This is the homepage of Bilkent Topology seminar (Spring 2023).
This term’s seminar will be a reading seminar on higher categories co-organized by Igor Sikora and Aziz Kharoof.
Feb 6: An introduction to infinity categories (David Blanc, University of Haifa)
In studying the homotopy theory of topological spaces it soon becomes apparent that the homotopy category itself is not sufficient, since many homotopy invariants cannot be described or calculated in that category. Since there are other settings, such as the chain complexes of homological algebra, in which this holds, Quillen proposed an axiomatization of such situations in terms of model categories. However, these turn out to be too restrictive for dealing with certain questions, in particular with homotopy commutative diagrams and the invariants (such as Toda brackets) which they encode. Dwyer and Kan suggested an alternative simplicial approach, which later devolved into several independent models for what we now call infinity categories, in terms of simplicially enriched categories, simplicial spaces, quasi-categories, and others. In the talk, we will provide examples of questions best addressed in this setting, and briefly describe the form they take in the different models, as time permits.
Feb 13: Simplicial sets (Aziz)
This talk aims to introduce and recall basic notions on simplicial sets. Apart from basic definitions, we would like to discuss the following notions: weak equivalences, Kan complexes, Kan fibrations, and geometric realization. Also, the adjunction between singular simplicial set and geometric realization should be covered.
References: [H]: Section 1, [W]: Chapter 8, [F]
Feb 20: Quasicategories (Aziz)
In this talk we will introduce the first model of infinity categories, namely quasicategories. We will discuss the motivation of this definition as a generalization of the nerve of a category. Then we will define basic categorical notions in the quasi category, and the homotopy relation there, showing how we can think about morphisms and composition up to homotopy. In addition, we will show how the hom-object from object x to object y contains all the higher order morphisms from x to y and the relation between them. Later we will talk about functors and natural maps between quasicategories.
References: [H]: Section 2, [G]: Section 1.1
Feb 27: Basic constructions in quasicategories (Igor)
The goal of this talk is the discussion of the basic notions and constructions in the theory of infinity categories. We want to discuss the following constructions: the product of quasicategries, homotopy category of a quasicategory, join, slices and, most importantly, colimits and limits.
Ref: [G]: Section 2, [H]: Section 2.3
Mar 13: Model categories I - basic definitions (Mustafa)
The goal of this talk is to provide basic definitions of the theory model categories. We would like to introduce the definition of a model category and its homotopy category. In particular, this will require a discussion of fibrations, cofibrations and weak equivalences, fibrant and cofibrant objects, cylinder and path objects. Then we will proceed to the notion of left and right homotopy and define the homotopy category of a model category. The whole theory will be shown using two examples: Quillen model structure on topological spaces and Quillen model structure on simplicial sets.
Ref: [DS], [H]: Section 3
Mar 20: Model Categories II - Derived functors and Quillen adjunctions (Igor)
Having the notion of a homotopy category, we will define the notion of a derived functor. Further on, we will proceed to the idea of comparing model structures and their homotopy categories by Quillen functors. Therefore we will cover Quillen functors, Quillen adjunctions and Quillen equivalences. We will also prove that Quillen model structures on simplicial sets and topological spaces are Quillen equivalent. The talk will finish with a model structure on simplicial sets which is relevant for the theory of quasicategories, i.e., the Joyal model structure.
Ref: [DS], [H]: Section 4, 5.3 for Joyal model structure, [BS]: Section 2 for inner fibrations
Mar 27: Simplicial categories I (Igor)
In this talk, we will discuss the second model of infinity categories: categories enriched over simplicial sets. We will start with a short overview of enriched categories and follow to the simplicial categories. We will also introduce simplicial functors and the homotopy category of a simplicial category. Then we will proceed with the Bergner model structure and sketch the proof of the fact that it is indeed a model structure.
Ref: [H]:Sections 5.1,5.2, [G]: Section 1.2
Apr 3: Simplicial categories II - Dwyer-Kan localizations(Aziz)
he goal of this talk will be to understand the idea of localization of a category with respect to the class of maps and see how Dwyer-Kan localization is an example of such. Therefore we will start with the notion of a localization of a category. Then we will proceed to several approaches to the Dwyer-Kan localization - as a derived functor with specific resolution and the hammock version, that gives a constructive description of the homotopy category. We will discuss the relation of DK localization of a simplicial model category and of its homotopy category.
Ref: [H]: Section 5.4
Apr 10: Segal spaces I (Özgün)
This talk will prepare a background for the third model of infinity categories: complete Segal spaces. Therefore the following topics should be discussed: bisimplicial sets, model structures on functor categories, Reedy model structure as an example of the injective model structure and Rezk nerve of a category.
Ref: [H]: Sections 6.1-6.5, [B]: Sections 4,5
Apr 17: Relizability of fusion systems (Bob Oliver, Université Paris 13)
For the abstract, click here
Note: This talk is not a part of the reading seminar.
Apr 24: Segal spaces II (Igor)
In this talk, we will continue introducing the third model of infinity categories: complete Segal spaces. The following notions will be covered: Segal spaces, homotopy category of Segal spaces, completeness of Segal spaces and CSS model structure.
Ref: [H]: Sections 6.6-6.9, [B]: Sections 4,5
May 2 (Tue)^{1}: The Homotopy Coherent Nerve (Redi)
The aim of this talk is to introduce the homotopy coherent nerve of a simplicially enriched category and discuss some related constructions. To the best of our capabilities, we will try to focus on the heuristic aspects of the bits of theory we present. The coherent nerve is part of an explanatory cycle which tells us that the simplicial model and the quasi-category model for (oo, 1)-categories are equivalent. We begin by informally discussing the central notion of homotopy coherence in a simplicial category. This way, the coherent nerve can be understood to be comprised of coherent simplices. The formal definition can be formulated via a Yoneda extension of a cosimplicial object in simplicially enriched categories. It follows from the construction that this nerve construction has an adjoint, which is not so easy to understand at first glance. The latter functor can be interpreted as a free (simplicially enriched) categorification and understood via the necklace construction of Dugger and Spivak. Simplicial categorification is of interest: for example, when applied to the nerve of a category it produces a cofibrant replacement in the Bergner model structure which allows us to have a precise definition of homotopy coherence.
Ref: [H]: Section 6.10, [B]: Section 7
May 16, 16:00: Complete Segal spaces and generalizations to higher $(\infty,n)$-categories (Julie Bergner, University of Virginia)
Complete Segal spaces provide one of the nicest models for $(\infty,1)$-categories from the perspective of homotopy theory, since the model structure can be obtained as a localization of the Reedy model structure on simplicial spaces. In this talk, we’ll recall complete Segal spaces and their model structure, and then compare them with other models. We will then look at some of the ways these comparisons can be generalized higher $(\infty,n)$-categories.
[B] J. Bergner, A survey of (infty,1) Categories
[BS] C. Barwick, J. Shah, Fibrations in infty-Category Theory
[DS] W. G. Dwyer and J. Spalinski, Homotopy theories and model categories
[F] G. Friedman, An elementary illustrated introduction to simplicial sets
[G] M. Groth, A Short Course on Infinity Categories
[H] V. Hinich, Lectures on Infinity categories
[W] C. Weibel, An introduction to homological algebra
Charles Rezk, Introduction to quasicategories
Kerodon website
D. Dugger, A Primer on Homotopy Colimits
May 1 is holiday. ↩