[HoTT] [CMU-HoTT] Special series of lectures — Cisinski, Nguyen, Walde on *Univalent Directed Type Theory*

[mathieu anel <mathieu.anel@gmail.com>]

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~*~*~*~*~*~*~*~* CMU HoTT Seminar Online *~*~*~*~*~*~*~*~


Don't miss lecture 3 !


Denis-Charles Cisinski (Universität Regensburg)

Hoang Kim Nguyen (Universität Regensburg)

Tashi Walde (Technische Universität München)


on *** Univalent Directed Type Theory ***

(abstract below)


Lecture 1 — video https://www.youtube.com/watch?v=5YOltuTcBK8

Slides intro
https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf

Slides talk
https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf


Lecture 2 — video https://www.youtube.com/watch?v=xWmELBvHMPo

Slides
https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf


*Next lecture*

*Lecture 3 — Monday March 27 (10am-12pm EDT / 4pm-6pm CEST)*



Abstract:


We will introduce a version of dependent type theory that is suitable to
develop a synthetic theory of 1‑categories. The axioms are both a fragment
and an extension of ordinary dependent type theory. The axioms are chosen
so that (∞,1)‑category theory (under the form of quasi-categories or
complete Segal spaces) gives a semantic interpretation, in a way which
extends Voevodsky's interpretation of univalent dependent type theory in
the homotopy theory of Kan complexes. More generally, using a slight
generalization of Shulman's methods, we should be able to see that the
theory of (∞,1)‑categories internally in any ∞‑topos (as developed by
Martini and Wolf) is a semantic interpretation as well (hence so is
parametrized higher category theory introduced by Barwick, Dotto, Glasman,
Nardin and Shah). There are of course strong links with ∞‑cosmoi of Riehl
and Verity as well as with cubical Hott (as strongly suggested by the work
of Licata and Weaver), or simplicial Hott (as in the work of Buchholtz and
Weinberger). We will explain the axioms in detail and have a glimpse at
basic theorems and constructions in this context (Yoneda Lemma, Kan
extensions, Localizations). We will also discuss the perspective of
reflexivity: since the theory speaks of itself (through directed
univalence), we can use it to justify new deduction rules that express the
idea of working up to equivalence natively (e.g. we can produce a logic by
rectifying the idea of having a locally cartesian type). In particular,
this logic can be used to produce and study semantic interpretations of
Hott.



Link for the Zoom Meeting

https://cmu.zoom.us/j/622894049?pwd=bkhtRlNxL3E3SnZCTU1oSFNHcHJNQT09

Meeting ID: 622 894 049

Passcode: ‘the Brunerie number’


Seminar webpage

https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313

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