Cyclic operads, dendroidal structures, higher categories

Abstract

The thesis consists of three parts, each of these parts contributes to the theory of operads with some new results. In the first part we study the homotopy theory of cyclic operads by employing methods developed by Berger and Moerdijk for the study of the homotopy theory of operads. We prove that under some mild assumptions the category of cyclic operads over a closed symmetric monoidal model category admits a model structure. Moreover, when these assumptions are met, we construct a cofibrant resolution for cyclic operads that extends the classical Boardman-Vogt resolution of operads to the cyclic case. In the second part of the thesis we prove a Dold-Kan type correspondence theorem between the suitably constructed category of dendroidal chain complexes and the category of dendroidal abelian groups. The result generalizes the classical Dold-Kan correspondence theorem between simplicial abelian groups and chain complexes to their respective analogs in the theory of operads. In the third part of the work we focus on comparing dendroidal notions of weak higher categories with the corresponding classical ones. We prove that for the purposes of homotopy theory, bicategories and tricategories are equivalent to their dendroidal analogs.