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# Timetable (HHHW01)

## 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 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, these being 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" form.In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided 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 "model-agnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent models.Related Linkshttp://www.math.jhu.edu/~eriehl/scratch.pdf - lecture notes from a similar series of four talks delivered at EPFLhttp://www.math.jhu.edu/~eriehl/elements.pdf - book in progress on the subject of these lectures 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)Isovariant homotopy theory An isovariant map between G-spaces 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 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, these being 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" form.In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided 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 "model-agnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent models.Related Linkshttp://www.math.jhu.edu/~eriehl/scratch.pdf - lecture notes from a similar series of four talks delivered at EPFLhttp://www.math.jhu.edu/~eriehl/elements.pdf - book in progress on the subject of these lectures 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)Thirteen ways of looking at an equivariant stable category Motivated both by new higher categorical foundations as well as the technology emerging from Hill-Hopkins-Ravenel, 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 N-infinity operads. In particular, I will describe incomplete stable categories of "O-spectra" associated to any N-infinity 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 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, these being 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" form.In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided 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 "model-agnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent models.Related Linkshttp://www.math.jhu.edu/~eriehl/scratch.pdf - lecture notes from a similar series of four talks delivered at EPFLhttp://www.math.jhu.edu/~eriehl/elements.pdf - book in progress on the subject of these lectures 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
 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, these being 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" form.In the second talk, we'll justify the framework introduced in the first talk 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 encode the universal properties of adjunction and of limits and colimits as equivalences of comma ∞-categories. We also introduce co/cartesian fibrations in both one-sided and two-sided 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 "model-agnostic” (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 that exploit the combinatorics of one particular model remain valid in the other biequivalent models.Related Linkshttp://www.math.jhu.edu/~eriehl/scratch.pdf - lecture notes from a similar series of four talks delivered at EPFLhttp://www.math.jhu.edu/~eriehl/elements.pdf - book in progress on the subject of these lectures 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))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 Schommer-Pries (University of Notre Dame)The Relative Tangle Hypothesis I will describe recent progress on a non-local 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 Boardman-Vogt 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 (Max-Planck-Institut 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 Douglas-Schommer-Pries-Snyder and Brochier-Jordan-Snyder on Turaev-Viro and Reshetikin-Turaev theories). We will discuss applications of these dualiability results: one is to construct examples of low-dimensional 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 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