We will continue online on Thursday, September 24, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:
Raffael Stenzel
Infinitycategorical comprehension schemes
Abstract: Comprehension schemes arose as crucial notions in the early work on the foundations of set theory, and hence they found expression in a considerable variety of foundational settings for mathematics. Particularly, they have been introduced to the context of categorical logic first by Lawvere and then by Benabou in the 1970s. In this talk we define and study a theory of comprehension schemes for fibered infinitycategories, generalizing Johnstone's respective notion for ordinary categories. This includes natural generalizations of all the fundamental instances originally defined by Benabou, and their application to Jacob's comprehension categories. Thereby, we can characterize  numerous categorical structures arising in higher topos theory  the notion of univalence  internal infinitycategories in terms of comprehension schemes, while some of the 1categorical counterparts fail to hold in ordinary category theory. As an application, we can show that the universal cartesian fibration is represented via externalization by the "freely walking chain" in the infinitycategory of small infinitycategories. In the end, if my time management permits, we take a look at the externalization construction of internal infinitycategories from a model categorical perspective and review some examples from the literature in this light.
