Cyclic operads: syntactic, algebraic and categorified aspects

Jovana Obradovic 1, 2
2 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : In this thesis, we examine different frameworks for the general theory of cyclic operads of Getzler and Kapranov. As suggested by the title, we set up theoretical grounds of syntactic, algebraic and categorified nature for the notion of a cyclic operad. In the syntactic treatment, we propose a λ-calculus-style formal language, called μ-syntax, as a lightweight representation of the entries-only cyclic operad structure. As opposed to the original exchangeable-output characterisation of cyclic operads, according to which the operations of a cyclic operad have inputs and an output that can be “exchanged” with one of the inputs, the entries-only cyclic operads have only entries (i.e. the output is put on the same level as the inputs). By employing the rewriting methods behind the formalism, we give a complete step-by-step proof of the equivalence between the unbiased and biased definitions of cyclic operads. Guided by the microcosm principle of Baez and Dolan and by the algebraic definitions of operads of Kelly and Fiore, in the algebraic approach we define cyclic operads internally to the category of Joyal’s species of structures. In this way, both the original exchangeable-output characterisation of Getzler and Kapranov, and the alternative entries-only characterisation of cyclic operads of Markl are epitomised as “monoid-like” objects in “monoidal-like” categories of species. (Strictly speaking, the two products on species, which capture the two ways of defining cyclic operads, are not monoidal, as they are not associative, but the induced structures arise in the same way as the one reflecting a specification of a monoid in a monoidal category. In particular, they are both subject to isomorphisms which fix the lack of associativity.) Relying on a result of Lamarche on descent for species, we use these “monoid-like” definitions to prove the equivalence between the exchangeable-output and entries-only points of view on cyclic operads. Finally, we establish a notion of categorified cyclic operad for set-based cyclic operads with symmetries, defined in terms of generators and relations. The categorifications we introduce are obtained by replacing sets of operations of the same arity with categories, by relaxing certain defining axioms, like associativity and commutativity, to isomorphisms, while leaving the equivariance strict, and by formulating coherence conditions for these isomorphisms. The coherence theorem that we prove has the form “all diagrams of canonical isomorphisms commute”. For entries-only categorified cyclic operads, our proof is of syntactic nature and relies on the coherence of categorified operads established by Dosen and Petric. We prove the coherence of exchangeable-output categorified cyclic operads by “lifting to the categorified setting” the equivalence between entries-only and exchangeable-output cyclic operads, set up previously in the algebraic approach.
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Jovana Obradovic. Cyclic operads: syntactic, algebraic and categorified aspects. Category Theory [math.CT]. Université Paris Diderot - Paris 7 - Sorbonne Paris Cité, 2017. English. ⟨tel-01676983⟩

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