Series

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Jérôme Scherer: Floyd’s manifold is a conjugation space

7th December, 2021

This is joint work with Wolfgang Pitsch. We illustrate how equivariant stable homotopy methods can help us recognize the structure of a conjugation space, as introduced by Hausmann, Holm, and Puppe. We first explain their definition and present a characterization in terms of purity (obtained in previous joint work with Nicolas Ricka). We then perform equivariantly Floyd's construction from the 1970s of a pair of 5- and 10-dimensional manifolds with four cells, relying on Lück and Uribe’s work on equivariant bundles. The 10-dimensional one is a conjugation space.

Fosco Loregian: Towards a formal category theory of derivators

23rd November, 2021

Derivator theory, initiated by Grothendieck and Heller in the '90s to correct the shortcomings of triangulated categories, motivated a lot of research regarding the foundation of (∞,1)-category theory, and its applications to algebraic geometry/topology.

For a 2-category theorist, a (pre)derivator is a familiar object - (a suitably co/complete) prestack on the category cat of small categories - and yet still little is known about the formal properties of the 2-category PDer. The present talk is motivated by the belief that time is ripe for a more conceptual look into the foundations of derivator theory, and that far from being a mere exercise in style, such a conceptualization yields many practical advantages.

After briefly outlining the essentials of "formal category theory'' (2-categories can be used to organize the theory of "categories with structure" just as category theory organizes the theory of "sets with structure"), I will report on a conjecture regarding the possibility to provide a "yoneda structure" or a "proarrow equipment" to the 2-category of pre/derivators. Under suitable assumptions, these are equivalent ways to equip PDer with a calculus of Kan extensions, and building on prior work of Di Liberti and myself, this allows to speak about "locally presentable" and "accessible" objects (showing that Adamek-Rosický and Renaudin's definitions eventually coincide); the overall goal is to provide a suitable form of special/general adjoint functor theorem for a morphism of prederivators (such a theorem would simplify a lot the life of the average algebraic geometer).

John Greenlees: The torsion Adams spectral sequence for rational torus-equivariant spectra

19th October, 2021

We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, 𝒜t(G) of finite injective dimension, a homology theory π∗𝒜t taking values in 𝒜t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence

Ext𝒜t(G)∗ , ∗ (π∗𝒜t(X), π∗𝒜t(Y)) → [X,Y]∗G

for rational G-spectra X and Y.

This approach should be viewed as an analogue of the Cousin complex in algebraic geometry. It is expected that a similar method will apply to other tensor triangulated categories with finite-dimensional Noetherian Balmer spectra.

Camillo De Lellis: What is the h-principle?

11th October, 2021

The honest answer to the question is that I actually do not know. I will therefore rather talk about several famous examples that are widely called 'h-principle results' and try to explain some of the ideas behind the ones I am most familiar with.

Clover May: Classifying perfect complexes of ℤ/2-Mackey functors

7th October, 2021

Mackey functors play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors.  In this talk, I will discuss joint work with Dan Dugger and Christy Hazel classifying perfect chain complexes of constant Mackey functors for G=ℤ/2.  Our decomposition leads to a computation of the Balmer spectrum of the derived category.  We extend these results to classify all finite modules over the equivariant Eilenberg-MacLane spectrum Hℤ/2. 

John Bourke: An orthogonal approach to algebraic weak factorization systems

30th September, 2021

Factorization systems (both weak and strong) are commonly defined as consisting of two classes of maps satisfying a certain orthogonality relation and a factorization axiom. The standard definition of algebraic weak factorization system, involving comonads and monads, is rather different. The goal of this talk will be to describe an equivalent definition of algebraic weak factorization system emphasising orthogonality and factorization.

Nicholas Meadows: Higher theories and monads

16th September, 2021

We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of ∞-categories, and exploit this to prove many classical theorems about monads in the ∞-categorical setting. Among other things, we prove that the category of algebras for an accessible monad on a locally presentable ∞ category is locally presentable. We also apply the result to construct examples of ∞-categorical monads from pretheories.

Yuri Berest: Spaces of quasi-invariants and homotopy Lie groups

22nd July, 2021

Quasi-invariants are natural algebraic generalizations of classical invariant polynomials of finite reflection groups. They first appeared in mathematical physics - in the work of O. Chalykh and A. Veselov on quantum integrable systems - in the early 1990s, and since then have found many interesting applications in other areas: most notably, representation theory, algebraic geometry and combinatorics. In this talk, I will explain how the algebras of quasi-invariants arise in topology: as cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result is a generalization of a well-known theorem of A. Borel that realizes the algebra of classical invariant polynomials of a Weyl group W(G) as the cohomology ring of the classifying space BG of the corresponding Lie group G. Perhaps most interesting here is the fact that our construction of spaces of quasi-invariants is purely homotopy-theoretic. It can therefore be extended to some non-Coxeter (p-adic pseudo-reflection) groups, in which case the compact Lie groups are replaced by the so-called p-compact groups (a.k.a. homotopy Lie groups).

Giulio Lo Monaco: Vopěnka’s principle in ∞-categories

24th June, 2021

Vopěnka's principle has arisen as a model-theoretical statement, provably independent of ZFC set theory. However, there are a number of categorical ways of formulating it, preventing the existence of proper classes of objects with some conditions in presentable categories, and these are what our attention will be focused on. In particular, we will look at analogous statements in the context of ∞-categories and we will ask how these new statements interact with the older ones. Moreover, some of the consequences of Vopěnka's principle on classes of subcategories of presentable categories are investigated and to some extent generalized to ∞-categories. A parallel discussion is undertaken about the similar but weaker statement known as weak Vopěnka's principle.

Leonardo Larizza: Lax factorization systems and categories of partial maps

17th June, 2021

Factorization systems describe morphisms in a category by factorizing them into pairs of composable morphisms. Their definition depends on a kind of orthogonality relation between morphisms, which entails the existence of some diagonal morphisms for certain squares. In this seminar we present the new notion of lax weak orthogonality between morphisms, which involves lax squares and the factorization systems it generates. Then we will introduce lax versions of functorial and algebraic weak factorization systems and some of their properties. These lax factorization systems are discussed, keeping the theory of ordinary factorization systems as a blueprint and providing useful properties. An overview of the examples of such lax factorization systems is presented in the context of partial maps. We conclude with a discussion of general constructions of these examples and their description in the particular case of sets with partial maps.

Raffael Stenzel: Proof relevance in higher topos theory

27th May, 2021

In the short course of its definition and early exploration, the theory of higher toposes (by which I specifically mean (∞,1)-toposes) has been found to exhibit various traits which appear rather odd from the perspective of ordinary topos theory. Motivated by the fact that the internal language of every higher (Grothendieck) topos is a univalent type theory - and hence is intrinsically 'proof relevant' - we reconsider the basic characteristic notions associated to a higher topos from a purely logical proof relevant point of view.

Given a small ∞-category C, this will motivate the notion of a logical structure sheaf on C whose ideals correspond exactly to the left exact localizations of the infinity-category [Cop, S] of presheaves over C. This in turn will naturally lead to a corresponding notion of generalized Grothendieck topologies on C which, first, capture all higher toposes embedded in [Cop, S], and second, correspond exactly to the classical notion of Grothendieck topologies in the monic (i.e. the proof irrelevant) context. We will see that these notions induce a Kripke-Joyal semantics valued in spaces (rather than in the classical subobject classifier) in obvious fashion as well. In the end of the talk we will take a look at a few examples of such topologies and, if time permits (which it rarely ever does, time appears to be pretty absolute when it comes to this), end with a discussion of some open questions.

Tashi Walde: Higher Segal spaces via Higher Excision

20th May, 2021

Starting from the classical Segal spaces, Dyckerhoff and Kapranov introduced a hierarchy of what they call higher Segal structures. While the first new level (2-Segal spaces) has been well studied in recent years, not much is known about the higher levels and the hierarchy as a whole.

In this talk I explain how this hierarchy can be understood conceptually in close analogy to the manifold calculus of Goodwillie and Weiss. I describe a natural 'discrete manifold calculus' on the simplex category and on the cyclic category, for which the polynomial functors are precisely the higher Segal objects. Furthermore, this perspective yields intrinsic categorical characterizations of higher Segal objects in the spirit of higher excision.

Walter Tholen: Spaces vs Categories, Perfect Maps vs Discrete Fibrations

6th May, 2021

We consider perfect maps of topological spaces and discrete cofibrations of categories to guide us into Burroni's notions of T-category and T-functor. In that environment we establish a so-called comprehensive factorization system that entails the classical Street-Walters system, as well as the (antiperfect, perfect) system for continuous maps of Tychonoff spaces known since the 1960s.

Hoang Kim Nguyen: Contravariant homotopy theories and Quillen’s Theorem A

29th April, 2021

In this talk I will show how to construct a model structure on a locally presentable category with a suitable cylinder object such that the model structure behaves in a 'covariant' or 'contraviariant' way with respect to the cylinder. Examples of such model structures include the covariant and contravariant model structures on simplicial sets and the cocartesian and cartesian model structures on marked simplicial sets modelling presheaves with values in ∞-groupoids and ∞-categories respectively.

The model structures come with an abstract notion of cofinal functor which recovers the usual definition of cofinal functor for ∞-categories when applied to the covariant and contravariant model structures on simplicial sets. When applied yo presheaves valued in n-types, one obtains a version of Quillen’s Theorem A for n-categories.

Michael Ching: Tangent ∞-categories and Goodwillie calculus

8th April, 2021

In 1984 Rosický introduced tangent categories in order to capture axiomatically some properties of the tangent bundle functor on the category of smooth manifolds and smooth maps. Starting in 2014 Cockett and Cruttwell have developed this theory in more detail to emphasize connections with cartesian differential categories and other contexts arising from computer science and logic.

In this talk I will discuss joint work with Kristine Bauer and Matthew Burke which extends the notion of tangent category to ∞-categories. To make this generalization we use a characterization by Leung of tangent categories as modules over a symmetric monoidal category of Weil-algebras and algebra homomorphisms. Our main example of a tangent ∞-category is based on Lurie's model for the tangent bundle to an ∞-category itself. Thus we show that there is a tangent structure on the ∞-category of (differentiable) ∞-categories. This tangent structure encodes all the higher derivative information in Goodwillie’s calculus of functors, and sets the scene for further applications of ideas from differential geometry to higher category theory.

Evgeny Mukhin: Supersymmetric analogues of partitions and plane partitions

8th April, 2021

We will explain combinatorics of various partitions arising in the representation theory of quantum toroidal algebras associated to Lie superalgebra 𝔤𝔩(m|n). Apart from being interesting in its own right, this combinatorics is expected to be related to crystal bases, fixed points of the moduli spaces of BPS states, equivariant K-theory of moduli spaces of maps, and other things.

Jiří Adámek: C-Varieties of Ordered and Quantitative Algebras

1st April, 2021

Mardare, Panangaden and Plotkin introduced C-varieties of algbebras on metric spaces. These are categories of metric-enriched algebras specified by equations in a context. A context puts restrictions on the distances of variables one uses. We prove that C-varieties are precisely the monadic categories over Met for countably accessible enriched monads preserving epimorphisms.

We analogously introduce C-varieties of ordered algebras as categories specified by inequalities in a context. Which means that conditions on inequalities between variables are imposed. We prove that C-varieties precisely correspond to enriched finitary monads on Pos preserving epimorphisms.

Karol Szumilo: ∞-groupoids in lextensive categories

18th March, 2021

I will discuss a construction of a new model structure on simplicial objects in a countably lextensive category (i.e., a category with well-behaved finite limits and countable coproducts). This builds on previous work on a constructive model structure on simplicial sets, originally motivated by modelling Homotopy Type Theory, but now applicable in a much wider context.