University of Cambridge > > Isaac Newton Institute Seminar Series > The model-independent theory of (∞,1)-categories (3)

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

Add to your list(s) Download to your calendar using vCal

If you have a question about this talk, please contact INI IT.

HHHW01 - Higher structures in homotopy theory

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, 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.

In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided variants, the latter of which are used to define “modules” between ∞-categories, of which comma ∞-categories are the prototypical example.

In the fourth talk, we'll prove that theory being developed isn’t just “model-agnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent models.

Related Links

This talk is part of the Isaac Newton Institute Seminar Series series.

Tell a friend about this talk:

This talk is included in these lists:

Note that ex-directory lists are not shown.


© 2006-2021, University of Cambridge. Contact Us | Help and Documentation | Privacy and Publicity