arXiv subjects

Simon Willerton

Publications and source records attributed to Simon Willerton.

At least 19 recordsLinked to original sources

arXiv2026

The microscopic weighting on a metric space

We introduce the microscopic weighting, a canonical signed measure of mass one that can be associated to almost any finite metric space. The microscopic weighting is obtained as the small-scale limit of the weightings used to define the magnitude function. We give general criteria for its existence, proving in particular that every finite space of strictly negative type admits a microscopic weighting; this includes every finite subset of Euclidean or hyperbolic space and every finite tree. Heuristically speaking, the microscopic weighting distributes its mass as widely as possible across a space, assigning greater weight to sparse or outlying regions and emphasizing points on the boundary. Indeed, we show that on a finite space of negative type the microscopic weighting can be characterized (when it exists) as an optimizing measure for an energy integral determined by the distance function. Alternatively, it can be characterized in terms of the geometry of the Schoenberg embedding. Each of these interpretations also clarifies the information carried by the derivative of the magnitude function at zero. Though our main focus in this paper is on finite metric spaces, we lay the groundwork to extend the theory to compact subsets of Euclidean space. In that setting, we observe that the microscopic weighting must be understood as a distribution rather than as a measure.

math.MG↗

arXiv2025

Extranatural transformations, adjunctions of two variables and conjugation

Adjunctions of two variables generalize the relationship between tensor product and the internal hom functor in a closed monoidal category. For a pair of ordinary adjunctions $(F\dashv U, F'\dashv U')$ conjugation relates natural transformations of the form $F\Rightarrow F'$ with natural transformations of the form $U' \Rightarrow U$. We look at conjugation for general two variable adjunctions. It is useful in the context of Grothendieck's six operations as we will show that this is an appropriate way to view the constructions of Fausk, Hu and May where they discuss things like the projection formula and internal adjunctions. Extensive use is made of surface diagram notation as this is a helpful way to keep track of the three dimensions of composition. This also places the work in the context of formal category theory as, for instance, closed monoidal categories are defined without reference to objects or morphisms inside them. In an appendix it is explained how an appropriate setting for this perspective is the monoidal double category of functors and profunctors, using the fact that the bicategory of profunctors has duals.

math.CT↗

arXiv2024

Metric-like spaces as enriched categories: three vignettes

This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a number -- the distance -- such that the triangle inequality is satisfied; a small category is a set of objects such that between each pair of objects there is a set -- the hom-set -- such that elements of the hom-sets can be composed. The analogy between the structures that can be made in to a common generalization of the two structures, so that both are examples of enriched categories. This gives a bridge between category theory and metric space theory. I will describe this and three examples from around mathematics where this perspective has been useful or interesting. The examples are related to the tight span, the magnitude and the Legendre-Fenchel transform.

math.CT↗

arXiv2020

The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials

Magnitude is an invariant of metric spaces with origins in category theory. Using potential theoretic methods, Barceló and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schröder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows.

math.MG↗

arXiv2018

On the magnitude of odd balls via potential functions

Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in arXiv:1507.02502, Barceló and Carbery gave a procedure for calculating the magnitude of balls in odd dimensional Euclidean spaces. In this paper their approach is modified in various ways: this leads to an explicit determinantal formula for the magnitude of odd balls and leads to the conjecturing of a simpler formula in terms of Hankel determinants. This latter formula is proved using a rather different approach in arXiv:1708.03227, but the current paper provides the reasoning that lead to the formula being conjectured. Finally, an empirically-tested Hankel determinant formula for the derivative of the magnitude is conjectured.

math.MG↗

arXiv2015

Categorifying the magnitude of a graph

The magnitude of a graph can be thought of as an integer power series associated to a graph; Leinster introduced it using his idea of magnitude of a metric space. Here we introduce a bigraded homology theory for graphs which has the magnitude as its graded Euler characteristic. This is a categorification of the magnitude in the same spirit as Khovanov homology is a categorification of the Jones polynomial. We show how properties of magnitude proved by Leinster categorify to properties such as a Kunneth Theorem and a Mayer-Vietoris Theorem. We prove that joins of graphs have their homology supported on the diagonal. Finally, we give various computer calculated examples.

math.CO↗

arXiv2015

The Legendre-Fenchel transform from a category theoretic perspective

The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this paper we show how it arises in the context of category theory using categories enriched over the extended real numbers $\overline{ \mathbb{R}}:=[-\infty,+\infty]$. A key ingredient is Pavlovic's 'nucleus of a profunctor' construction. The pairing between a vector space and its dual can be viewed as an $\overline {\mathbb{R}}$-profunctor; the construction of the nucleus of this profunctor is the construction of a lot of the theory of the Legendre-Fenchel transform. For a relation between sets viewed as a $\{\mathrm{true},\mathrm{false}\}$-valued profunctor, the construction of the nucleus is the construction of the Galois connection associated to the relation. One insight given by this approach is that the relevant structure on the function spaces involved in the Legendre-Fenchel transform is something like a metric but is asymmetric and can take negative values. This '$\overline {\mathbb{R}}$-structure' is a considerable refinement of the usual partial order on real-valued function space and it allows a natural interpretation of Toland-Singer duality and of the two tropical module structures on the set of convex functions.

math.CT↗

arXiv2015

Spread: a measure of the size of metric spaces

Motivated by Leinster-Cobbold measures of biodiversity, the notion of the spread of a finite metric space is introduced. This is related to Leinster's magnitude of a metric space. Spread is generalized to infinite metric spaces equipped with a measure and is calculated for spheres and straight lines. For Riemannian manifolds the spread is related to the volume and total scalar curvature. A notion of scale-dependent dimension is introduced and seen, numerically, to be close to the Hausdorff dimension for approximations to certain fractals.

math.MG↗

arXiv2013

Tight spans, Isbell completions and semi-tropical modules

In this paper we consider the categorical Isbell completion construction for generalized metric spaces in the sense of Lawvere. We show that this is an analogue of the tight span construction of classical metric spaces, and that the Isbell completion coincides with the directed tight span of Hirai and Koichi. The notions of categorical completion and cocompletion are related to the existence of semi-tropical module structure, and it is shown that the Isbell completion (hence the directed tight span) has two different semi-tropical module structures.

math.CT↗

arXiv2013

On the magnitude of spheres, surfaces and other homogeneous spaces

In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier work with Leinster in arXiv:0908.1582. An explicit formula for the magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary homogeneous Riemannian manifold the leading terms of the asymptotic expansion of the magnitude are calculated and expressed in terms of the volume and total scalar curvature of the manifold. In the particular case of a homogeneous surface the form of the asymptotics can be given exactly up to vanishing terms and this involves just the area and Euler characteristic in the way conjectured for subsets of Euclidean space in previous work.

math.DG↗

arXiv2012

On the asymptotic magnitude of subsets of Euclidean space

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.

math.MG↗

arXiv2009

Heuristic and computer calculations for the magnitude of metric spaces

The notion of the magnitude of a compact metric space was considered in arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line segments, circles and Cantor sets. In this paper more evidence is presented for a conjectured relationship with a geometric measure theoretic valuation. Firstly, a heuristic is given for deriving this valuation by considering 'large' subspaces of Euclidean space and, secondly, numerical approximations to the magnitude are calculated for squares, disks, cubes, annuli, tori and Sierpinski gaskets. The valuation is seen to be very close to the magnitude for the convex spaces considered and is seen to be 'asymptotically' close for some other spaces.

math.MG↗

arXiv2008

A diagrammatic approach to Hopf monads

Given a Hopf algebra in a symmetric monoidal category with duals, the category of modules inherits the structure of a monoidal category with duals. If the notion of algebra is replaced with that of monad on a monoidal category with duals then Bruguieres and Virelizier showed when the category of modules inherits this structure of being monoidal with duals, and this gave rise to what they called a Hopf monad. In this paper it is shown that there are good diagrammatic descriptions of dinatural transformations which allows the three-dimensional, object-free nature of their constructions to become apparent.

math.CT↗

arXiv2007

The Mukai pairing, I: a categorical approach

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.

math.AG↗

arXiv2006

On the Rozansky-Witten weight systems

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold X gives rise to Vassiliev weight systems. In this paper we study these weight systems by using D(X), the derived category of coherent sheaves on X. The main idea (stated here a little imprecisely) is that D(X) is the category of modules over the shifted tangent sheaf, which is a Lie algebra object in D(X); the weight systems then arise from this Lie algebra in a standard way. The other main results are a description of the symmetric algebra, universal enveloping algebra, and Duflo isomorphism in this context, and the fact that a slight modification of D(X) has the structure of a braided ribbon category, which gives another way to look at the associated invariants of links. Our original motivation for this work was to try to gain insight into the Jacobi diagram algebras used in Vassiliev theory by looking at them in a new light, but there are other potential applications, in particular to the rigorous construction of the (1+1+1)-dimensional Rozansky-Witten TQFT, and to hyperkaehler geometry.

math.DG↗

arXiv2005

The twisted Drinfeld double of a finite group via gerbes and finite groupoids

The twisted Drinfeld double (or quasi-quantum double) of a finite group with a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3-cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3-dimensional topological quantum field theory. In particular the representation category of the twisted Drinfeld double is viewed as the `space of sections' associated to a transgressed gerbe over the loop groupoid.

math.QA↗

arXiv2004

Gerbes and homotopy quantum field theories

For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev's 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories.

math.AT↗

arXiv2002

An almost-integral universal Vassiliev invariant of knots

A `total Chern class' invariant of knots is defined. This is a universal Vassiliev invariant which is integral `on the level of Lie algebras' but it is not expressible as an integer sum of diagrams. The construction is motivated by similarities between the Kontsevich integral and the topological Chern character.

math.GT↗