Tuesday 1st June 2021 at 14:00
Speaker: Kathryn Hess (École Polytechnique Fédérale de Lausanne)
Title: A comonadic machine for creating calculi
Abstract: Abstracting the framework common to most flavors of functor calculus, one can define a calculus on a category M equipped with a distinguished class of weak equivalences to be a functor that associates to each object x of M a tower of objects in M that are increasingly good approximations to x, in some well defined, Taylor-type sense. Such calculi could be applied, for example, to testing whether morphisms in M are weak equivalences.
In this talk, after making the definition above precise, I will describe a machine for creating calculi on functor categories Fun(C, M) that is natural in both the source C and the target M. Our calculi arise by comparison of the source category C with a tower of test categories, equipped with cubical structure of progressively higher dimension, giving rise to sequences of resolutions of functors from C to M, built from comonads derived from the cubical structure on the test categories. The stages of the towers of functors that we obtain measure how far the functor we are analyzing deviates from being a coalgebra over each of these comonads. The naturality of this construction makes it possible to compare both different types of calculi on the same functor category, arising from different towers of test categories, and the same type of calculus on different functor categories, given by a fixed tower of test categories.
This is joint work with Brenda Johnson.
Tuesday 25th May 2021 at 10:00
Speaker: Michael Batanin (CRM)
Title: Operadic categories as generalisation of operads
Abstract: Operadic categories were formally defined in 2015 by Batanin and Markl in a paper on a duoidal generalisation of Deligne’s conjecture. One can develop an impression that the only purpose of this notion is to be a convenient tool for bookkeeping of various classical and exotic multivariable structures. In my talk I’ll try to convince you that operadic category structure itself is a rich new algebraic structure which generalises classical operads.
More precisely, I am going to show that the category of classical symmetric operads in Set (where sets of colours can vary) is a reflective subcategory of the category of operadic categories. The inclusion is given by the operadic form of the Grothendieck construction, and its left adjoint (reflector) is a generalised symmetrisation functor, whose explicit expression can be written but requires a separate talk to explain. Nevertheless I’ll provide some examples related to the theory of En-algebras where this functor is better understood.
Finally, if time permits I’ll show as an application how one can interpret and generalise the Baez–Dolan plus-construction using the above relationships between operads and operadic categories.
Tuesday 18th May 2021 at 16:00
Speaker: Owen Gwilliam (University of Massachusetts)
Title: Centers of higher enveloping algebras and bulk-boundary systems
Abstract: The universal enveloping algebra of a Lie algebra plays a key role in representation theory (for obvious reasons) and in physics, particularly in encoding symmetries of quantum systems. But it is just one in a family of higher enveloping algebras: each dg Lie algebra 𝔤 has an enveloping En algebra Un(𝔤). (Here En refers to "n-dimensional algebras" in the sense of the little n-disks operad.) This construction admits a nice presentation via factorization algebras, by work of Knudsen, and we will discuss how it relates to symmetries of quantum field theories. We will discuss a model for the center of Un(𝔤) and how this framework encodes the observables of a bulk-boundary system where the bulk is topological BF theory for the Lie algebra 𝔤 and the boundary encodes "topological currents". This is joint work with Greg Ginot and Brian Williams.
Tuesday 11th May 2021 at 10:00
Speaker: Birgit Richter (Universität Hamburg)
Title: Detecting and describing ramification for structured ring spectra
Abstract: This is a report on joint work with Eva Höning. For rings of integers in an extension of number fields there are classical methods for detecting ramification and for identifying ramification as being tame of wild. Noether's theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the norm map. We take the latter fact and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. I will discuss several examples in the context of topological K-theory and modular forms.
Tuesday 4th May 2021 at 10:00
Speaker: Martin Markl (Czech Academy of Sciences)
Title: Operadic categories and their operads: May versus Markl
Abstract: Traditional operads exist in two disguises which are, under some quite standard assumptions, equivalent: in the form where the compositions in all inputs are made simultaneously; this is how they were introduced by May —and in a form, introduced much later by Markl, whose structure compositions are binary. The crucial advantage of the latter is that free Markl operads are naturally graded by the length of the chain of compositions. The same dichotomy holds also for operads over operadic categories, though the relation between these two approaches is much subtler than in the traditional case.
The talk will be devoted to theory of Markl operads and its relation to the original formulation of operad theory over operadic categories by Batanin–Markl. As in the classical case, free Markl operads are more structured than (the analogs of) May operads. This is crucial in the Koszul duality theory for operads over operadic categories, which is the theme of our current work.
Tuesday 27th April 2021 at 10:00
Speaker: Simona Paoli (University of Leicester)
Title: Weakly globular double categories and weak units
Abstract: Weakly globular double categories are a model of weak 2-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani 2-categories. Fair 2-categories, introduced by J. Kock, model weak 2-categories with strictly associative compositions and weak unit laws. In this talk I will illustrate how to establish a direct comparison between weakly globular double categories and fair 2-categories and prove they are equivalent after localisation with respect to the 2-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.
Tuesday 20th April 2021 at 16:00
Speaker: Mathieu Anel (Carnegie Mellon University)
Title: The enveloping ∞-topos of a presheaf 1-topos
Abstract: The ∞-topos enveloping of a 1-topos is a higher analogue of the 1-topos envelope of a frame. It is a surprising fact that the envelope of a presheaf 1-topos need not be a presheaf ∞-topos. We shall review a counter-example and give a necessary and sufficient condition under which the envelope stays a presheaf ∞-category.
Tuesday 13th April 2021 at 10:00
Speaker: Bashar Saleh (Universitat de Barcelona)
Title: Rational homotopy theory of homotopy automorphisms of manifolds
Abstract: The homotopy type of the monoid of homotopy automorphisms of a CW-complex X is interesting since it classifies X-fibrations up to equivalence. In a recent article by A. Berglund and I. Madsen, techniques for rationally modelling homotopy automorphisms of manifolds are developed. This opened up for different types of stability results for homotopy automorphisms of iterated connected sums of manifolds.
Tuesday 6th April 2021 at 10:00
Speaker: Claudia Scheimbauer (TU München)
Title: Higher Morita categories and extensions
Abstract: In this talk I will explain higher Morita categories of En-algebras and bimodules and discuss dualizability therein. Important examples are a 3-category of fusion categories and a 4-category of modular tensor categories. Then we will discuss why these do not suffice for Reshetikin–Turaev theories and I will give an outlook on work-in-progress with Freed and Teleman on how to remedy this.
Tuesday 23rd March 2021 at 10:00
Speaker: Vladimir Dotsenko (Université de Strasbourg)
Title: F-manifolds and pre-Lie algebras
Abstract: The notion of an F-manifold was introduced by Hertling and Manin as a weakening of the notion of a Frobenius manifold; it naturally appears in singularity theory, for instance. This notion leads to a very interesting algebraic structure defined by identities that are quadratic and cubic (and thus definitely outside the scope of the Koszul duality theory). That algebraic structure weakens the notion of a Poisson algebra, and the main result of this talk will indicate that this happens in a very remarkable way. Namely, Poisson algebras are quasi-classical limits of associative algebras, and I shall outline how to prove that F-manifold algebras are, in the same sense, quasi-classical limits of pre-Lie algebras.
Tuesday 16th March 2021 at 10:00
Speaker: Carles Casacuberta (Universitat de Barcelona)
Title: Lifting homotopical (co)localizations to algebras over monads
Abstract: For a monad T acting on a model category M, suppose that the category MT of T-algebras admits a transferred model structure (by the way, we can display an example where it does not). In joint work with Oriol Raventós and Andrew Tonks, we found necessary and sufficient conditions under which a homotopy localization L or a homotopy colocalization C on M lifts to Ho(M)T, and apparently stronger conditions for the existence of a lifting to Ho(MT). Although the categories Ho(M)T and Ho(MT) are far from equivalent in general, we discuss situations in which the two lifting criteria turn out to be equivalent, including those cases in which the forgetful functor commutes with homotopy colimits.
Tuesday 9th March 2021 at 10:00
Speaker: Clemens Berger (Université Nice Sophia Antipolis)
Title: Moment categories and operads
Abstract: Almost half a century ago operads have been introduced by May and Boardman–Vogt. Since then they are used with great success in different areas of mathematics and even outside. I will present a new approach based on the concept of moment category.
Starting point is an active/inert factorisation system giving rise to a family of split idempotent endomorphisms, called moments. Segal's category Γ plays a universal role here. The inert/active factorisation system on the dual category of finite sets and partial maps is the cornerstone of Lurie's theory of infinity operads.
Moment structures on Γ, Δ and Θn encode the structure of symmetric operad, non-symmetric operad and globular n-operad (Batanin) respectively. There is an analog of the plus construction of Baez–Dolan in our setting. It takes a moment category to a hypermoment category such that operads for the former get identified with monoids for the latter. These monoids are presheaves satisfying certain Segal conditions strictly.
We show that the plus construction of Γ may be identified with a full subcategory of the dendroidal category Ω of Moerdijk–Weiss. This is one possible explanation of the fact that dendroidal Segal spaces are models for infinity operads.
Tuesday 2nd March 2021 at 10:00
Speaker: Pedro Boavida de Brito (Universidade de Lisboa)
Title: Galois symmetries of knot spaces
Abstract: I'll describe how the absolute Galois group of the rationals acts on a space which is closely related to the space of all knots. The path components of this space form a finitely generated abelian group which is, conjecturally, a universal receptacle for integral finite-type knot invariants. The added Galois symmetry allows us to extract new information about its homotopy and homology beyond characteristic zero. This is joint work with Geoffroy Horel.
Tuesday 23rd February 2021 at 10:00
Speaker: Amar Hadzihasanovic (Tallinn University of Technology)
Title: The smash product of monoidal theories
Abstract: The smash product of pointed spaces is a classical construction of topology. The tensor product of props, which extends both the Boardman–Vogt product of symmetric operads and the tensor product of Lawvere theories, seems firmly like a piece of universal algebra. In this talk, I will show that the two are facets of the same construction: a “smash product of pointed directed spaces”. Here “directed spaces” are modelled by combinatorial structures called diagrammatic sets, while the cartesian product of spaces is replaced by a form of Gray product. Most interestingly, the smash product applies to presentations of higher-dimensional theories and systematically produces oriented equations and higher-dimensional coherence data (oriented syzygies). This introduces a synthetic, compositional method in rewriting on higher structures. The talk is based on my preprint arXiv:2101.10361 with the same title.
Tuesday 16th February 2021 at 16:00
Speaker: André Joyal (Université du Québec à Montréal)
Title: Higher sheaves
Abstract: The notion of sheaf is central in topos theory. We introduce a notion of higher sheaf in higher topos theory (joint work with M. Anel, G. Biedermann and E. Finster).