Title: The quasi-category of homotopy coherent monads in a (∞, 2)-category.
Abstract: Homotopy coherent diagrams in a simplicial category K can be encoded as simplicial functors C → K, where C is a well chosen simplicial category. This idea goes back at least to Cordier and Porter (Math. Proc. Cambridge Philos. Soc. 1986) and originated in earlier work of Vogt (Math. Z. 134, 1973) on homotopy coherent diagrams. For instance, the homotopy coherent nerve is constructed in this way. In Riehl and Verity’s paper (Adv. Math. 2016), C is the universal 2-category containing the object of study, either a monad or an adjunction. For instance they define homotopy coherent monads as simplicial functors Mnd → K, where K = qCat∞ , the category of quasi-categories enriched over itself, and where Mnd is the universal 2-category containing a monad.
In this talk, we define a cosimplicial object Mnd[-] in 2-categories which induces a nerve NMnd : sCat → sSet. When K is a 2-category, NMnd(K) = NMnd(K), where Mnd(K) is the 1-category of monads in K, as defined by Street in (JPAA 1972). We show that when K is enriched in quasi-categories and sufficiently complete, NMnd (K) is a quasi-category whose objects are the homotopy coherent monads in K.
Title: Not even wrong!
Abstract:
I will share some thoughts on the foundations of string theory. This is joint work in progress with M. Ando.
Title: Sullivan diagrams and homological stability
Abstract:
In string topology one studies the algebraic structures of the chains of the free loop space of a manifold by defining operations on them. Recent results show that these operations are parametrized by certain graph complexes that compute the homology of compatifications of the Moduli space of Riemann surfaces. Finding non-trivial homology classes of these compactifications is related to finding non-trivial string operations. However, the homology of these complexes is largely unknown. In this talk I will describe one of these complexes: the chain complex of Sullivan diagrams and give some computational results and further work on this direction.
Title: Moduli spaces of bialgebras, higher Hochschild cohomology and formality
Abstract:
Algebras over props provide a good formalism to parametrize various structures of bialgebras as well as their homotopy version, which naturally appears in problems related to topology, geometry and mathematical physics. A relevant idea to understand their deformation theory is to gather them in a moduli space of algebraic structures in the setting of Toen-Vezzosi’s derived algebraic geometry. Infinitesimal deformations of such structures are then controlled by tangent Lie (or L-infinity) algebras naturally providing the corresponding cohomology theories and obstruction theories.
In a work in collaboration with Gregory Ginot, we apply these results to several open problems relating the deformation theory of E₂-algebras with the deformation theory of associative and coassociative bialgebras. In particular, relying on the higher Deligne conjecture (now a theorem), we solve two longstanding conjectures of Gerstenhaber-Schack and Kontsevich: the existence of an E₃-structure refining the deformation complex of a dg bialgebra, and the E₃-formality of this deformation complex in the case of a symmetric bialgebra. This E₃-formality theorem provides, in turn, a new proof of Etingof-Kazdhan deformation quantization independent from the choice of a Drinfeld associator.
Title: What do homotopy algebras form?
Abstract:
In this talk, I will describe joint work with V. Dolgushev in which we use ideas from deformation theory to construct an enriched category whose objects are homotopy algebras of a fixed type, e.g., L∞, A∞, or Ger∞ algebras. The enrichment is over a certain symmetric monoidal category of L∞-algebras. Roughly, this is a “non-abelian” analogue of the fact that chain complexes are enriched over themselves. From this L∞-enriched category, we obtain a simplicial category by using a non-abelian analogue of the Dold-Kan functor. We show that the mapping spaces in this simplicial category are, in fact, Kan complexes, and that this construction produces an explicit model for the ∞-category of homotopy algebras.
Title: Operadic convolution in probability theory
Abstract: The algebra of random variables associated to a probability space is a toy model for the observables of an algebraic quantum field theory. In this context, operadic language provides a concise encapsulation of the relationship between the moments and the cumulants of the random variables. I hope to talk about two aspects of this relationship related to non-commutative probability theory as an advertisement of potentially fertile ground for exploration by experts in this language. First, over a commutative ground ring, the framework leads to an operad in some ways reminiscent of the cactus operad. Second, over a non-commutative ground-ring, the combinatorics of the relationship between moments and cumulants uses composition along trees and is even more intimately tied to operadic language.
Title: Chain models for moduli space operads.
Abstract: First, we will review several operads built from moduli spaces of punctured spheres and their compactifications. In particular we will recall that algebraic structures arising in homological mirror symmetry and string topology may be represented by such operads. I will then discuss the problem of lifting these structures to the chain level and describe some recent progress.
Title: Hierarchical networks and coloured modular operads
Abstract:
In applications across science and technology, there is an increasing interest in hierarchical structures of complex networks. My work aims to develop a general conceptual framework to capture the structure and function of complex networks at multiple spatio-temporal scales in one single model.
In this talk, I will first explain how coloured modular operads, and, in particular, a sheaf-theoretic approach similar to that of Joyal and Kock (2009), provide a suitable formalism for this problem. I will conclude with a short impression of how these ideas are being received by applied scientists, and some of the collaborations on ‘real world’ problems that are currently in the pipeline.
Title: On the K-theory of monoid algebras
Abstract:
I will discuss work in progress with G. Cortiñas, M. Walker and C. Weibel regarding the algebraic K-theory of commutative monoid algebras. The ingredients I will focus on in this talk are a computation in (topological) cyclic homology, and some results in the algebraic geometry of monoids.
Title: Etale Homotopy Theory for Higher Stacks
Abstract:
Etale homotopy theory, as originally introduced by Artin and Mazur in the late 60s, is a way of associating to a suitably nice scheme a pro-object in the homotopy category of spaces, and can be used as a tool to extract topological invariants of the scheme in question. It is a celebrated theorem of theirs that, after profinite completion, the etale homotopy type of an algebraic variety of finite type over the complex numbers agrees with the homotopy type of its underlying topological space equipped with the complex analytic topology. We will present work of ours which offers a refinement of this construction which produces a pro-object in the infinity-category of spaces (rather than its homotopy category) and applies to a much broader class of objects, including all algebraic stacks. We will also present a generalization of the previously mentioned theorem of Artin-Mazur, which holds in much greater generality than the original result.
Title: Weakly globular structures in homotopy theory and higher category theory.
Abstract: n-types are spaces whose homotopy groups vanish in dimension higher than n. They are the building blocks of spaces thanks to the Postnikov decomposition. The search for combinatorial structures to model algebraically n-types leads to higher categorical structures. In this talk we discuss this problem using a novel approach, based on iterated internal categories and the notion of weak globularity. We discuss the resulting structure, called weakly globular n-fold categories, and its relevance to homotopy theory and to higher category theory.
Title: Operads and polynomial 2-monads
Abstract: Polynomial 2-monads provide a framework for the discussion of operadic structures, and have been used by Batanin-Berger in the study of when transferred model structures on categories of operads exist. In this talk the theory of polynomial functors will be recalled, it will be explained how operads can be seen as polynomial monads in a few ways, and how a lot of operadic theory can be recovered from more general ideas in 2-dimensional monad theory.