Virtual Workshop on Higher Topos Theory, November 20/21-2020

Finster:	I Introduction to Homotopy Type Theory
	In this talk, I will describe how in the past decade, homotopy theory has come to play an important role in our understanding of constructive mathematics, particularly as implemented by Martin-Löf's theory of dependent types. These ideas have lead to surprising connections between such seemingly unrelated areas as computer-assisted mathematics and oo-topos theory, and prompted Vladimir Voevodsky to suggest that type theory could serve as a new foundation for mathematics based entirely on homotopy theory. I will describe some of the basic intuitions of this program and attempt to give a flavor for how mathematics is carried out in this style.
	II Modalities and the Generalized Blakers-Massey Theorem
	Building on the ideas introduced in the first lecture about the connection between homotopy theory and logic, I will describe the notion of a modality, a concept which has its origins in modal logic, but turns out to have been discovered quite independently (though clearly not in the same language) by homotopy theorists thinking about the theory of localization. Furthermore, I will describe how modalities allow us to give a refined statement of a classical result of algebraic topology, the Blakers-Massey connectivity theorem.
	A generalized Blakers-Massey Theorem arXiv version
Forero:	Introduction into Decomposition Spaces I/II
	Decomposition spaces are simplicial infinity-groupoids subject to certain exactness conditions, needed to induce a coalgebra structure on the space of arrows. In these talks, we briefly explain the motivation and some basic notions, and we finish with a conjecture about a universal property of a certain decomposition space of intervals.
Joyal:	Introduction to Higher Topos Theory I: Grothendieck topos theory
	In this first talk, I will introduce the notion of (Grothendieck) topos with examples. Topoi (or toposes) were introduced in algebraic geometry by Grothendieck. l will use the term "logos" instead of "topos" for reasons that will be become clear at the end. The category Set of sets is a logos. If E is a logos, then so is the category E^C for any small category C, and the slice category E/B for any object B of E. A logos has colimits and finite limits. The colimit operations in a logos are like the addition operations in a commutative ring; the finite limit operations are like the multiplication operations. The operations are related by distributive laws. A homomorphism of logoi is a functor which preserves colimits and finite limits. The category Set is like the ring of integers. There are a logos of polynomial functors Set[X] in one variable object X and a logos Set[A] freely generated by a ring object A, or any kind of algebraic object. The quotients of a logos E are the left exact localisations of E. The category of topoi is the opposite​ of the category of logoi. Here are the slides of the first talk.
	Introduction to Higher Topos Theory II: Higher categories and higher topos theory
	In this second talk, I will discuss some basic aspects of the theory of ∞-categories and of ∞-topoi. A basic example of ∞-category is the category S of (topological) spaces. Another example is the category of Kan complexes, or any category enriched over Kan complexes. A third example is a quasi-category. The nerve of a category is a quasi-category. Category theory can be wholly extended to quasi-categories. In a quasi-category one can define the following notions: an inital object, a diagram in a quasi-category, a limit of a diagram, a pullback square, and a pushout square. Higher topoi were introduced by Charles Rezk. They have applications in homotopy theory and derived algebraic geometry (Bertrand Töen and Gabrielle Vezzosi). Higher topos theory was developed systematically by Jacob Lurie. The ∞-category of spaces S is an ∞-logos. If E is an ∞-logos, then so is the ∞-category E^C for any small category C, and the slice ∞-category E/B for any object B of E. An ∞-logos has colimits and finite limits. The relations between these operations are expressed by the Rezk​ descent principle​ which can be thought of distributive laws. A homomorphism of ∞-logoi is a functor preserving colimits and finite limits. There is an ∞-logos of polynomial functors S[X] in one variable object X. The quotients of an ∞-logos E are the left exact localisations of E. It was discovered (independently) by Georg Biedermann and Charles Rezk that the category of finitary n-excisives functors S--->S introduced by Goodwillie is a logos S[X(n)]. The category of ∞-topoi is the opposite​ of the category of ∞-logoi.

Registration:


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