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A proof of the model-independence of $\infty$-category theory

Banff International Research Station for Mathematical Innovation and Discovery

In joint work with Dominic Verity we prove that four models of (â ,1)-categories â quasi-categories, complete Segal spaces, Segal categories, and 1-complicial sets â are equivalent for the purpose of developing â -category theory. To prove this we first introduce the notion of an â -cosmos, a category in which â -categories live as objects, an example of which is given by each of the four models mentioned above. We then explain how the category theory of â -categories can be developed inside any â -cosmos; eg, we define right adjoints and limits and prove that the former preserve the latter. We conclude by arguing that the four above mentioned â -cosmoi all biequivalent, the upshot being that â -categorical structures are preserved, reflected, and created by a number of â change-of-modelâ functors. More precisely, we show that each of these â -cosmoi have a biequivalent â calculus of modules,â modules between â -categories being a vehicle to express â -categorical universal properties.

Mathematics; Algebraic topology; Category theory; homological algebra

video/mp4

Author affiliation: Johns Hopkins University

2019-03-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69196

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/