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Timetable (HHHW01)

Higher structures in homotopy theory

Monday 2nd July 2018 to Friday 6th July 2018

Monday 2nd July 2018
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (INI Director)
10:00 to 11:00 Emily Riehl (Johns Hopkins University)
The model-independent theory of (∞,1)-categories (1)
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, that is, 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" notion.

In the second talk, we'll justify the framework introduced in the first 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 present a framework for expressing the universal properties of adjunctions, limits and colimits, and also introduce co/cartesian fibrations in both one-sided and two-sided variants.

In the fourth talk, we'll prove that theory being developed isn’t just "model-independent” (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 about quasi-categories remain valid in the three other models.

Related Links
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Thomas Nikolaus (Universität Münster)
Higher categories and algebraic K-theory (1)
INI 1
12:30 to 13:30 Buffet Lunch at INI
13:30 to 14:30 Tobias Dyckerhoff (Universität Bonn)
Higher Segal spaces (1)
INI 1
14:30 to 15:30 John Francis (Northwestern University)
Factorization homology (1)
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Sarah Yeakel (University of Maryland, College Park)
tba
INI 1
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 3rd July 2018
10:00 to 11:00 Emily Riehl (Johns Hopkins University)
The model-independent theory of (∞,1)-categories (2)
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, that is, 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" notion.

In the second talk, we'll justify the framework introduced in the first 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 present a framework for expressing the universal properties of adjunctions, limits and colimits, and also introduce co/cartesian fibrations in both one-sided and two-sided variants.

In the fourth talk, we'll prove that theory being developed isn’t just "model-independent” (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 about quasi-categories remain valid in the three other models.

Related Links
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Thomas Nikolaus (Universität Münster)
Higher categories and algebraic K-theory (2)
INI 1
12:30 to 13:30 Buffet Lunch at INI
13:30 to 14:30 Tobias Dyckerhoff (Universität Bonn)
Higher Segal spaces (2)
INI 1
14:30 to 15:30 John Francis (Northwestern University)
Factorization homology (2)
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Andrew Blumberg (University of Texas at Austin)
tba
INI 1
Wednesday 4th July 2018
10:00 to 11:00 Emily Riehl (Johns Hopkins University)
The model-independent theory of (∞,1)-categories (3)
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, that is, 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" notion.

In the second talk, we'll justify the framework introduced in the first 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 present a framework for expressing the universal properties of adjunctions, limits and colimits, and also introduce co/cartesian fibrations in both one-sided and two-sided variants.

In the fourth talk, we'll prove that theory being developed isn’t just "model-independent” (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 about quasi-categories remain valid in the three other models.

Related Links
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Thomas Nikolaus (Universität Münster)
Higher categories and algebraic K-theory (3)
INI 1
12:30 to 13:30 Buffet Lunch at INI
13:30 to 17:00 Free Afternoon
19:30 to 22:00 Formal Dinner at Emmanuel College
Thursday 5th July 2018
10:00 to 11:00 Emily Riehl (Johns Hopkins University)
The model-independent theory of (∞,1)-categories (4)
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, that is, 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" notion.

In the second talk, we'll justify the framework introduced in the first 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 present a framework for expressing the universal properties of adjunctions, limits and colimits, and also introduce co/cartesian fibrations in both one-sided and two-sided variants.

In the fourth talk, we'll prove that theory being developed isn’t just "model-independent” (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 about quasi-categories remain valid in the three other models.

Related Links
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Thomas Nikolaus (Universität Münster)
Higher categories and algebraic K-theory (4)
INI 1
12:30 to 13:30 Buffet Lunch at INI
13:30 to 14:30 Rune Haugseng (Københavns Universitet (University of Copenhagen))
tba
INI 1
14:30 to 15:30 Christopher Schommer-Pries (University of Notre Dame)
tba
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Kathryn Hess (EPFL - Ecole Polytechnique Fédérale de Lausanne)
Configuration spaces of fiber bundles
INI 1
Friday 6th July 2018
10:00 to 11:00 Claudia Scheimbauer (Max-Planck-Institut für Mathematik, Bonn)
tba
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Markus Spitzweck (Universität Osnabrück)
Hermitian K-theory for Waldhausen infinity categories with genuine duality
In the talk we will introduce the concept of an infinity category with genuine duality, a refinement of the notion of a duality on an infinity category. We will then define the infinity category of Waldhausen infinity categories with genuine duality and study the hermitian/real K-theory functor on this category. In particular we will state and indicate a proof of the Additivity theorem in this context. This is joint work with Hadrian Heine and Paula Verdugo.
INI 1
12:30 to 13:30 Buffet Lunch at INI
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons