Polycategories via pseudo-distributive laws

AbstractIn this paper, we give a novel abstract description of Szabo's polycategories. We use the theory of double clubs – a generalisation of Kelly's theory of clubs to ‘pseudo’ (or ‘weak’) double categories – to construct a pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ of this pseudo-distributive law are precisely symmetric polycategories

LNL polycategories and doctrines of linear logic

We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as LNL polycategories, including LNL adjunctions, linear exponential comonads, LNL multicategories, IL-indexed categories, linearly distributive categories with storage, commutative and strong monads, CBPV-structures, models of polarized calculi, Freyd-categories, and skew multicategories, as well as ordinary cartesian, symmetric, and planar multicategories and monoidal categories, symmetric polycategories, and linearly distributive and *-autonomous categories. To study such classes of structures uniformly, we define a notion of LNL doctrine, such that each of these classes of structures can be identified with the algebras for some such doctrine. We show that free algebras for LNL doctrines can be presented by a sequent calculus, and that every morphism of doctrines induces an adjunction between their 2-categories of algebras

Compositional frameworks for supermaps and causality

Quantum supermaps are transformations of quantum processes, and have found many applications in quantum foundations and quantum information theory in the past two decades, particularly in the study of causality. Whilst the concept of a supermap is a simple and intuitive one, the current state-of-the-art formalisations of supermaps cannot be applied to arbitrary Hilbert spaces or Operational Probabilistic Theories (OPTs). We review the standard approaches to defining supermaps in quantum theory, wherever possible highlighting the background compositional principles at play using diagrammatic languages and referring to their algebraic formalisation in the field of category theory. The core argument of this thesis is that a more principled and general approach to defining quantum supermaps exists, using a definition of locally-applicable transformation, which can be applied to any symmetric monoidal category. As a consequence this approach can be applied to all quantum processes on general quantum degrees of freedom and to all transformations in OPTs. We identify key compositional features for entire theories of supermaps and show that the supermaps of those theories are always operationally described by locally-applicable transformations. Two tests for a good construction of supermaps on symmetric monoidal categories are identified, recovery of standard physicists definitions for quantum supermaps when applied to categories of standard quantum processes, and existence of key compositional features. By the end of the thesis we find a way to strengthen locally-applicable transformations to construct the theory of polyslots, which passes both tests. Applications of this new general framework for the study of quantum causality and quantum information theory are identified as future potential research directions

Bifibrations of polycategories and classical multiplicative linear logic

In this thesis, we develop the theory of bifibrations of polycategories. We start by studying how to express certain categorical structures as universal properties by generalising the shape of morphism. We call this phenomenon representability and look at different variations, namely the correspondence between representable multicategories and monoidal categories, birepresentable polycategories and $\ast$-autonomous categories, and representable virtual double categories and double categories. We then move to introduce (bi)fibrations for these structures. We show that it generalises representability in the sense that these structures are (bi)representable when they are (bi)fibred over the terminal one. We show how to use this theory to lift models of logic to more refined ones. In particular, we illustrate it by lifting the compact closed structure of the category of finite dimensional vector spaces and linear maps to the (non-compact) $\ast$-autonomous structure of the category of finite dimensional Banach spaces and contractive maps by passing to their respective polycategories. We also give an operational reading of this example, where polylinear maps correspond to operations between systems that can act on their inputs and whose outputs can be measured/probed and where norms correspond to properties of the systems that are preserved by the operations. Finally, we recall the B\'enabou-Grothendieck correspondence linking fibrations to indexed categories. We show how the B-G construction can be defined as a pullback of virtual double categories and we make use of fibrational properties of vdcs to get properties of this pullback. Then we provide a polycategorical version of the B-G correspondence.Comment: 250 pages, 15 figures, PhD thesis in the Theory Group at the Computer Science School of the University of Birmingham under the supervision of Noam Zeilberger and Paul Lev

The algebra of entanglement and the geometry of composition

String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces". Operations of polygraphs modelled on operations of topological spaces are used as the foundation of a compositional universal algebra, where sliding moves arise from tensor products of polygraphs. We reconstruct several higher algebraic theories in this framework. In this regard, the standard formalism of polygraphs has some technical problems. We propose a notion of regular polygraph, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension. We define a category of non-degenerate shapes, and show how to calculate their tensor products. Then, we introduce a notion of weak unit to recover weakly degenerate boundaries in low dimensions, and prove that the existence of weak units is equivalent to a representability property. We then turn to applications of diagrammatic algebra to quantum theory. We re-evaluate the category of Hilbert spaces from the perspective of categorical universal algebra, which leads to a bicategorical refinement. Then, we focus on the axiomatics of fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has several advantages over ZX calculi, including a computationally meaningful normal form, and a fragment whose diagrams can be read as setups of fermionic oscillators. Moreover, its generators reflect an operational classification of entangled states of 3 qubits. We conclude with generalisations of the ZW calculus to higher-dimensional systems, including the definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures; University of Oxford doctoral thesi

Bifibrations of polycategories and classical multiplicative linear logic

In this thesis, we develop the theory of bifibrations of polycategories. We start by studying how to express certain categorical structures as universal properties by generalising the shape of morphism. We call this phenomenon representability and look at different variations, namely the correspondence between representable multicategories and monoidal categories, birepresentable polycategories and *-autonomous categories, and representable virtual double categories and double categories. We then move to introduce (bi)fibrations for these structures. We show that it generalises representability in the sense that these structures are (bi)representable when they are (bi)fibred over the terminal one. We show how to use this theory to lift models of logic to more refined ones. In particular, we illustrate it by lifting the compact closed structure of the category of finite dimensional vector spaces and linear maps to the (non-compact) *-autonomous structure of the category of finite dimensional Banach spaces and contractive maps by passing to their respective polycategories. We also give an operational reading of this example, where polylinear maps correspond to operations between systems that can act on their inputs and whose outputs can be measured/probed and where norms correspond to properties of the systems that are preserved by the operations. Finally, we recall the Bénabou-Grothendieck correspondence linking fibrations to indexed categories. We show how the B-G construction can be defined as a pullback of virtual double categories and we make use of fibrational properties of vdcs to get properties of this pullback. Then we provide a polycategorical version of the B-G correspondence

Higher operads, higher categories

Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. This is the first book on the subject and lays its foundations. Many examples are given throughout. There is also an introductory chapter motivating the subject for topologists.Comment: Book, 410 page

A Monadic Approach to Polycategories

Polycategories should form a rather natural generalization of multicategories: besides the domains also the codomains of morphisms are allowed to be strings of objects. But while small multicategories can be elegantly characterized as monoids in a bicategory of set -spans with free set -monoids as domains, no such description of small polycategories seems to have been known so far. To addres