Luciana Basualdo Bonatto

An infinity operad of normalized cacti

Normalized cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By introducing a new topological operad of bracketed trees, we show that this operation is associative up-to all higher homotopies and that normalized cacti form an ∞-operad in the form of a dendroidal space satisfying a weak Segal condition. In particular, this provides one of the few examples in the literature of infinity operads that are not a nerve of an actual operad.

Nathaniel Bottman

The relative 2-operad of 2-associahedra in symplectic geometry

The Fukaya A_{∞}-category Fuk(M) is a rich invariant of a symplectic manifold M, and its manipulation and computation is a core focus of current symplectic geometry. Building on work of Wehrheim and Woodward, I have proposed that the correct way to encode the functoriality properties of Fuk is by defining an “(A_{∞},2)-category” called Symp, in which the objects are symplectic manifolds and hom(M,N) is defined to be Fuk(M^{−} × N). Underlying the new notion of an (A_{∞},2)-category is a family of abstract polytopes called 2-associahedra, which form a “relative 2-operad” (another new notion, which is related to Batanin’s theory of higher operads). I will describe all of these constructions from scratch, without assuming any knowledge of symplectic geometry. This talk is based partly on joint work with Shachar Carmeli, and I will mention related joint work with Alexei Oblomkov.

Anna Cepek

Configuration spaces of ℝ^{n} and Joyal’s category 𝝧_{n}

We examine configurations of finite subsets of Euclidean space within the homotopy-theoretic context of ∞-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of these configuration spaces in terms of the category 𝝧_{n}.

Michael Ching

Goodwillie calculus and operads

The goal of this talk is to survey the role of operads in Goodwillie’s calculus of functors. A key observation is that the derivatives of the identity functor, on a suitable pointed ∞-category *C*, admit an operad structure which in the case of pointed spaces recovers a spectral version of the Lie operad. I will give a couple different ways to construct the operad structure in general, and then focus on the case where *C* is itself the ∞-category of algebras over some (stable, non-unital) operad *P*. In that case, the derivatives of the identity functor on *C* recover, in some form, the operad *P*.

Richard Garner

Sweedler duals for monads

While the linear dual of any coalgebra is an algebra, the converse is not true; however, there is an adjoint to the coalgebra-to-algebra functor, given by the so-called Sweedler dual.

There is a notion of “linear dual” for an endofunctor of Set, given by homming into the identity functor for the Day convolution structure. Again, this sends comonads to monads, but not vice versa; but again, there is an adjoint. This “Sweedler dual” comonad of a monad was introduced by Katsumata, Rivas and Uustalu in 2019.

The purpose of this talk is to give an explicit construction of the Sweedler dual comonad of any monad on Set. The category of coalgebras for the Sweedler dual turns out to be a presheaf category, whose indexing category can be described explicitly in terms of a kind of computational dynamics of the monad. If time permits, we also describe the source-etale topological category which classifies the topological Sweedler dual comonad of a monad on Set; in particular, this recovers all kinds of etale topological groupoids of interest in the study of combinatorial C^{*}-algebras.

Syunji Moriya

A spectral sequence for cohomology of knot spaces

This talk is based on the preprint arXiv:2003.03815. Let Emb(𝘚¹,𝘔) be the space of embeddings from 𝘚¹ to a closed manifold 𝘔 (space of knots in 𝘔). Recently, this space is studied by Arone–Szymik, Budney–Gabai, and Kupers, using Goodwillie–Weiss embedding calculus. In this talk, we introduce a spectral sequence for cohomology of Emb(𝘚¹,𝘔) whose 𝘌₂-term has an algebraic presentation, using Sinha’s cosimplicial model which is derived from the calculus. This converges to the correct target if 𝘔 is simply connected and of dimension ≥ 4 for general coefficient ring. Using this, we see a computation of 𝘏^{*}(Emb(𝘚¹,𝘚^{k}×𝘚^{l})) in low degrees under some assumption on *k,l* and an isomorphism 𝜋₁(Emb(𝘚¹,𝘔)) ≅ 𝘏₂(𝘔,ℤ) for some simply connected 4-dimensional 𝘔.

Our main idea of the construction is to replace configuration spaces in the cosimplicial model with fat diagonals via Poincaré Lefschetz duality. To do this, we use a notion of a (co)module over an operad. A somewhat curious point is that we need spectra (in stable homotopy) even though our concern is singular cohomology.

Joost Nuiten

Moduli problems for operadic algebras

A classical principle in deformation theory asserts that any formal deformation problem over a field of characteristic zero is classified by a differential graded Lie algebra. This principle has been described more precisely by Lurie and Pridham, who establish an equivalence between dg-Lie algebras and formal moduli problems indexed by Artin commutative dg-algebras. I will discuss an extension of this result to more general pairs of Koszul dual operads over a field of characteristic zero. For example, there is an equivalence of infinity-categories between pre-Lie algebras and formal moduli problems indexed by permutative algebras. Under this equivalence, permutative deformations of a trivial algebra are classified by the usual pre-Lie structure on its deformation complex. In the case of the coloured operad for nonunital operads, a relative version of Koszul duality yields an equivalence between nonunital operads and certain kinds of operadic formal moduli problems. This is joint work with D. Calaque and R. Campos.

Dan Petersen

Lie, associative and commutative quasi-isomorphism

Let A and A' be commutative dg algebras over Q. There are two a priori different notions of what it means for them to be quasi-isomorphic: one could ask for a zig-zag of quasi-isomorphisms in the category of commutative dg algebras, or a zig-zag in the larger category of not necessarily commutative dg algebras. Our first main result is that these two notions coincide. The second main result is Koszul dual to the first, and states that if two dg Lie algebras over Q have quasi-isomorphic universal enveloping algebras, then the derived completions of the two dg Lie algebras are quasi-isomorphic. The latter result is new even for classical Lie algebras concentrated in degree zero. Both results have immediate consequences in rational homotopy theory. (Joint with Campos, Robert-Nicoud, Wierstra)

Sophie Raynor

Unpacking the combinatorics of modular operads

Whilst operads are governed by trees, undirected graphs of arbitrary genus are needed in order to describe modular operads. And this can get complicated. Especially if we’re interested in understanding notions of modular operads, such as Joyal and Kock’s *compact symmetric multicategories*, where the combination of the contraction operation and a unital operadic composition presents particular challenges.

I’ll describe how to first break the problem into its constituent parts, and then use the classical theory of distributive laws to put the pieces back together. The decomposition allows us to apply Weber’s theory to get a fully faithful nerve via completely abstract methods. More interestingly, the proof method makes the combinatorics of modular operads, and especially the fiddly stuff, completely explicit. Hence it provides a roadmap for developing the theory, and the possibility for gaining new conceptual insights into the structures described.

María Ronco

Order and substitution on graph associahedra

M. Carr and S. Devadoss introduced in [1] associated a finite partially ordered set to any simple finite graph, whose geometric realization is a convex polytope 𝒦Γ, the graph-associahedron. Their construction include many well-known families of polytopes, liked permutahedra, associahedra, cyclohedra and the standard simplexes.

The goal of the present work is to give an *algebraic* description of graph associahedra. We introduce a substitution operation on Carr and Devadoss tubings, which allows us to describe graph associahedra as a free object on the set of all connected simple graphs, for a type of colored operad generated by pairs of a finite connected graph and a connected subgraph of it.

We show that substitution of tubings may be understood in the context of M. Batanin and M. Markl's operadic categories. We describe an order on the faces of graph-associahedra, different from the one given by Carr and Devadoss, which allows us to construct a standard triangulation of graph associahedra, following [2].

(joint work with Stefan Forcey)

[1] M. Carr, S. Devadoss, *Coxeter complexes and graph associahedra*, Topol. and its Applic. 153 (1-2) (2006) 2155–2168.

[2] J.-L. Loday, *Parking functions and triangulation of the associahedron*, Proceedings of the Street’s fest 2006, Contemp. Math. AMS 431 (2007), 327–340.

Dmitry Vaintrob

The operad of framed formal curves and a program of Kontsevich

It has long been conjectured (originally formalized by Kontsevich) that the operad of framed little disks can be enriched to an operad in an appropriate category of motives (in the sense of Grothendieck and Voevodsky). I will explain such a construction, in a motivic category associated to logarithmic schemes (or more generally, stratified formal schemes) and explain how (via ideas of Kontsevich, Tamarkin, Beilinson and others), this construction leads to a systematic resolution of several formality and deformation theoretic results, including the Deligne formality conjecture and deformation-quantization previously proven via more transcendental techniques.

Bruno Vallette

Deformation theory of cohomological field theories

I will explain how to develop the deformation theory of cohomological field theories as a special case of a general deformation theory of morphisms of modular operads. Two cases will be considered: a classical and a quantum one. Using ideas of Merkulov–Willwacher based on graphs complexes, I will introduce and develop a new universal deformation group which acts functorially via explicit formulas on the moduli spaces of gauge equivalence classes of morphisms of modular operads. In the classical case, the action is trivial; but in the quantum case, this group contains the prounipotent Grothendieck–Teichmüller group and its action is highly non-trivial even in the simplest case. Then, I will enrich these graph complexes with characteristic classes coming from the geometry of the moduli spaces of curves and obtain in this way (rather surprisingly) a natural homotopy extension to Givental group action in the classical case, and in the quantum case, a huge group that includes both Givental and Grothendieck–Teichmüller groups.

It is a joint work with Volodya Dotsenko, Sergey Shadrin, and Arkady Vaintrob (arXiv:2006.01649).