Tom Leinster


I am a mathematician at the University of Edinburgh. I am also a member of the Boyd Orr Centre for Population and Ecosystem Health and a host of The n-Category Café.


New post: Magnitude 2023, with copious notes from the talks Cover of Entropy and Diversity
New talks: The many faces of magnitude (Osaka colloquium) and Magnitude homology equivalence of Euclidean sets (for Magnitude 2023)
New talk: Magnitude cohomology of graphs (on work of Richard Hepworth)
New talk: Magnitude homology (for Category Theory 2023)
Updated paper: A categorical derivation of Lebesgue integration; associated post
New talk: Magnitude homology (for Clemens Berger's birthday conference)
Why are we on strike?
Two new talks: Measuring diversity (RIKEN, Japan)
Updated book: Entropy and Diversity: The Axiomatic Approach (new arXiv version correcting a small number of minor errors)
New book! Entropy and Diversity: The Axiomatic Approach, now in bookshops and online; celebratory announcement post and arXiv announcement post

Most of my work over the last decade has been part of a large, wide-reaching project on magnitude. Here's a bibliography of all the works on magnitude that I know about.

Campaigning etc:

Why are we on strike?
Graph of Covid-19 cases at the University of Edinburgh
Here I am on the front pages of the National, the Herald and the Scotsman (and way back on p.21 of the Guardian), demonstrating in support of St Andrews professor and former Catalan education minister Clara Ponsatí. What's going on in Catalonia? Try this short explanation and, sadly, this.
My writings on mathematics and mass surveillance: list of articles here and my New Scientist article here




Books (all available free online):

Entropy and Diversity: The Axiomatic Approach was published by Cambridge University Press in April 2021
Basic Category Theory was published by Cambridge University Press in 2014
Higher Operads, Higher Categories was published by Cambridge University Press in 2004

Here are my papers, grouped by subject.


Spaces of extremal magnitude (with Mark Meckes), arXiv:2112.12889, 7 pages, 2021
The maximum entropy of a metric space (with Emily Roff), arXiv:1908.11184; also Quarterly Journal of Mathematics 72 (2021), 1271–1309
Entropy modulo a prime, arXiv:1903.06961; also Communications in Number Theory and Physics 15(2) (2021), 279–314; discussion
A short characterization of relative entropy, arXiv:1712.04903; also Journal of Mathematical Physics 60 (2019), Issue 2, 023302; discussion
Magnitude homology of enriched categories and metric spaces (with Michael Shulman), arXiv:1711.00802 (with major rewrite in 2020); also Algebraic & Geometric Topology 21 (2021), 2175–2221; discussion
The magnitude of a metric space: from category theory to geometric measure theory (with Mark Meckes), arXiv:1606.00095; also in Nicola Gigli (ed.), Measure Theory in Non-Smooth Spaces, de Gruyter Open, 2017
Maximizing diversity in biology and beyond (with Mark Meckes), arXiv:1512.06314; also Entropy 18 (2016), article 88
On the magnitude of a finite dimensional algebra, (with Joseph Chuang and Alastair King), arXiv:1505.04281; also Theory and Applications of Categories 31 (2016), 63–72
The magnitude of a graph, arXiv:1401.4623; also Mathematical Proceedings of the Cambridge Philosophical Society 166 (2019), 247–264; discussions (1, 2, 3)
Codensity and the ultrafilter monad, arXiv:1209.3606; also Theory and Applications of Categories 28 (2013), 332–370; discussions (1, 2, 3, 4)
Notions of Möbius inversion, arXiv:1201.0413; also Bulletin of the Belgian Mathematical Society—Simon Stevin 19 (2012), 911–935; discussion
Measuring diversity: the importance of species similarity (with Christina Cobbold), Ecology 93 (2012), 477–489
A characterization of entropy in terms of information loss (with John Baez and Tobias Fritz), arXiv:1106.1791; also Entropy 13 (2011), no. 11, 1945–1957; discussion
A multiplicative characterization of the power means, arXiv:1103.2574; also Bulletin of the London Mathematical Society 44 (2012), 106–112; discussion
The magnitude of metric spaces, arXiv:1012.5857; also Documenta Mathematica 18 (2013), 857–905; discussion
Integral geometry for the 1-norm, arXiv:1012.5881; also Advances in Applied Mathematics 49 (2012), 81–96; discussion
A maximum entropy theorem with applications to the measurement of biodiversity, arXiv:0910.0906, 27 pages, 2009
On the asymptotic magnitude of subsets of Euclidean space (with Simon Willerton), arXiv:0908.1582; Geometriae Dedicata 164 (2013), 287–310; discussion
The Euler characteristic of a category as the sum of a divergent series (with Clemens Berger), arXiv:0707.0835; also Homology, Homotopy and Applications 10(1) (2008), 41–51; discussion
The Euler characteristic of a category, math.CT/0610260; also Documenta Mathematica 13 (2008), 21–49; nice description here and discussion here and here

Self-similarity and recursion:

A general theory of self-similarity, arXiv:1010.4474; also Advances in Mathematics 226 (2011), 2935–3017. Supersedes the preprints 'A general theory of self-similarity I' and 'A general theory of self-similarity II'
General self-similarity: an overview, math.DS/0411343, 10 pages, 2004; also in Real and Complex Singularities (Proceedings of the Australian-Japanese Workshop, Sydney, 2005), World Scientific (2007)
A general theory of self-similarity II: recognition, math.DS/0411345, 28 pages, 2004. An updated version of this, merged with part I, is published as 'A general theory of self-similarity' (above)
A general theory of self-similarity I, math.DS/0411344, 49 pages, 2004. An updated version of this, merged with part II, is published as 'A general theory of self-similarity' (above)
Objects of categories as complex numbers (with Marcelo Fiore), math.CT/0212377; also Advances in Mathematics 190 (2005), 264-277
An objective representation of the Gaussian integers (with Marcelo Fiore), math.RA/0211454; also Journal of Symbolic Computation 37 (2004), no. 6, 707-716

Category theory in algebra:

An abstract characterization of Thompson's group F (with Marcelo Fiore), math.GR/0508617; also Semigroup Forum 80 (2010), 325–340; informal explanation and discussion
Are operads algebraic theories?, math.CT/0404016; also Bulletin of the London Mathematical Society 38 (2006), no. 2, 233–238

Higher category theory:

Weak ∞-categories via terminal coalgebras (with Eugenia Cheng), arXiv:1212.5853; also Theory and Applications of Categories 34 (2019), no. 34, 1073–1133
A survey of definitions of n-category, math.CT/0107188; also Theory and Applications of Categories 10 (2002), no. 1, 1–70
Topology and higher-dimensional category theory: the rough idea, math.CT/0106240, 15 pages, 2001
Operads in higher-dimensional category theory (PhD thesis), math.CT/0011106; also Theory and Applications of Categories 12 (2004), no. 3, 73–194
Generalized enrichment of categories, math.CT/0204279; also Journal of Pure and Applied Algebra 168 (2002), 391–406
fc-multicategories, math.CT/9903004, 8 pages, 1999
Generalized enrichment for categories and multicategories, math.CT/9901139, 79 pages, 1999
Basic bicategories, math.CT/9810017, 11 pages, 1998
Structures in higher-dimensional category theory, math.CT/0109021, 81 pages, 1998
General operads and multicategories, math.CT/9810053, 35 pages, 1997

(Polished versions of most of the material in these unpublished papers can be found in my book.)

Homotopy algebra:

Homotopy algebras for operads, math.QA/0002180, 101 pages, 2000
Up-to-homotopy monoids, math.QA/9912084, 8 pages, 1999


The eventual image, arXiv:2210.00302, 39 pages, 2022
Isbell conjugacy and the reflexive completion (with Tom Avery), arXiv:2102.08290; also Theory and Applications of Categories 36 (2021), 306–347
A categorical derivation of Lebesgue integration, arXiv:2011.00412, 24 pages, 2020; also Journal of the London Mathematical Society 107 (6) (2023), 1959–1982; discussions (1, 2)
The probability that an operator is nilpotent, arXiv:1912.12562; also American Mathematical Monthly 128 (4) (2021), 371–375
The bijection between projective indecomposable and simple modules, arXiv:1410.3671, 10 pages, 2014; also Bulletin of the Belgian Mathematical Society—Simon Stevin 22 (2015), 725–735
Notes on commutation of limits and colimits (with Marie Bjerrum, Peter Johnstone and Will Sawin), arXiv:1409.7860, 5 pages, 2014; also Theory and Applications of Categories 30 (2015), 527–532
Rethinking set theory, arXiv:1212.6543, 8 pages, 2012; also American Mathematical Monthly 121 (2014), no. 5, 403–415
An informal introduction to topos theory, arXiv:1012.5647, 27 pages; also Publications of the nLab 1 (2011), no. 1
Perfect numbers and groups, math.GR/0104012, 12 pages, 1996ish; associated Sloane's integer sequence

For other writing (reviews, magazine articles, blog posts, ...), see Notes below.




Here are slides and notes from some talks, grouped by subject. Within each subject, the most recent are listed first.


Magnitude cohomology of graphs (for a mixed mathematical audience)
Magnitude homology (short talk for category theorists)
Magnitude homology (long for category theorists and topologists)
Measuring diversity (two talks for a mixed scientific audience)
The mathematics of diversity (for mathematicians)
What is the uniform distribution? (for statisticians and data scientists)
New invariants of metric spaces: magnitude and maximum diversity (for geometers)
The magnitude of graphs and finite metric spaces (for topologists and neuroscientists)
Unexpected connections (for undergraduates)
Magnitude (for category theorists)
Magnitude and magnitude homology (for pure mathematicians)
Magnitude homology (for applied algebraic topologists)
Magnitude in geometry: invariants old and new (for a general mathematical audience)
Maximizing biological diversity (for information theorists and biologists)
The Euler characteristic of an algebra (for algebraists)
The Euler characteristic of an algebra (for category theorists)
The place of diversity in pure mathematics (for a life sciences audience)
The many faces of magnitude (for a general mathematical audience)
The Convex Magnitude Conjecture (for a general mathematical audience)
Entropy, diversity and magnitude (for a general mathematical audience)
Measuring diversity: the importance of species similarity (for ecologists)
Integral geometry for the 1-norm (for convex and integral geometers)
Notions of Möbius inversion (for category theorists)
The magnitude of metric spaces I (for integral geometers)
Magnitude and diversity: how an invariant from category theory solves a problem in mathematical ecology (for category theorists)
Size (for a pure-mathematical audience, mostly model theorists)
Counting, measure and metrics (for a general mathematical audience)
How to measure almost anything (for a general scientific audience)
The cardinality of a metric space (shorter)
The cardinality of a metric space (longer)
New perspectives on Euler characteristic (for a general mathematical audience)
The Euler characteristic of a category (for category theorists)
Another look at Euler characteristic (for a general pure-mathematical audience)

Self-similarity and recursion:

The categorical origins of Lebesgue integration
The eventual image
Terminal coalgebras via modules
Coalgebraic topology
Periodicity of spaces of walks
Jónsson-Tarski toposes
Self-similarity and recursion
A universal Banach space

Category theory in algebra:

Codensity monads
Codensity and the ultrafilter monad
The Thompson groups
Nerves of algebras (see also this discussion)

Higher category theory:

Introduction to higher (especially globular) operads
A survey of the theory of bicategories
Operads (90-minute tutorial)
Higher-dimensional algebra (for politicians)


Large sets (mostly for category theorists)
The functoriality of the reflexive completion (for category theorists)
Cayley's theorem and its linearization
The Snowden revelations, five years on (for Ethics in Mathematics 1, Cambridge)
The reflexive completion (joint work with Tom Avery)
The legacy of Turing's work at Bletchley Park (short talk for general public)
The mathematics of biodiversity (public lecture)
Unexpected connections (for beginning PhD students)
Rethinking set theory
The power of abstract thinking (for prospective PhD students)
The peculiar traits of human mathematics

Talk-related, but not actually talks:

The Edinburgh category theory seminar
The categorical late lunch
The categorical lunch
The Scottish Category Theory Seminar
Research programme: The mathematics of biodiversity at the Centre de Recerca Matemàtica, Barcelona, 18 June–20 July 2012. Activities page here. The programme included a five-day exploratory conference (2–6 July)
Extremely short introduction to Beamer
Conference in celebration of the 60th birthday of my PhD supervisor, Martin Hyland
The 2008 Rankin Lectures, given in Glasgow by John Baez
The 83rd Peripatetic Seminar on Sheaves and Logic (held in Glasgow in 2006); includes some notes from the talks
Category theory seminars in Cambridge, 1996-2002




I'm one of the hosts of The n-Category Café, a research blog on mathematics, physics and philosophy. Here are some of my posts (most recent first):

Magnitude 2023
The eventual image, eventually
Probability monads as codensity monads (on work of Ruben Van Belle)
Ergodic set theory is trivial
Spaces of extremal magnitude
What is the uniform distribution?
Diversity and the mysteries of counting
Entropy and Diversity is out!
Borel determinacy does not require replacement
Large sets 13: summary
Large sets 12: replacement
Large sets 11: measurability, continued
Large sets 10: measurability
Large sets 9: inaccessibility
Large sets 8: cardinal arithmetic
Large sets 7: beth fixed points
Large sets 6: beths
Large sets 5: alephs
Large sets 4: the index of a set
Large sets 3: well-ordered sets
Large sets 2: limits
Large sets 1: introduction
The Just Mathematics Collective
Algebraic closure
The lie of "It's Just Math" (after a blog post by Jade Master)
The uniform measure
Magnitude homology of enriched categories and metric spaces
The categorical origins of Lebesgue integration, revisited
Quotienting out the degenerate
Counting nilpotents: a short paper
A Joyal-type proof of the number of nilpotents
A visual telling of Joyal's proof of Cayley's formula
Counting nilpotents
Geodesic spheres and Gram matrices
What happened at the magnitude workshop
How much work can it be to add some things up?
Entropy mod p
A well ordering is a consistent choice function
The Fisher metric will not be deformed
Entropy modulo a prime (continued)
Entropy modulo a prime
Magnitude homology in Sapporo
Functional equations, entropy and diversity: a seminar course
Functional equations XI: the diversity of a metacommunity
Functional equations X: value
Functional equations IX: entropy on a metric space
Functional equations VIII: measuring biodiversity
Functional equations VII: the p-norms
Functional equations VI: using probability theory to solve functional equations
Functional equations V: expected surprise
Functional equations IV: a simple characterization of relative entropy
Variance in a metric space
Functional equations III: explaining relative entropy
Functional equations II: Shannon entropy
The Heilbronn Institute and the University of Bristol
Functional equations I: Cauchy's equation
The Brunn–Minkowski inequality
Category theory in Barcelona
Basic Category Theory free online
Field notes on the behaviour of a large assemblage of ecologists
Magnitude homology
Monoidal categories with projections
A survey of magnitude
In praise of the Gershgorin disc theorem
How the simplex is a vector space
Weil, venting
Where does the spectrum come from?
Rainer Vogt
How do you handle your email?
Five quickies
More on the AMS and NSA
The AMS must justify its support of the NSA
Effective sample size
Turing's legacy
Graph colouring and cartesian closed categories
Jaynes on mathematical courtesy
New evidence of the NSA deliberately weakening encryption
The atoms of the module world
Why it matters
Holy crap, do you know what a compact ring is?
Wrestling with tight spans
Basic Category Theory
The place of diversity in pure mathematics
Math and mass surveillance: a roundup
The categorical origins of Lebesgue integration
Should mathematicians cooperate with GCHQ? Part 3 (response to Richard Pinch and Malcolm MacCallum)
Categories vs. algebras
Should mathematicians cooperate with GCHQ? Part 2
New Scientist article
Fourier series and flipped classrooms
Should mathematicians cooperate with GCHQ?
The deteriorating relationship between academics and the NSA
The magnitude of a graph
The Electronic Frontier Foundation at the Joint Meetings
Academics Against Mass Surveillance
The long grind of writing a book
Commuting limits and colimits over groups
Severing ties with the NSA
Who ordered that?
Linear operators done right
The Shannon capacity of a graph, 2
The Shannon capacity of a graph, 1
The first commutative diagram?
Whitney twists
Tutte polynomials and magnitude functions
Colouring a graph
Carleson's theorem
Rethinking set theory
Almost all of the first 50 billion groups have order 1024
The Zorn identity
The curious dependence of set theory on order theory
Where do ultraproducts come from?
Where do linearly compact vector spaces come from?
Where do ultrafilters come from?
Where do monads come from?
Jech (1973), page 118
Integrating against the Euler characteristic
log|x| + C
The eventual image, part 2
On the law of large numbers (such as 60)
The eventual image
Measuring diversity, plus version for non-mathematicians on John Baez's blog
Do you know this idempotent?
Spectra of operators and rings
Universal measures
Hadwiger's theorem, part 2
Mixed volume
Definitions of ultrafilter
Hadwiger's theorem, part 1
The magnitude of an enriched category
Möbius inversion for categories
An operadic introduction to entropy
Entropies vs. means
Which graphs can be given a category structure?
Characterizing the generalized means
Characterizing the p-norms
Magnitude of metric spaces: a roundup
An informal introduction to topos theory
The Boyd Orr Centre, or: what is a severed horse leg?
What is integral geometry?
Benoît Mandelbrot
Fetishizing p-values
The difference between measure zero and empty interior
What is the Langlands Programme?
The Dold–Kan Theorem: two questions
A perspective on higher category theory
Sheaves do not belong to algebraic geometry (with proof)
F and the shibboleth
What you're doing is good for you
An adventure in analysis
Asymptotics of the magnitude of metric spaces
Entropy, diversity and cardinality (part 2)
Entropy, diversity and cardinality (part 1)
The cardinality of a metric space
How I learned to love the nerve construction
On Linear Algebra Done Right

Here are a couple of my comments at the n-Category Café:

A short explanation of the Central Limit Theorem and how to view it as a result about maximum entropy
Informal introduction to classifying toposes

And here are some odds and ends:

Review by Stefan Forcey of my book Entropy and Diversity, in the European Mathematical Society Magazine
Graph of Covid-19 cases at the University of Edinburgh
I am a person
Working seminar on persistent homology
Review of the book Coherence in Three-Dimensional Category Theory by Nick Gurski (published in the Bulletin of the London Mathematical Society)
Article for New Scientist magazine: Ethical calculus (also syndicated in Slate), plus associated post and follow-on articles in Boing Boing, Slashdot, Mediapart (free version here), Zeit Online, and Spiegel Online
Two articles for the London Mathematical Society Newsletter, both entitled Should mathematicians cooperate with GCHQ?: first (appears in the April 2014 edition, p.34) and second (appears in the July 2014 edition, p.42)
A review of the popular mathematics book The Colours of Infinity
An interview with me in the December 2008 issue of The Reasoner, largely about higher categories
Doing without diagrams: how to take a proof that uses elements, utter some magic words, and conclude that it's valid in any category
My favourite proof of the Fundamental Theorem of Algebra: argument learned from Graeme Segal and notes extracted from a course I taught
Coproducts of operads, and the W-construction: an observation that plays a part in the story of the operad of phylogenetic trees.




Edinburgh 4th year Galois Theory 2022–23
Edinburgh Linear Algebra 2018–19: accelerated course for undergraduates entering directly into the second year
Tips on writing projects, for my undergraduate project students
Edinburgh Functional equations seminar course, 2017; see also this post
Edinburgh 4th year General Topology 2014–15, plus introductory lecture and problem sheets
Edinburgh 4th year Fourier Analysis 2013–14 (mostly Fourier series), plus problem sheets
Tips on mathematical writing for undergraduates
Glasgow M.Sci. Category Theory 2007–8
Glasgow Category Theory 2004
Cambridge Part III Category Theory 2000
Cambridge Part IA/IB Linear Maths: some notes on the minimal polynomial and Jordan canonical form



This page was last changed on 8 December 2023. Photo