We watch the first of four talks given by Emily Riehl at the Issac Newton Institute on the model-independent theory of (∞,1)-categories.
Abstract: 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, these being 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" form.
The video for this talk is available here: https://www.newton.ac.uk/event/hhhw01/timetable