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**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.

**Title**: Categorical Semantics of the ZX-calculus

**Abstract**: The ZX-calculus is a graphical language for qubit quantum circuits. In other words, it is a presentation for the full subcategory of complex matrices under the bilinear tensor product, where the objects are powers of 2. A consequence of Zanasi’s thesis is that the prop of linear spans over F_2 is equivalent to the phase free fragment of the ZX-calculus. We extend this correspondence to the affine and nonlinear cases. In the former case, we show that the fragment of the ZX-calculus with one pi-phase is a presentation for the full subcategory of spans of finite dimensional F_2-affine vector spaces, where the objects are non-empty affine vector spaces. In the latter case, we show that the fragment of the ZH-calculus with natural number H-boxes is a presentation for the full subcategory of spans of sets of finite functions where the objects are powers of 2 (the ZH -calculus is equivalent to the ZX-calculus). We must consider these full subcategories of spans because in these cases, because unlike in the linear case, the full categories of spans are not themselves props, having too many objects. These results are proven as modularly as possible, incrementally adding generators via pushout and distributive laws of props.

**Title**: Characterizing Cofree Cartesian Differential Categories

**Abstract**: Cartesian differential categories come equipped with a differential operator which formalizes the derivative from multivariable calculus. There has recently been renewed interest in cofree Cartesian differential categories. For any Cartesian left additive category X there exists a cofree Cartesian differential category Faa(X) over it, which satisfies the expected couniversal property, and this construction is known as the Faa di Bruno construction. A natural question to ask is whether well-known examples of Cartesian differential categories?

Therefore, we would like to answer the following: starting with only an arbitrary Cartesian differential category, how can we check if it is a cofree Cartesian differential category without knowing the base Cartesian left additive category?

In this talk, we will provide a characterization of cofree Cartesian differential categories using only internal structure, that is, as categories enriched over complete ultrametric spaces (where the metric is similar to that of power series) and whose base Cartesian left additive category is induced by maps whose derivative is zero. A consequence of this result is that the induced cofree Cartesian differential category comonad is of effective descent type. Furthermore, we also explain how many well-known Cartesian differential categories are NOT cofree.

This talk should be accessible to everyone! Even those unfamiliar with differential categories.

**Title**: Measurement for Mixed Unitary Categories

**Abstract:**

Mixed Unitary Categories (MUCs) [1] provide a generalization for the finite-dimensional categorical quantum mechanic framework of dagger compact closed categories (dagger KCCs) by introducing dagger structure to Linearly Distributive Categories (LDCs). The goal of this generalization is to develop a framework that will accommodate quantum systems of arbitrary dimensions without forgoing the rich structures of dagger-KCCs. In our previous work, we demonstrated that one can describe quantum processes a.k.a. completely positive maps in the MUC framework. In this talk, I will show how one can describe quantum measurements with this framework. We observe that in the MUC framework, a measurement occurs in two steps – compaction into a unitary core followed by traditional measurement. We also note that, while compacting, structures on the domain type can be transferred to the codomain type. Finally, in alignment with the purpose of MUCs, we note that in the presence of free exponential modalities, every pair of complementary measurements within a unitary core, arises as a compaction of a ‘linear bialgbera’ on exponential modalities.

References:

[1]. Cockett, Robin, Cole Comfort, and Priyaa Srinivasan. “Dagger linear logic for categorical quantum mechanics.” *arXiv preprint arXiv:1809.00275* (2018).