arXiv subjects

Brendan Fong

Publications and source records attributed to Brendan Fong.

At least 19 recordsLinked to original sources

arXiv2022

Temporal Landscapes: A Graphical Logic of Behavior

We present an elementary introduction to a new logic for reasoning about behaviors that occur over time. This logic is based on temporal type theory. The syntax of the logic is similar to the usual first-order logic; what differs is the notion of truth value. Instead of reasoning about whether formulas are true or false, our logic reasons about temporal landscapes. A temporal landscape may be thought of as representing the set of durations over which a statement is true. To help understand the practical implications of this approach, we give a wide variety of examples where this logic is used to reason about autonomous agents.

math.LO↗

arXiv2022

Extracting Mathematical Concepts from Text

We investigate different systems for extracting mathematical entities from English texts in the mathematical field of category theory as a first step for constructing a mathematical knowledge graph. We consider four different term extractors and compare their results. This small experiment showcases some of the issues with the construction and evaluation of terms extracted from noisy domain text. We also make available two open corpora in research mathematics, in particular in category theory: a small corpus of 755 abstracts from the journal TAC (3188 sentences), and a larger corpus from the nLab community wiki (15,000 sentences).

cs.CL↗

arXiv2021

Regular Calculi I: Graphical Regular Logic

What is ergonomic syntax for relations? In this first paper in a series of two, to answer the question we define regular calculi: a suitably structured functor from a category representing the syntax of regular logic to the category of posets, that takes each object to the poset of relations on that type. We introduce two major classes of examples, regular calculi corresponding to regular theories, and regular calculi corresponding to regular categories. For working in regular calculi, we present a graphical framework which takes as primitive the various moves of regular logic. Our main theorem for regular calculi is a syntax-semantics $2$-dimensional adjunction to regular categories.

math.CT↗

arXiv2021

Behavioral Mereology: A Modal Logic for Passing Constraints

Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos, providing maximal generality; however, by using only its internal logic we can hide the details and readers may assume a completely elementary set-theoretic discussion. We consider the relationship between various parts of a whole in terms of how behavioral constraints are passed between them, and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetric accessibility.

cs.LO↗

arXiv2020

String Diagrams for Regular Logic (Extended Abstract)

Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free regular category FRg(T) on a set T. From this point of view, regular theories are certain monoidal 2-functors from a suitable 2-category of contexts -- the 2-category of relations in FRg(T) -- to that of posets. Such functors assign to each context the set of formulas in that context, ordered by entailment. We refer to such a 2-functor as a regular calculus because it naturally gives rise to a graphical string diagram calculus in the spirit of Joyal and Street. We shall show that every natural category has an associated regular calculus, and conversely from every regular calculus one can construct a regular category.

cs.LO↗

arXiv2020

Behavioral Mereology (Proofs and Properties)

math.LO↗

arXiv2020

Supplying bells and whistles in symmetric monoidal categories

It is common to encounter symmetric monoidal categories $\mathcal{C}$ for which every object is equipped with an algebraic structure, in a way that is compatible with the monoidal product and unit in $\mathcal{C}$. We define this formally and say that $\mathcal{C}$ supplies the algebraic structure. For example, the category $\mathsf{Rel}$ of relations between sets has monoidal structures given by both cartesian product and disjoint union, and with respect to either one it supplies comonoids. We prove several facts about the notion of supply, e.g. that the associators, unitors, and braiding of $\mathcal{C}$ are automatically homomorphisms for any supply, as are the coherence isomorphisms for any strong symmetric monoidal functor that preserve supplies. We also show that any supply of structure in a symmetric monoidal category can be extended to a supply of that structure on its strictification.

math.CT↗

arXiv2019

Regular and relational categories: Revisiting 'Cartesian bicategories I'

Regular logic is the fragment of first order logic generated by $=$, $\top$, $\wedge$, and $\exists$. A key feature of this logic is that it is the minimal fragment required to express composition of binary relations; another is that it is the internal logic of regular categories. The link between these two facts is that in any regular category, one may construct a notion of binary relation using jointly-monic spans; this results in what is known as the bicategory of relations of the regular category. In this paper we provide a direct axiomatization of bicategories of relations, which we term relational po-categories, reinterpreting the earlier work of Carboni and Walters along these lines. Our main contribution is an explicit proof that the 2-category of regular categories is equivalent to that of relational po-categories. Throughout, we emphasize the graphical nature of relational po-categories.

math.CT↗

arXiv2019

Graphical Regular Logic

Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free regular category $\mathsf{FRg}(\mathrm{T})$ on a set $\mathrm{T}$. From this point of view, regular theories are certain monoidal 2-functors from a suitable 2-category of contexts---the 2-category of relations in $\mathsf{FRg}(\mathrm{T})$---to the 2-category of posets. Such functors assign to each context the set of formulas in that context, ordered by entailment. We refer to such a 2-functor as a regular calculus because it naturally gives rise to a graphical string diagram calculus in the spirit of Joyal and Street. Our key aim to prove that the category of regular categories is essentially reflective in that of regular calculi. Along the way, we demonstrate how to use this graphical calculus.

math.CT↗

arXiv2019

Backprop as Functor: A compositional perspective on supervised learning

A supervised learning algorithm searches over a set of functions $A \to B$ parametrised by a space $P$ to find the best approximation to some ideal function $f\colon A \to B$. It does this by taking examples $(a,f(a)) \in A\times B$, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.

math.CT↗

arXiv2019

Lenses and Learners

Lenses are a well-established structure for modelling bidirectional transformations, such as the interactions between a database and a view of it. Lenses may be symmetric or asymmetric, and may be composed, forming the morphisms of a monoidal category. More recently, the notion of a learner has been proposed: these provide a compositional way of modelling supervised learning algorithms, and again form the morphisms of a monoidal category. In this paper, we show that the two concepts are tightly linked. We show both that there is a faithful, identity-on-objects symmetric monoidal functor embedding a category of asymmetric lenses into the category of learners, and furthermore there is such a functor embedding the category of learners into a category of symmetric lenses.

cs.LG↗

arXiv2019

Hypergraph Categories

Hypergraph categories have been rediscovered at least five times, under various names, including well-supported compact closed categories, dgs-monoidal categories, and dungeon categories. Perhaps the reason they keep being reinvented is two-fold: there are many applications---including to automata, databases, circuits, linear relations, graph rewriting, and belief propagation---and yet the standard definition is so involved and ornate as to be difficult to find in the literature. Indeed, a hypergraph category is, roughly speaking, a "symmetric monoidal category in which each object is equipped with the structure of a special commutative Frobenius monoid, satisfying certain coherence conditions". Fortunately, this description can be simplified a great deal: a hypergraph category is simply a "cospan-algebra". The goal of this paper is to remove the scare-quotes and make the previous statement precise. We prove two main theorems. First is a coherence theorem for hypergraph categories, which says that every hypergraph category is equivalent to an objectwise-free hypergraph category. Second, we prove that the category of objectwise-free hypergraph categories is equivalent to the category of cospan-algebras.

math.CT↗

arXiv2018

A recipe for black box functors

The task of constructing compositional semantics for network-style diagrammatic languages, such as electrical circuits or chemical reaction networks, has been dubbed the black boxing problem, as it gives semantics that describes the properties of each network that can be observed externally, by composition, while discarding the internal structure. One way to solve these problems is to formalise the diagrams and their semantics using hypergraph categories, with semantic interpretation a hypergraph functor, called the black box functor, between them. Reviewing a principled method for constructing hypergraph categories and functors, known as decorated corelations, in this paper we construct a category of \emph{decorating data}, and show that the decorated corelations method is itself functorial, with a universal property characterised by a left Kan extension. We then argue that the category of decorating data is a good setting in which to construct any hypergraph functor, giving a new construction of Baez and Pollard's black box functor for reaction networks as an example.

math.CT↗

arXiv2018

A Compositional Framework for Passive Linear Networks

Passive linear networks are used in a wide variety of engineering applications, but the best studied are electrical circuits made of resistors, inductors and capacitors. We describe a category where a morphism is a circuit of this sort with marked input and output terminals. In this category, composition describes the process of attaching the outputs of one circuit to the inputs of another. We construct a functor, dubbed the "black box functor", that takes a circuit, forgets its internal structure, and remembers only its external behavior. Two circuits have the same external behavior if and only if they impose same relation between currents and potentials at their terminals. The space of these currents and potentials naturally has the structure of a symplectic vector space, and the relation imposed by a circuit is a Lagrangian linear relation. Thus, the black box functor goes from our category of circuits to the category of symplectic vector spaces and Lagrangian linear relations. We prove that this functor is We prove that this functor is symmetric monoidal and indeed a hypergraph functor. We assume the reader is familiar with category theory, but not with circuit theory or symplectic linear algebra.

math.CT↗

arXiv2018

Seven Sketches in Compositionality: An Invitation to Applied Category Theory

This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each pairing an evocative application, such as databases, electric circuits, or dynamical systems, with the exploration of a categorical structure, such as adjoint functors, enriched categories, or toposes. No prior knowledge of category theory is assumed. A feedback form for typos, comments, questions, and suggestions is available here: https://docs.google.com/document/d/160G9OFcP5DWT8Stn7TxdVx83DJnnf7d5GML0_FOD5Wg/edit

math.CT↗

arXiv2018

Universal Constructions for (Co)Relations: categories, monoidal categories, and props

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature and are relevant for program semantics, quantum computation and control theory.

cs.LO↗

arXiv2017

A Universal Construction for (Co)Relations

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature, including the cases of linear corelations (relevant for the semantics of electrical circuits), of partial equivalence relations and of linear subspaces (semantics of signal flow graphs and of the phase-free ZX calculus).

cs.LO↗

arXiv2017

Decorated Corelations

Let $\mathcal C$ be a category with finite colimits, and let $(\mathcal E,\mathcal M)$ be a factorisation system on $\mathcal C$ with $\mathcal M$ stable under pushouts. Writing $\mathcal C;\mathcal M^{\mathrm{op}}$ for the symmetric monoidal category with morphisms cospans of the form $\stackrel{c}\to \stackrel{m}\leftarrow$, where $c \in \mathcal C$ and $m \in \mathcal M$, we give method for constructing a category from a symmetric lax monoidal functor $F\colon (\mathcal C; \mathcal M^{\mathrm{op}},+) \to (\mathrm{Set},\times)$. A morphism in this category, termed a \emph{decorated corelation}, comprises (i) a cospan $X \to N \leftarrow Y$ in $\mathcal C$ such that the canonical copairing $X+Y \to N$ lies in $\mathcal E$, together with (ii) an element of $FN$. Functors between decorated corelation categories can be constructed from natural transformations between the decorating functors $F$. This provides a general method for constructing hypergraph categories---symmetric monoidal categories in which each object is a special commutative Frobenius monoid in a coherent way---and their functors. Such categories are useful for modelling network languages, for example circuit diagrams, and such functors their semantics.

math.CT↗