Higher structures in homotopy theory
Monday 2nd July 2018 to Friday 6th 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 modelindependent theory of (∞,1)categories (1)
Coauthor: Dominic Verity (Macquarie University) In these talks we use the nickname "∞category" to refer to either a quasicategory, a complete Segal space, a Segal category, or 1complicial set (aka a naturally marked quasicategory)  these terms referring to Quillen equivalent models of (∞,1)categories, that is, weak infinitedimensional 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 2category like that of (strict 1)categories, functors, and natural transformations. In the first talk, we'll use standard 2categorical 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 2category of ∞categories, ∞functors, and ∞natural transformations. In the 2category of quasicategories 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 2categories. 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 onesided and twosided variants. In the fourth talk, we'll prove that theory being developed isn’t just "modelindependent” (in the sense of applying equally to the four models mentioned above) but invariant under changeofmodel functors. As we explain, it follows that even the "analyticallyproven" theorems about quasicategories 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 Ktheory (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 
10:00 to 11:00 
Emily Riehl (Johns Hopkins University) The modelindependent theory of (∞,1)categories (2)
Coauthor: Dominic Verity (Macquarie University) In these talks we use the nickname "∞category" to refer to either a quasicategory, a complete Segal space, a Segal category, or 1complicial set (aka a naturally marked quasicategory)  these terms referring to Quillen equivalent models of (∞,1)categories, that is, weak infinitedimensional 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 2category like that of (strict 1)categories, functors, and natural transformations. In the first talk, we'll use standard 2categorical 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 2category of ∞categories, ∞functors, and ∞natural transformations. In the 2category of quasicategories 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 2categories. 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 onesided and twosided variants. In the fourth talk, we'll prove that theory being developed isn’t just "modelindependent” (in the sense of applying equally to the four models mentioned above) but invariant under changeofmodel functors. As we explain, it follows that even the "analyticallyproven" theorems about quasicategories 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 Ktheory (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 
10:00 to 11:00 
Emily Riehl (Johns Hopkins University) The modelindependent theory of (∞,1)categories (3)
Coauthor: Dominic Verity (Macquarie University) In these talks we use the nickname "∞category" to refer to either a quasicategory, a complete Segal space, a Segal category, or 1complicial set (aka a naturally marked quasicategory)  these terms referring to Quillen equivalent models of (∞,1)categories, that is, weak infinitedimensional 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 2category like that of (strict 1)categories, functors, and natural transformations. In the first talk, we'll use standard 2categorical 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 2category of ∞categories, ∞functors, and ∞natural transformations. In the 2category of quasicategories 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 2categories. 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 onesided and twosided variants. In the fourth talk, we'll prove that theory being developed isn’t just "modelindependent” (in the sense of applying equally to the four models mentioned above) but invariant under changeofmodel functors. As we explain, it follows that even the "analyticallyproven" theorems about quasicategories 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 Ktheory (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 
10:00 to 11:00 
Emily Riehl (Johns Hopkins University) The modelindependent theory of (∞,1)categories (4)
Coauthor: Dominic Verity (Macquarie University) In these talks we use the nickname "∞category" to refer to either a quasicategory, a complete Segal space, a Segal category, or 1complicial set (aka a naturally marked quasicategory)  these terms referring to Quillen equivalent models of (∞,1)categories, that is, weak infinitedimensional 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 2category like that of (strict 1)categories, functors, and natural transformations. In the first talk, we'll use standard 2categorical 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 2category of ∞categories, ∞functors, and ∞natural transformations. In the 2category of quasicategories 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 2categories. 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 onesided and twosided variants. In the fourth talk, we'll prove that theory being developed isn’t just "modelindependent” (in the sense of applying equally to the four models mentioned above) but invariant under changeofmodel functors. As we explain, it follows that even the "analyticallyproven" theorems about quasicategories 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 Ktheory (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 SchommerPries (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 
10:00 to 11:00 
Claudia Scheimbauer (MaxPlanckInstitut für Mathematik, Bonn) tba 
INI 1 
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Markus Spitzweck (Universität Osnabrück) Hermitian Ktheory 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 Ktheory 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 