arXiv subjects

Tom Leinster

Publications and source records attributed to Tom Leinster.

At least 19 recordsLinked to original sources

arXiv2025

Equivalence via surjections

Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ with $B$ when there exists a strictly structure-preserving map $A \to B$ that is genuinely (not just essentially) surjective in each dimension and faithful in the top dimension. We prove this principle for four types of structure: categories, monoidal categories, bicategories and double categories. The last of these theorems suggests that the right notion of equivalence between double categories is Campbell's gregarious double equivalence, a conclusion also reached for different reasons in recent work of Moser, Sarazola and Verdugo.

math.CT↗

arXiv2024

Galois Theory

These are the notes for an undergraduate course at the University of Edinburgh, 2021-2023. Assuming basic knowledge of ring theory, group theory and linear algebra, the notes lay out the theory of field extensions and their Galois groups, up to and including the fundamental theorem of Galois theory. Also included are a section on ruler and compass constructions, a proof that solvable polynomials have solvable Galois groups, and the classification of finite fields.

math.NT↗

arXiv2024

Magnitude homology equivalence of Euclidean sets

Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense that there exist back-and-forth maps inducing mutually inverse maps in homology? We give a concrete geometric necessary and sufficient condition in the case of closed Euclidean sets. Along the way, we introduce the convex-geometric concepts of inner boundary and core, and prove a strengthening for closed convex sets of the classical theorem of Carath\'eodory.

math.MG↗

arXiv2024

The eventual image

In a category with enough limits and colimits, one can form the universal automorphism on an endomorphism in two dual senses. Sometimes these dual constructions coincide, as in the categories of finite sets, finite-dimensional vector spaces, and compact metric spaces. There, beginning with an endomorphism $f$, there is a doubly-universal automorphism on $f$ whose underlying object is the eventual image $\bigcap_n \mathrm{im}(f^n)$. Our main theorem unifies these examples, stating that in any category with a factorization system satisfying certain axioms, the eventual image has two dual universal properties. A further theorem characterizes the eventual image as a terminal coalgebra. In all, nine characterizations of the eventual image are given, valid at different levels of generality.

math.CT↗

arXiv2023

A categorical derivation of Lebesgue integration

We identify simple universal properties that uniquely characterize the Lebesgue $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0, 1]$, equipped with a small amount of extra structure, is initial as such. The second states that the $L^p$ functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces $\ell^p$ and $c_0$, as well as the functor $L^2$ taking values in Hilbert spaces.

math.FA↗

arXiv2021

Spaces of extremal magnitude

Magnitude is a numerical invariant of compact metric spaces. Its theory is most mature for spaces satisfying the classical condition of being of negative type, and the magnitude of such a space lies in the interval $[1, \infty]$. Until now, no example with magnitude $\infty$ was known. We construct some, thus answering a question open since 2010. We also give a sufficient condition for the magnitude of a space to converge to 1 as it is scaled down to a point, unifying and generalizing previously known conditions.

math.MG↗

arXiv2021

Isbell conjugacy and the reflexive completion

The reflexive completion of a category consists of the Set-valued functors on it that are canonically isomorphic to their double conjugate. After reviewing both this construction and Isbell conjugacy itself, we give new examples and revisit Isbell's main results from 1960 in a modern categorical context. We establish the sense in which reflexive completion is functorial, and find conditions under which two categories have equivalent reflexive completions. We describe the relationship between the reflexive and Cauchy completions, determine exactly which limits and colimits exist in an arbitrary reflexive completion, and make precise the sense in which the reflexive completion of a category is the intersection of the categories of covariant and contravariant functors on it.

math.CT↗

arXiv2020

The maximum entropy of a metric space

We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies generalise the Shannon and Rényi entropies of information theory. We prove that on any space X, there is a single probability measure maximising all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows; its asymptotics determine geometric information about X, including the volume and dimension. We also study the large-scale limit of the maximising measure itself, arguing that it should be regarded as the canonical or uniform measure on X. Primarily we work not with entropy itself but its exponential, called diversity and (in its finite form) used as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.

math.MG↗

arXiv2020

Entropy and Diversity: The Axiomatic Approach

This book brings new mathematical rigour to the ongoing vigorous debate on how to quantify biological diversity. The question "what is diversity?" has surprising mathematical depth, and breadth too: this book involves parts of mathematics ranging from information theory, functional equations and probability theory to category theory, geometric measure theory and number theory. It applies the power of the axiomatic method to a biological problem of pressing concern, but the new concepts and theorems are also motivated from a purely mathematical perspective. The main narrative thread requires no more than an undergraduate course in analysis. No familiarity with entropy or diversity is assumed.

q-bio.PE↗

arXiv2020

Entropy modulo a prime

Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability distribution in which the "probabilities" are integers modulo a prime p. The entropy, too, is an integer mod p. Entropy mod p is shown to be uniquely characterized by a functional equation identical to the one that characterizes ordinary Shannon entropy. We also establish a sense in which certain real entropies have residues mod p, connecting the concepts of entropy over R and over Z/pZ. Finally, entropy mod p is expressed as a polynomial which is shown to satisfy several identities, linking into work of Cathelineau, Elbaz-Vincent and Gangl on polylogarithms.

math.NT↗

arXiv2020

Magnitude homology of enriched categories and metric spaces

Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories, which we call magnitude homology (in fact, it is a special sort of Hochschild homology), whose graded Euler characteristic is the magnitude. Magnitude homology of metric spaces generalizes the Hepworth--Willerton magnitude homology of graphs, and detects geometric information such as convexity.

math.AT↗

arXiv2020

The probability that an operator is nilpotent

Choose a random linear operator on a vector space of finite cardinality N: then the probability that it is nilpotent is 1/N. This is a linear analogue of the fact that for a random self-map of a set of cardinality N, the probability that some iterate is constant is 1/N. The first result is due to Fine, Herstein and Hall, and the second is essentially Cayley's tree formula. We give a new proof of the result on nilpotents, analogous to Joyal's beautiful proof of Cayley's formula. It uses only general linear algebra and avoids calculation entirely.

math.CO↗

arXiv2019

Weak $\infty$-categories via terminal coalgebras

Higher categorical structures are often defined by induction on dimension, which a priori produces only finite-dimensional structures. In this paper we show how to extend such definitions to infinite dimensions using the theory of terminal coalgebras, and we apply this method to Trimble's notion of weak n-category. Trimble's definition makes explicit the relationship between n-categories and topological spaces; our extended theory produces a definition of Trimble infinity-category and a fundamental infinity-groupoid construction. Furthermore, terminal coalgebras are often constructed as limits of a certain type. We prove that the theory of Batanin-Leinster weak infinity-categories arises as just such a limit, justifying our approach to Trimble infinity-categories. In fact we work at the level of monads for infinity-categories, rather than infinity-categories themselves; this requires more sophisticated technology but also provides a more complete theory of the structures in question.

math.CT↗

arXiv2017

A short characterization of relative entropy

We prove characterization theorems for relative entropy (also known as Kullback-Leibler divergence), q-logarithmic entropy (also known as Tsallis entropy), and q-logarithmic relative entropy. All three have been characterized axiomatically before, but we show that earlier proofs can be simplified considerably, at the same time relaxing some of the hypotheses.

cs.IT↗

arXiv2016

Basic Category Theory

This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.

math.CT↗

arXiv2016

How to partition diversity

Diversity measurement underpins the study of biological systems, but measures used vary across disciplines. Despite their common use and broad utility, no unified framework has emerged for measuring, comparing and partitioning diversity. The introduction of information theory into diversity measurement has laid the foundations, but the framework is incomplete without the ability to partition diversity, which is central to fundamental questions across the life sciences: How do we prioritise communities for conservation? How do we identify reservoirs and sources of pathogenic organisms? How do we measure ecological disturbance arising from climate change? The lack of a common framework means that diversity measures from different fields have conflicting fundamental properties, allowing conclusions reached to depend on the measure chosen. This conflict is unnecessary and unhelpful. A mathematically consistent framework would transform disparate fields by delivering scientific insights in a common language. It would also allow the transfer of theoretical and practical developments between fields. We meet this need, providing a versatile unified framework for partitioning biological diversity. It encompasses any kind of similarity between individuals, from functional to genetic, allowing comparisons between qualitatively different kinds of diversity. Where existing partitioning measures aggregate information across the whole population, our approach permits the direct comparison of subcommunities, allowing us to pinpoint distinct, diverse or representative subcommunities and investigate population substructure. The framework is provided as a ready-to-use R package to easily test our approach.

q-bio.QM↗

arXiv2016

The magnitude of a metric space: from category theory to geometric measure theory

Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral geometry and geometric measure theory, including volume, capacity, dimension, and intrinsic volumes. This paper will give an overview of the theory of magnitude, from its category-theoretic genesis to its connections with these geometric quantities.

math.MG↗

arXiv2016

On the magnitude of a finite dimensional algebra

There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic and dimension. Here we establish its significance in an algebraic context. Specifically, in the representation theory of an associative algebra A, a central role is played by the indecomposable projective A-modules, which form a category enriched in vector spaces. We show that the magnitude of that category is a known homological invariant of the algebra: writing chi_A for the Euler form of A and S for the direct sum of the simple A-modules, it is chi_A(S, S).

math.RA↗