arXiv subjects

Matthew Di Meglio

Publications and source records attributed to Matthew Di Meglio.

7 recordsLinked to original sources

arXiv2025

Dagger categories of relations: The equivalence of dilatory dagger categories and epi-regular independence categories

Several categories look like categories of relations, but do not fit the established theory of relations in regular categories. They include the category of surjective multivalued functions, the category of injective partial functions, the category of finite probability spaces and stochastic matrices, and the category of Hilbert spaces and linear contractions. To explain these anomalous examples, we develop a parallel theory of relations in epi-regular independence categories. Just as regular categories correspond to tabular allegories, epi-regular independence categories correspond to dilatory dagger categories. The equivalence maps epi-regular independence categories to their associated dagger category of relations, and dilatory dagger categories to their wide subcategory of coisometries.

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arXiv2025

Hilbert $*$-categories: Where limits in analysis and category theory meet

This article introduces Hilbert $*$-categories: an abstraction of categories having algebraic and analytic properties similar to those of the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include categories of Hilbert W*-modules and of unitary representations of groupoids. Hilbert $*$-categories are "analytically" complete in two ways: every bounded increasing sequence of Hermitian endomorphisms has a supremum, and every suitably bounded orthogonal family of parallel morphisms is summable. These "analytic" completeness properties are not assumed outright; rather, they are derived, respectively, from two new universal constructions: codirected $\ell^2$-limits of contractions and $\ell^2$-products. In turn, these are built from directed colimits in the wide subcategory of isometries.

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arXiv2024

Dagger categories and the complex numbers: Axioms for the category of finite-dimensional Hilbert spaces and linear contractions

We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and linear contractions in terms of simple category-theoretic structures and properties that do not refer to norms, continuity, or real numbers. This builds on Heunen, Kornell, and Van der Schaaf's easier characterisation of the category of all Hilbert spaces and linear contractions.

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arXiv2023

Pre-Hilbert $*$-categories: The Hilbert-space analogue of abelian categories

This article introduces pre-Hilbert $*$-categories: an abstraction of categories exhibiting "algebraic" aspects of the theory of Hilbert spaces. Notably, finite biproducts in pre-Hilbert $*$-categories can be orthogonalised using the Gram-Schmidt process, and generalised notions of positivity and contraction support variants of Douglas' lemma and Sz.-Nagy's unitary dilation theorem. Underpinning these generalisations is the structure of an involutive identity-on-objects contravariant endofunctor, which encodes adjoints of morphisms. The axioms for pre-Hilbert $*$-categories are otherwise inspired by those for abelian categories, comprising a few simple properties of products and kernels. Additivity is not assumed, but nevertheless follows. In fact, the similarity with abelian categories runs deeper: pre-Hilbert $*$-categories are quasi-abelian and thus also homological. Examples include the $*$-category of unitary representations of a groupoid, the $*$-category of finite-dimensional inner product modules over an ordered division $*$-ring, and the $*$-category of self-dual Hilbert modules over a W*-algebra.

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arXiv2023

Universal Properties of Lens Proxy Pullbacks

A comprehensive account of the categorical properties of the category of small categories and asymmetric delta lenses is given in the recent works of Chollet et al. and Di Meglio. An important construction for proving many of these properties is Johnson and Rosebrugh's "pullback" of lenses, which we call the proxy pullback of lenses. We give a new treatment of the proxy pullback in terms of compatibility, a stronger notion of commutativity for squares of lenses. The proxy pullback is sometimes, but not always, a real pullback. Using new notions of sync-minimal and independent lens spans, we characterise when a lens span that forms a commuting square with a lens cospan has a comparison lens to a proxy pullback of the cospan.

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arXiv2022

Coequalisers under the Lens

Lenses encode protocols for synchronising systems. We continue the work begun by Chollet et al. at the Applied Category Theory Adjoint School in 2020 to study the properties of the category of small categories and asymmetric delta lenses. The forgetful functor from the category of lenses to the category of functors is already known to reflect monos and epis and preserve epis; we show that it preserves monos, and give a simpler proof that it preserves epis. Together this gives a complete characterisation of the monic and epic lenses in terms of elementary properties of their get functors. Next, we initiate the study of coequalisers of lenses. We observe that not all parallel pairs of lenses have coequalisers, and that the forgetful functor from the category of lenses to the category of functors neither preserves nor reflects all coequalisers. However, some coequalisers are reflected; we study when this occurs, and then use what we learned to show that every epic lens is regular, and that discrete opfibrations have pushouts along monic lenses. Corollaries include that every monic lens is effective, every monic epic lens is an isomorphism, and the class of all epic lenses and the class of all monic lenses form an orthogonal factorisation system.

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arXiv2022

An introduction to enriched cofunctors

Cofunctors are a kind of map between categories which lift morphisms along an object assignment. In this paper, we introduce cofunctors between categories enriched in a distributive monoidal category. We define a double category of enriched categories, enriched functors, and enriched cofunctors, whose horizontal and vertical 2-categories have 2-cells given by enriched natural transformations between functors and cofunctors, respectively. Enriched lenses are defined as a compatible enriched functor and enriched cofunctor pair; weighted lenses, which were introduced by Perrone, are precisely lenses enriched in weighted sets. Several other examples are also studied in detail.

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