#### Statement on Black Lives Matter

**Title:** The Diamond Lemma through homotopical algebra.

**Abstract:**

The Diamond Lemma is a result indispensable to those studying associative (and other types of) algebras defined by generators and relations. In this talk, I will explain how to approach this celebrated result through the lens of homotopical algebra: we will see how every multigraded resolution of a monomial algebra leads to “its own” Diamond Lemma, which is hard-coded into the Maurer-Cartan equation of its tangent complex. For the reader familiar with homotopical algebra, we hope to provide a conceptual explanation of a very useful but perhaps technical result that guarantees uniqueness of normal forms through the analysis of “overlapping ambiguities”. For a reader familiar with Gröbner bases or term rewriting theory, we hope to offer some intuition behind the Diamond Lemma and at the same time a framework to generalize it to other algebraic structures and optimise it. This is joint work with Vladimir Dotsenko (arXiv:2010.14792).

**Title:** PD operads and partition Lie algebras.

**Abstract:**

Partition Lie algebras, introduced by Brantner and Mathew, are homotopy theoretic refinements of Lie algebras that appear in deformation theory in positive characteristic. However, since partition Lie algebras are defined in a purely ∞-categorical way, it is hard to get one’s hands on them. For instance, it is not so clear how to describe them explicitly in terms of point-set models, because there is no operad whose algebras are partition Lie algebras.

In this talk, I will discuss a homotopy-theoretic generalization of the notion of an operad, called a “PD operad”, whose algebras can also carry certain kinds of divided power operations. The homotopy theory of such PD operads and their algebras can be understood quite explicitly in terms of chain complexes, using some homological algebra for representations of the symmetric groups. As a particular example, I will describe the PD operad that controls partition Lie algebras. Based on joint work with Lukas Brantner and Ricardo Campos.

**Title**: Equivariant Categories on Varieties

**Abstract**:

The derived category of sheaves on a variety is a central object of study in arithmetic geometry and the representation theory of algebraic groups. However, while the derived category of a variety is quite important, people who study groups acting on spaces have needed an equivariant version of the derived category so as to give a natural place from which to derive equivariant cohomology. It wasn’t until the nineties that a good definition of the equivariant derived category was discovered in a usable fashion by taking descent data through the derived categories of resolutions of an action on the base space. However, the constructions of equivariant derived categories differ greatly from the equivariant categories of sheaves that precede them and consequently make it difficult to see how to build categories of equivariant objects over a variety in a uniform way.

In this talk I will introduce a formalism that shows how to give a uniform and more general construction of equivariant categories overtop a variety by using certain pseudofunctors to construct equivariant categories. After introducing these categories, I will also show how this construction allows us natural ways to deduce properties of the equivariant category simply by looking at properties of the pseudofunctor. Finally, I’ll show that the functors and natural transformations that arise between equivariant categories that come from equivariant data are induced by pseudonatural transformations and modifications which give us techniques for lifting adjoints in a 2-category of pseudofunctors to what I call equivariant adjoints between equivariant categories. If there is any time left, I’ll also give some comments as to how these techniques allow us to produce triangulations on certain families of equivariant categories that contain the equivariant derived category.

**Title**: Bifold algebras

**Abstract**:

Given V-enriched algebraic theories T and U for a system of arities J in the sense of [1], commuting pairs of T- and U-algebra structures on the same object may be described equivalently as bifunctors that preserve J-cotensors in each variable separately, and these we call bifold algebras. Bifold algebras may be described equivalently as algebras for a theory called the tensor product of T and U, provided that J and V satisfy certain conditions that we do not assume in this talk. Every bifold algebra has two underlying algebras, which we call its left and right faces. (By an algebra, here we mean a pair consisting of a theory T and a T-algebra A.)

In this talk, we construct a two-sided fibration [2] of bifold algebras over various theories, and we show that the notion of commutant for algebras [3, 4] arises via universal constructions in this two-sided fibration. Using this method, we develop a functorial treatment of commutants for algebras over various theories. On this basis, we study bifold algebras in which one face is the commutant of the other, and vice versa, and we discuss examples in algebra, order theory, and topology.

[1] R. B. B. Lucyshyn-Wright, Enriched algebraic theories and monads for a system of arities. Theory and Applications of Categories 31 (2016), 101-137.

[2] R. Street, Fibrations and Yoneda’s lemma in a 2-category. Lecture Notes in Mathematics 420 (1974), Springer.

[3] R. B. B. Lucyshyn-Wright, Commutants for enriched algebraic theories and monads. Applied Categorical Structures 26 (2018), 559-596.

[4] R. B. B. Lucyshyn-Wright, Functional distribution monads in functional-analytic contexts. Advances in Mathematics 322 (2017), 806-860.

**Title**: Homotopy theory for Kan simplicial manifolds and a smooth analog of Sullivan’s realization functor

**Abstract**: Kan simplicial manifolds, also known as “Lie infinity-groupoids”, are simplicial Banach manifolds which satisfy conditions similar to

the horn filling conditions for Kan simplicial sets. Group-like Lie infinity-groupoids (a.k.a “Lie infinity-groups”) have been used to construct geometric models for the higher stages of the Whitehead tower of the orthogonal group. With this goal in mind, Andre Henriques developed a smooth analog of Sullivan’s realization functor from rational homotopy theory which produces a Lie infinity-group from certain commutative dg-algebras (i.e. L-infinity algebras).

In this talk, I will describe a homotopy theory for both these commutative dg-algebras and for Lie infinity-groups, and discuss some examples that demonstrate the compatibility between the two. Conceptually, this work can be interpreted either as a smooth analog of classical results of Bousfield and Gugenheim in rational homotopy theory, or as a homotopy-theoretic analog of classical theorems from Lie theory.

**Title**: The Grothendieck construction for lenses

**Abstract**: The Grothendieck construction, which describes an equivalence between functors into Cat and split opfibrations, may be generalised in several ways. One such generalisation is the equivalence between lax double functors into Span, from a small category B, and ordinary functors into B. Delta lenses are a generalisation of split opfibrations, where the chosen lifts need not be opcartesian. The purpose of this talk is to investigate the question: is there also some kind of generalised Grothendieck construction which yields delta lenses? The main result establishes an equivalence between lax double functors from a small category B into sMult (the double category of split multi-valued functions) and delta lenses into B. We will also see how several examples of delta lenses, including split opfibrations, may be understood from this perspective.

**Title**: The monoidal fibered category of Beck modules

**Abstract**: In his 1967 thesis, Beck proposed a notion of module over an object in a category C. This provided a natural notion of coefficient module for André-Quillen (co)homology of any algebraic structure, generalizing the original case of commutative rings. Motivated by Quillen homology, I will discuss the tensor product of Beck modules. As one varies the object in C, the categories of Beck modules over different objects assemble into a fibered category over C, sometimes called the tangent category of C. I will describe how this fibered category interacts with the tensor product. Lastly, I will sketch work in progress on the homotopy theory of simplicial Beck modules over simplicial objects, generalizing some work of Quillen on simplicial commutative rings.

Title: (Commutative) theories and (monoidal) monads

Abstract: This is an expository talk on the monad/theory correspondence, following the recent work of Garner-Bourke in the enriched setting and Berger-Mellies-Weber in Set-based categories. If time permits, we will consider how this interacts with Day convolution to demonstrate a generalized commutative monad/commutative theory correspondence.

**Title**: Field theories in synthetic differential geometry

**Abstract: **

A proper mathematical treatment of field theories requires a good understanding and handling of the involved differentiable structures. In this talk we will explore the possibility to formulate field theories in the language of synthetic differential geometry. For (nonlinear) scalar field theories this approach was suggested and carried out by Marco Benini and Alexander Schenkel. They chose the Cahier topos as a well adapted model and were able to induce a natural differentiable structure on the solution space of the theories in question. I will review some of their ideas. On the way I will explain some basics of field theory in order to give a context for those who are unfamiliar with it.

The Cahiers topos is a category of set valued sheafs. This is not the right data to describe the fields of Yang Mills theory with. Following ideas of Benini, Schenkel and Urs Schreiber we will explore how a natural formulation would involve groupoid valued presheaves satisfying a homological descent condition.

**Title**: Discrete Double Fibrations

**Abstract: **Discrete fibrations over a small category correspond to presheaves on that small category by a category of elements construction. R. Paré proposes that certain lax, span-valued double functors serve as the double categorical analogue of ordinary presheaves and gives an associated category of elements construction. The question thus arises as to whether there is a corresponding notion of “discrete double fibration” and what kind of equivalence between these and lax, span-valued double functors can be obtained. In this talk, we shall review some versions of known elements constructions, see how the double category of elements fits into this pattern, study its fibration properties and see how these lead to a definition.