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, these being 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" form. In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞categories. We also introduce co/cartesian fibrations in both onesided and twosided variants, the latter of which are used to define "modules" between ∞categories, of which comma ∞categories are the prototypical example. In the fourth talk, we'll prove that theory being developed isn’t just "modelagnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent 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) Isovariant homotopy theory
An isovariant map between Gspaces is an equivariant map which preserves isotropy groups. Isovariant homotopy theory appears in situations where homotopy is applied to geometric problems, for example, in surgery theory. We will describe some new results in isovariant homotopy theory, including two model structures and an application to intersection theory. This is work in progress, based on conversations with Cary Malkiewich, Mona Merling, and Kate Ponto.

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, these being 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" form. In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞categories. We also introduce co/cartesian fibrations in both onesided and twosided variants, the latter of which are used to define "modules" between ∞categories, of which comma ∞categories are the prototypical example. In the fourth talk, we'll prove that theory being developed isn’t just "modelagnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent 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) Thirteen ways of looking at an equivariant stable category
Motivated both by new higher categorical foundations as well as the technology emerging from HillHopkinsRavenel, there has been a lot of recent work by various authors on formal characterizations of the equivariant stable category. In this talk, I will give an overview of this story, with focus on the perspective coming from the framework of Ninfinity operads. In particular, I will describe incomplete stable categories of "Ospectra" associated to any Ninfinity operad O as well as some indications of what we can say in the case when G is an infinite compact Lie group.

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, these being 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" form. In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞categories. We also introduce co/cartesian fibrations in both onesided and twosided variants, the latter of which are used to define "modules" between ∞categories, of which comma ∞categories are the prototypical example. In the fourth talk, we'll prove that theory being developed isn’t just "modelagnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent 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, these being 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" form. In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞categories. We also introduce co/cartesian fibrations in both onesided and twosided variants, the latter of which are used to define "modules" between ∞categories, of which comma ∞categories are the prototypical example. In the fourth talk, we'll prove that theory being developed isn’t just "modelagnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent 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)) Higher categories of higher categories
I will describe a construction of a higher category of enriched (infinity,n)categories, and attempt to convince the audience that this is interesting.

INI 1  
14:30 to 15:30 
Christopher SchommerPries (University of Notre Dame) The Relative Tangle Hypothesis
I will describe recent progress on a nonlocal variant of the cobordism hypothesis for higher categories of bordisms embedded into finite Euclidean spaces.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Kathryn Hess (EPFL  Ecole Polytechnique Fédérale de Lausanne) A Künneth theorem for configuration spaces of products
Bill Dwyer, Ben Knudsen, and I recently constructed a model for the configuration space of a product of parallelizable manifolds in terms of the derived BoardmanVogt tensor product of right modules over the operads of little cubes of the appropriate dimensions. In this talk, after recalling our earlier work, I will introduce a spectral sequence Ben and I have developed for computing the homology of this model.

INI 1 
10:00 to 11:00 
Claudia Scheimbauer (MaxPlanckInstitut für Mathematik, Bonn) Dualizability in the higher Morita category
In this talk I will explain how one can use geometric arguments to obtain results on dualizablity in a ``factorization version’’ of the Morita category. We will relate our results to previous dualizability results (by DouglasSchommerPriesSnyder and BrochierJordanSnyder on TuraevViro and ReshetikinTuraev theories). We will discuss applications of these dualiability results: one is to construct examples of lowdimensional field theories “relative” to their observables. An example will be given by Azumaya algebras, for example polynomial differential operators (Weyl algebra) in positive characteristic and its center. (This is joint work with Owen Gwilliam.)

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 