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**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).

Title: Isotropy Groups of Quasi-Equational Theories

Abstract: In [2], my PhD supervisors (Pieter Hofstra and Philip Scott) and I studied the new topos-theoretic phenomenon of isotropy (as introduced in [1]) in the context of single-sorted algebraic theories, and we gave a logical/syntactic characterization of the

isotropy group of any such theory, thereby showing that it encodes a notion of inner automorphism or conjugation for the theory. In the present talk, I will summarize the results of my recent

PhD thesis, in which I build on this earlier work by studying the isotropy groups of (multi-sorted) quasi-equational theories (also known as essentially algebraic, cartesian, or finite limit theories). In particular, I will show how

to give a logical/syntactic characterization of the isotropy group of any such theory, and that it encodes a notion of inner automorphism or conjugation for the theory. I will also describe how I have used this characterization to exactly

characterize the ‘inner automorphisms’ for several different examples of quasi-equational theories, most notably the theory of strict monoidal categories and the theory of presheaves valued in a category of models. In particular, the latter example provides a characterization of the (covariant) isotropy group of a category of set-valued presheaves, which had been an open question in the theory of categorical isotropy.

[1] J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012.

[2] P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018.

**Title**: Paradigms of composition

**Abstract**: Scientists and engineers manage the blistering complexity of the natural world by telling simplified stories which we call systems. Every different field has a different notion of system tailored the kinds of questions its practitioners are asking, and the kinds of interventions they seek to make. Electrical engineers use circuit diagrams, epidemiologists and chemists use Petri nets, biologists use continuous time Markov processes, physicists use Lagrangians and Hamiltonians, computer scientists use deterministic automata, decision theorists use Markov decision processes, and everyone uses systems of differential equations.

These fields also have a sense of how their systems can be put together — composed — to form more complex systems. Circuits may be plugged into one another, Petri nets may share species, Markov processes may share places, Lagrangian and Hamiltonian systems may share variables, the output of one automaton may be fed into the input of another, and parameters of differential equations may be set by the variables of other differential equations. In each of these paradigms of composition, there are many different specific doctrines of system: circuits, Petri nets, Markov processes in the port plugging paradigm; Hamiltonian and Lagrangian mechanics in the variable sharing paradigm; deterministic automata, Markov decision processes, and systems of differential equations in the input/output paradigm; etc.

Applied category theorists have set out to make formal the ways these systems can be composed to study the way the behaviors — trajectories, solutions, steady states, operational semantics, etc — of composite systems relate to the behaviors of component systems. In this talk, I will argue that the basic algebra of composing open systems is a doubly indexed category (a.k.a. double copresheaf), and give examples of doubly indexed categories of systems. These examples are produced through a single, abstract cartesian 2-functor: the vertical slice construction, taking a double functor to a “vertical slice” doubly indexed category. We will then see how behaviors of systems can be organized into lax doubly indexed functors by constructing representables.

Title: “Sheaf representation of monoidal categories”

Abstract: Wouldn’t it be great if monoidal categories were nice and easy? They are! We will discuss how a monoidal category embeds into a “nice” one, and how a “nice” monoidal category consists of global sections of a sheaf of “easy” monoidal categories. Here “nice” means that the idempotent subobjects of the tensor unit have joins that are respected by tensor products, and “easy” means that the topological space of which these subobjects are the opens is local. This theorem subsumes sheaf representation results for toposes, its proof is entirely concrete, and it cleanly separates “spatial” and “temporal” directions of monoidal categories. We will focus mostly on explaining these statements and placing them in context. Joint work with Rui Soares Barbosa and others.

Title: The etale category associated to an algebraic theory

Abstract: Equational algebraic theories are typically used in

mathematics to specify notions such as group, ring, lattice, and so on.

In computer science, by contrast, they are often used to specify

programming language features such as input/output, state, stack, and so

on.

For the mathematical applications, one typically cares about set-based

models. For the computer science applications, greater weight is placed

on the set-based comodels (i.e., models in Set^op), which can be seen as

state machines providing an environment for interpreting the given

language features. This is an idea due to Plotkin, Power and Shkaravska

(in various combinations).

The objective of this talk is to explain that:

1) the category of set-based comodels of an algebraic theory is always a

presheaf category;

2) the category of topological comodels of an algebraic theory is always

a category of continuous presheaves on an etale topological category.

By way of illustration, we explain how certain etale topological

categories, well-known from the study of C*-algebras, arise in this

manner.

Title: Pictures of finite limits

Abstract: In (classical) Lawvere theories, the central role is played by categories with finite products. For example, the free category with finite products on one object (FinSet^op) is the Lawvere theory of the empty algebraic theory, and the free category with finite products on a signature of an algebraic theory has a concrete description as a category of terms.

In recent joint work with Ivan di Liberti, Fosco Loregian and Chad Nester, we developed a Lawvere-style approach to algebraic theories with partially defined operations. It turns out that in this setting, instead of categories with finite products, the relevant concept is discrete cartesian restriction categories. We developed the technology to describe free such categories. It is known that splitting idempotents yields categories with finite limits. After describing the main steps of the above narrative I will focus on giving examples of the resulting string diagrammatic calculus for free categories with finite limits.