(∞,n)-Categories and their applications

Abstracts of the invited talks

Fernando Abellán, Enhanced Segal objects and the Gray tensor product of (∞,2)-categories.
The notion of a lax unital functor is one of the most fundamental concepts in the study of (∞,n)-categories which is native to the case n≥2. The importance of this concept can be justified by the fact that relevant constructions in category theory, (such as the Gray tensor product, lax monoidal functors, span categories,etc) can be characterized by universal properties which are specified in terms of lax unital functors. 

Even though it was known that the main models for the theory of (∞,2)-categories are equivalent, such comparisons were not capable of translating the notion of a lax unital functor across models. As a consequence, it was a priori not clear that constructions such as the Gray tensor product studied by Gaitsgory-Rozenblyum in the model of 2-fold complete Segal spaces (or complete Segal objects in (∞,1)-categories) agree with the version studied by Gagna-Harpaz-Lanari in the model of scaled simplicial sets. 

In this talk, I will address this problem by introducing the notion of an enhanced (complete) n-fold Segal object which is a 2-dimensional upgrade to the theory of n-fold (complete) Segal objects internal to an (∞,2)-category. When specialized to the (∞,2)-category of (∞,1)-categories functors and natural transformations, our construction will yield an (∞,2)-category which models the (∞,2)-category of (∞,n+1)-categories, lax unital functors and globular lax natural transformations. In the case n=1, this new technology will allow us to show that both versions of the Gray tensor product discussed above are equivalent.

Dimitri Ara, Gray-functorialities of the comma construction for strict ω-categories.
In this talk, I will investigate the functorialities of the comma construction in the world of strict ω-categories. We will see that these functorialities are captured by the language of Gray-ω-categories, that is, categories enriched in the category of strict ω-categories endowed with the Gray tensor product. To express these functorialities, we will need to define slices for Gray-ω-categories. As a particular case, we will get Gray-functorialities for the Grothendieck construction. (This talk is based on joint work with Léonard Guetta.)

Tim Campion, An (∞,∞)-categorical pasting theorem.
We identify a reasonably large class of pushouts of strict n-categories which are preserved by the "inclusion" functor into weak (∞,n)-categories. These include the pushouts used to assemble from generating cells the objects of Joyal's category Theta, the orientals, the Gray cubes, or more generally any parity complex (technically, torsion-free complex in the sense of Forest), including loop-free Steiner complexes. We give a few applications:

• Regarding a parity complex as a "pasting diagram", we deduce that such diagrams may be used interchangeably in either the weak or strict context. This generalizes work of Hackney, Ozornova, Riehl and Rovelli in dimension 2.

• There result explicit descriptions of a new family of models of (∞,∞)-categories as presheaves of spaces on parity complexes. These are related to models studied by Barwick and Schommer-Pries, and others studied by Henry.

• Combined with results of Ara and Lucas of a strict omega-categorical nature, we deduce the existence of a biclosed Gray monoidal structure on (∞,∞)-categories, and characterize it uniquely by the requirement that the inclusion of the strict Gray cubes be strong monoidal.

Tobias Dyckerhoff, Lax additivity, lax matrices, and mutation.
I will explain the concept of lax additivity which is an (∞,2)-categorical analog of the familiar 1-categorical notion of additivity. In this context, direct sums get replaced by lax sums leading to lax variants of matrices along with rules for how to multiply them. We will illustrate the resulting methods by investigating periodicitiy phenomena for mutations of semiorthogonal decompositions.

Based on joint work with Christ-Walde and Kapranov-Schechtman.

Dennis Gaitsgory, The character of Deligne-Lusztig representations via 2-categories.

David Gepner, Generators and relations for (∞,∞)-categories.
There are many full subcategories of the (∞,1)-category of (∞,∞)-categories which generate under colimits. We will discuss some of these, as well as some of the advantages or disadvantages inherent in working with one choice over another.

Rune Haugseng, Unfolding of symmetric monoidal (∞,n)-categories.
In Lurie's article on the Cobordism Hypothesis, he discusses a description of symmetric monoidal (∞,n)-categories with duals for objects and certain adjoints as "chain complexes" of symmetric monoidal ∞-categories with duals, but does not give a proof. I will discuss an approach to proving this based on a general description of closed symmetric monoidal V-enriched ∞-categories as lax symmetric monoidal functors to V. This is work in progress with Thomas Nikolaus.

Sil Linskens, Cocompleting parametrised higher categories.
In this talk, I will discuss the question of cocompleting parametrized higher categories. Because of the abundance of possible colimit diagrams, this is a richer and more subtle process than the non-parametrized case. As the main result I will explain how in many cases, one can derive a formula for the cocompletion in terms of partially lax limits, certain universal constructions in (∞,2)-category theory. Focusing in on an example, I will discuss the theory of categories parametrized over finite groupoids. In this case, the result tells us various interesting things about globally equivariant homotopy theory, which I will explain.

Félix Loubaton, The lax Grothendieck construction.
The Grothendieck construction is a fundamental operation in category theory. In this talk, I will give a generalization of this result for (∞,∞)-categories. To this end, I will present some important concepts in the theory of (∞,∞)-categories, such as semi-lax transformations and cartesian fibrations. 

Finally, I will explain how this result can be used to give explicit computations of lax Kan extensions.

Yuki Maehara, Equivalences in and between algebraic weak ω-categories.
There are many different approaches to weak higher-dimensional categories. One proposed by Leinster, based on an idea of Batanin's, defines weak ω-categories as the algebras for a particular monad on the category of globular sets. One of its advantages over the other, non-algebraic models is that, it is much more amenable to “take the argument from the strict case and replace every equality by an equivalence” kind of proof.

This talk consists of three parts. The first part is devoted to understanding Leinster’s definition as encoding (only) the existence part of the pasting theorem for globular pasting diagrams. The uniqueness of pasting turns out to be a consequence of the existence part if uniqueness is understood as that up to coinductive equivalence; these equivalences will be the main focus of the second part. In the final part, I will sketch how to actually “replace every equality in the strict case by a (coinductive) equivalence” and prove the 2-out-of-3 property for the class of weak equivalences between weak ω-categories.

This talk is based on joint work with Soichiro Fujii (Macquarie University) and Keisuke Hoshino (Kyoto University).

Aaron Mazel-Gee, An (∞,4)-category.
In this talk, I will introduce an (∞,4)-category of which I'm both fond and proud, constructed in recent joint work with Leon Liu, David Reutter, Catharina Stroppel, and Paul Wedrich. It arises as a braided monoidal (∞,2)-category, which can very roughly be thought of as that of "categorified representations of categorified quantum sl_∞". We intend for it and its cousins to play an analogous role for knot homologies (e.g. Khovanov homology) that ordinary braided monoidal categories of quantum group representations play for knot polynomials (e.g. the Jones polynomial), inspired by the longstanding open problem of generalizing knot homology theories from knots in R^3 to knots in arbitrary 3-manifolds.

Lyne Moser, (∞,n)-Limits III: Comparison across models.
This is the last talk of the 3-part talk series on a joint project with Nima Rasekh and Martina Rovelli. The goal of this talk is to compare the two different definitions of (∞,n)-limits presented in the first talk: the one for strictly enriched categories in (∞,n-1)-categories with the one for complete Segal objects in (∞,n-1)-categories. For this, we start by constructing a homotopy coherent categorification-nerve adjunction which yields a Quillen equivalence between these two models of (∞,n)-categories (replacing complete Segal objects with the homotopically equivalent notion of Segal category objects). Using this equivalence, we can build a straightening-unstraightening relating strictly enriched functors valued in the (∞,n)-category of (∞,n-1)-categories with the double (∞,n-1)-right fibrations introduced in the second talk. This gives us the desired technology to translate the universal properties of (∞,n)-limits across the two settings. 

Paige North, Higher categories via homotopy type theory.
The Equivalence Principle is an informal principle asserting that equivalent mathematical objects have the same properties. For example, group theory has been developed so that isomorphic groups have the same group-theoretic properties, and category theory has been developed so that equivalent categories have the same category-theoretic properties (though sometimes other, ‘evil’ properties are considered). Vladimir Voevodsky established Univalent Foundations as a foundation of mathematics (based on dependent type theory) in which the Univalence Principle, and thus the Equivalence Principle, for types (the basic objects of type theory) is a theorem. Later, versions of the Equivalence Principle for set-based structures such as groups and categories were shown to be theorems in Univalent Foundations.

In joint work with Ahrens, Shulman, and Tsementzis, we formulate and prove versions of the Equivalence Principle for a large class of categorical and higher categorical structures in Univalent Foundations. Our work encompasses (higher) categorical structures such as bicategories, dagger categories, opetopic categories, and more.

The Equivalence Principle in Univalent Foundations rely on the fact that the basic objects -- the types -- can be regarded as spaces. That is, Univalent Foundations can be viewed as an axiomatization of homotopy theory and as such is closely related to Quillen model category theory. Univalent Foundations can also be viewed as a foundation of mathematics based not on sets, but on spaces. It is the homotopical content of this foundation of mathematics that allows us prove something like the Equivalence Principle, something which is not possible in set-based foundations of mathematics, such as ZFC. If time allows, I will also talk about work to develop an extension of Univalent Foundations in which the basic objects are not spaces but higher categories.

Joost Nuiten, The infinitesimal tangle hypothesis.
The tangle hypothesis is a variant of the cobordism hypothesis that considers cobordisms embedded in some finite-dimensional Euclidean space (together with framings). Such tangles of codimension d can be organized into an E_d-monoidal n-category, where n is the maximal dimension of the tangles. The tangle hypothesis then asserts that this category of tangles is the free E_d-monoidal n-category with duals generated by a single object.

In this talk, based on joint work in progress with Yonatan Harpaz, I will describe an infinitesimal version of the tangle hypothesis: Instead of showing that the E_d-monoidal category of tangles is freely generated by an object, we show that its cotangent complex is free of rank 1. This provides support for the tangle hypothesis (of which it is a direct consequence), but can also be used to reduce the tangle hypothesis to a statement at the level of E_d-monoidal (n+1, n)-categories by means of obstruction theory.

Viktoriya Ozornova, Strictness in the ∞-world.
In this talk, I’d like to discuss the subtle relationship between strict and weak n-categories and (∞,n)-categories. For n=1, this is somewhat classical. For n=2, I will discuss in particular the results of a joint work with Moser and Rovelli. If time permits, I aim for some thoughts and conjectures for n>2.

Nima Rasekh, (∞,n)-Limits II: Fibrational approach and properties.
This is the second talk of the 3-part talk series on a joint project with Lyne Moser and Martina Rovelli. In the previous talk we defined limits in (∞,n)-categories using double-categorical methods. In this talk we develop the homotopy theory of double (∞,n-1)-right fibrations over a given (∞,n)-category, and in particular prove important properties about equivalences of such fibrations, such as an analogue to Quillen's Theorem A. Building on our theory of fibrations, we prove that our definition of (∞,n)-limits satisfies a range of essential properties intrinsic to a working theory of limits. This includes, but is not limited to, using cofinality techniques to facilitate computations of limits in (∞,n)-categories.

Martina Rovelli, (∞,n)-Limits I: Definition and first consistency results.
This is the first talk of a 3-part talk series on a joint project with Lyne Moser and Nima Rasekh. The general goal of the first episode is to correctly identify the universal property for the limit of a diagram valued in an (∞,n)-category. We will first discuss the description that naturally arises from enriched category theory, and mention some of its shortcoming in its usability. We will then propose an alternative viewpoint, which is more accessible in practice, and phrase the notion of a limit for a diagram valued in an (∞,n)-category in terms of an appropriate double (∞,n-1)-category of cones. We will show that this definition of (∞,n)-limit is compatible with the established notion of homotopy 2-limit in the context of 2-categories and the notion of (∞,1)-limit in the context of (∞,1)-categories, and with itself across different values of n.

Claudia Scheimbauer, Semiadditivity in higher categories and its application to topological field theories.
I will start with recalling the classical notion of semiadditive categories and its generalization called “m-semiadditivity” of ∞-categories. Then we will discuss its application to constructing finite gauge theories as topological field theories following Freed-Hopkins-Lurie-Teleman and Harpaz. Finally, I will discuss a (even) higher categorical version thereof, namely an (∞,k)-categorical version. This uses a loop-deloop adjunction for m-semiadditive enriched categories. This is joint work in progress with Tashi Walde.

Chris Schommer-Pries, Recognizing pushouts of (∞,n)-categories.
There are several models of (∞,n)-categories as certain presheaves of spaces or sets. In these models limits may often be computed in the underlying presheaf category, while colimits are notoriously difficult to compute. Nevertheless many important results in higher category theory depend on not only knowing that certain colimits of (∞,n)-categories exist, but also on being able to recognize them via small simple models. A famous example of this appears in Lurie’s approach to the bordism hypothesis. It rests on identifying certain bordism categories via a series of (homotopy) pushouts.

The goal of this talk is to describe new results that provide a general method/recognition schema for identifying pushouts of (∞,n)-categories. We hope to illustrate the technique with some simple examples.

Abstracts of the contributed talks