Higher Categorical Structures as Universal Fixed Points

Abstract

Let Cat_(oo,oo) denote the (oo,1)-category of (oo,oo)-categories with weakly inductive equivalences. The main objective of this thesis is to demonstrate that Cat_(oo,oo) satisfies universal properties with respect to homotopy-coherent internalisation and enrichment. To achieve these universal properties, we extend the theory of endofunctor algebras to the (oo,1)-categorical setting, and establish an analogue of Adámek’s free algebra construction. For any oo-topos X, we define an (oo,1)-category Sh_(n,r)(X) of sheaves of (n,r)-categories over X, where 0 <= n <= oo, and 0 <= r <= n+2, and relate these categories through a general construction of complete Segal space objects over X, and observe that presheaves of (n,r)-categories admit a well-defined notion of sheafification. By realising the construction of complete Segal space objects as an endofunctor over an appropriately-defined (oo,1)-category of distributors, we use our generalised theory of endofunctor algebras to prove that Sh_(oo,oo)(X) is the universal distributor that is invariant under the construction of complete Segal space objects. We then study the theory of (oo,1)-categorical enrichment and analyse the continuity of this construction to prove similarly that Cat_(oo,oo) is the initial object among presentably symmetric monoidal (oo,1)-categories that are invariant under enrichment.