Monoid-like Definitions of Cyclic Operad

Guided by the microcosm principle of Baez-Dolan and by the algebraic definitions of operads of Kelly and Fiore, we introduce two “monoid-like” definitions of cyclic operads, one for the original, “exchangable-output” characterisation of GetzlerKapranov, and the other for the alternative “entries-only” characterisation, both within the category of Joyal’s species of structures. Relying on a result of Lamarche on descent for species, we use these monoid-like definitions to prove the equivalence between the “exchangable-output” and “entries-only” points of view on cyclic operads. Introduction A species of structures S associates to each finite set X a set S(X) of combinatorial structures on X that are invariant under renaming the elements of X in a way consistent with composition of such renamings. The notion, introduced in combinatorics by Joyal in [J81], has been set up to provide a description of discrete structures that is independent from any specific format these structures could be presented in. For example, S(X) could be the set of graphs whose vertices are given by X, the set of all permutations of X, the set of all subsets of X, etc. Categorically speaking, a species of structures is simply a functor C : Bij → Set, wherein Set is the category of sets and functions, and Bij is the category of finite sets and bijections. Species can be combined in various ways into new species and these “species algebras” provide the category of species with different notions of “tensor product”. Some of these products allow to redefine operads internally to the category of species, as monoids. A definition given in this framework is usually referred to as algebraic. A definition of an operad as a collection of abstract operations of different arities that can be suitably composed will be called componential in this paper. Kelly [K05] has given an algebraic definition of a symmetric operad corresponding to the original componential definition of May [May72]. This definition is referred to as the monoidal definition of operads, since the involved product on species bears a monoidal structure. The second definition, which characterises operads with partial composition, has been recently established by Fiore in [F14]. The pre-Lie product of Fiore’s definition is not monoidal, but the inferred structure arises by the same kind of principle as the one reflecting a specification of a monoid in a monoidal category (which is why we call this definition the monoid-like definition of operads). This is a typical example of what has I would like to thank Pierre-Louis Curien and François Lamarche for useful discussions. Received by the editors 2016-07-08 and, in final form, 2017-03-10. Transmitted by Clemens Berger. Published on 2017-03-13. 2010 Mathematics Subject Classification: 18D50, 18D10, 05A99.