- 17 views
- 23 downloads
Higher Categorical Structures as Universal Fixed Points
-
- Author / Creator
- Goldthorpe, Zachariah PL
-
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.
-
- Subjects / Keywords
-
- Graduation date
- Fall 2024
-
- Type of Item
- Thesis
-
- Degree
- Doctor of Philosophy
-
- License
- This thesis is made available by the University of Alberta Library with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.