On the theory of higher Segal spaces

Abstract

This thesis contains three chapters, each dealing with one particular aspect of the theory of higher Segal spaces introduced by Dyckerhoff and Kapranov: <br /> (1) By exhibiting the simplex category as an ∞-categorical localization of the dendrex category of Moerdijk and Weiss, we identify the homotopy theory of 2-Segal spaces with that of invertible ∞-operads. <br /> (2) Inspired by a heuristic analogy with the manifold calculus of Goodwillie and Weiss, we characterize the various higher Segal conditions in terms of purely categorical conditions of higher weak excision on the simplex category and on Connes’ cyclic category. <br /> (3) We establish a large class of ∞-categorical Morita-equivalences of Dold–Kan type. As an application we describe higher Segal simplicial objects in the additive context as truncated coherent chain complexes; in the stable context, we identify higher Segal Γ-objects with polynomial functors in the sense of Goodwillie.