of the history of the field.
2.1 Monads and theories
Historically, algebraic theories such as monoids, groups, Lie algebras . . . used to be presented by means of generators and relations between them. In his doctoral dissertation [10], Lawvere introduced in 1963 an alternative method of specification of algebraic theories using categories.
Definition 1 (Lawvere theory). A Lawvere theory is a category L with finite products, in which every object is isomorphic to a finite cartesian power x n of a distinguished object x, called the generic object of the theory T .
There is a category of Lawvere theories, with morphisms the product-preserving functors between these theories such that the distinguished object of the source category is sent to the distinguished object of the target category. The idea behind Lawvere theories is to represent every n-ary operation of a theory L as a morphism from x n to x -here, the relations of the usual presentation of algebraic structures are encoded in the composition law of the category L. Moreover, there is no notion of “primitive” operation in a Lawvere theory: there are no specified generators, but rather all the operations are given in L. Just like a given group is a model of the algebraic theory of groups, there is a notion of model of a Lawvere theory: Definition 2 (Model of a Lawvere theory). A model of a Lawvere theory L in a category C
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