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The model-independent theory of (∞,1)-categories (3)

Presented by: 
Emily Riehl Johns Hopkins University
Wednesday 4th July 2018 - 10:00 to 11:00
INI Seminar Room 1
Co-author: Dominic Verity (Macquarie University)

In these talks we use the nickname "∞-category" to refer to either a quasi-category, a complete Segal space, a Segal category, or 1-complicial set (aka a naturally marked quasi-category) - these terms referring to Quillen equivalent models of (∞,1)-categories, that is, weak infinite-dimensional categories with all morphisms above dimension 1 weakly invertible. Each of these models has accompanying notions of ∞-functor, and ∞-natural transformation and these assemble into a strict 2-category like that of (strict 1-)categories, functors, and natural transformations.

In the first talk, we'll use standard 2-categorical techniques to define adjunctions and equivalences between ∞-categories and limits and colimits inside an ∞-category and prove that these notions relate in the expected ways: eg that right adjoints preserve limits. All of this is done in the aforementioned 2-category of ∞-categories, ∞-functors, and ∞-natural transformations. In the 2-category of quasi-categories our definitions recover the standard ones of Joyal/Lurie though they are given here in a "synthetic" rather than their usual "analytic" notion.

In the second talk, we'll justify the framework introduced in the first by giving an explicit construction of these 2-categories. This makes use of an axiomatization of the properties common to the Joyal, Rezk, Bergner/Pellissier, and Verity/Lurie model structures as something we call an ∞-cosmos.

In the third talk, we'll present a framework for expressing the universal properties of adjunctions, limits and colimits, and also introduce co/cartesian fibrations in both one-sided and two-sided variants.

In the fourth talk, we'll prove that theory being developed isn’t just "model-independent” (in the sense of applying equally to the four models mentioned above) but invariant under change-of-model functors. As we explain, it follows that even the "analytically-proven" theorems about quasi-categories remain valid in the three other models.

Related Links
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons