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Richard Garner

Publications and source records attributed to Richard Garner.

At least 19 recordsLinked to original sources
arXiv2026

Absolute colimits

In the context of enriched category theory, we give necessary and sufficient conditions for a module morphism $\alpha \dd M \to \CC(F,Z)$ to exhibit a functor $Z\dd \CA\to \CC$ as an absolute $M$-weighted colimit of a functor ${F\dd \CB\to \CC}$. We also review, with short proofs, the various criteria for the weight $M$ itself to be absolute, in the sense that any $M$-weighted colimit is absolute. Finally, we prove that any absolute $M$-weighted colimit can be viewed as a colimit weighted by an absolute weight $M'$.

math.CT↗
arXiv2026

Stone Duality for Monads

We introduce a contravariant idempotent adjunction between (i) the category of ranked monads on $\mathsf{Set}$; and (ii) the category of internal categories and internal retrofunctors in the category of locales. The left adjoint takes a monad $T$-viewed as a notion of computation, following Moggi-to its localic behaviour category $\mathsf{LB}T$. This behaviour category is understood as "the universal transition system" for interacting with $T$: its "objects" are states and the "morphisms" are transitions. On the other hand, the right adjoint takes a localic category $\mathsf{LC}$-similarly understood as a transition system-to the monad $\Gamma\mathsf{LC}$ where $(\Gamma\mathsf{LC})A$ is the set of $A$-indexed families of local sections to the source map which jointly partition the locale of objects. The fixed points of this adjunction consist of (i) hyperaffine-unary monads, i.e., those monads where term $t$ admits a read-only operation $\bar{t}$ predicting the output of $t$; and (ii) ample localic categories, i.e., whose source maps are local homeomorphisms and whose locale of objects are strongly zero-dimensional. The hyperaffine-unary monads arise in earlier works by Johnstone and Garner as a syntactic characterization of those monads with Cartesian closed Eilenberg-Moore categories. This equivalence is the Stone duality for monads; so-called because it further restricts to the classical Stone duality by viewing a Boolean algebra $B$ as a monad of $B$-partitions and the corresponding Stone space as a localic category with only identity morphisms.

cs.LO↗
arXiv2025

A unified treatment of commuting tensor products of categories, operads, symmetric multicategories and their bimodules

We provide a unified treatment of several commuting tensor products considered in the literature, including the tensor product of enriched categories and the Boardman-Vogt tensor product of operads and symmetric multicategories, subsuming work of Elmendorf and Mandell. We then show how a commuting tensor product extends to bimodules, generalising results of Dwyer and Hess. In particular, we construct a double category of symmetric multicategories, symmetric multifunctors and bimodules and show that it admits a symmetric oplax monoidal structure. These applications are obtained as instances of a general construction of commuting tensor products on double categories of monads, monad morphisms and bimodules.

math.CT↗
arXiv2025

Free differential modalities

Categorical models of the exponential modality of linear logic will often, but not always, support an operation of differentiation. When they do, we speak of a monoidal differential modality; when they do not, we have merely a monoidal coalgebra modality. More generally, there are notions of differential modality and coalgebra modality which stand in the same relation, but which move outside the realm of linear logic; they model important structures such as smooth differentiation on Euclidean space. In this paper, we show that, in a suitably well-behaved k-linear symmetric monoidal category, each (monoidal) coalgebra modality can be freely completed to a (monoidal) differential modality. In particular, we prove the existence of an initial monoidal differential modality, which, even in simple examples such as the category of sets and relations, yields new models of differential linear logic. Key to our proofs is the concept of an algebraically-free commutative monoid in a symmetric monoidal category. A commutative monoid $M$ is said to be algebraically-free on an object $X$ if actions by the monoid $M$ correspond to self-commuting actions by the mere object $X$. Along the way, we study the theory of algebraically-free commutative monoids and self-commuting actions, and their interplay with differential modality structure.

math.CT↗
arXiv2023

Cartesian closed varieties I: the classification theorem

In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories -- whose models are sets equipped with an action by a monoid M -- and all hyperaffine theories -- whose models are sets with an action by a Boolean algebra B. We improve on Johnstone's result by showing that an equational theory is cartesian closed just when its operations have a unique hyperaffine-unary decomposition. It follows that any non-degenerate cartesian closed variety is a variety of sets equipped with compatible actions by a monoid M and a Boolean algebra B; this is the classification theorem of the title.

math.LO↗
arXiv2023

Cartesian closed varieties II: links to algebra and self-similarity

This paper is the second in a series investigating cartesian closed varieties. In first of these, we showed that every non-degenerate finitary cartesian variety is a variety of sets equipped with an action by a Boolean algebra B and a monoid M which interact to form what we call a matched pair [B|M]. In this paper, we show that such pairs [B|M] are equivalent to Boolean restriction monoids and also to ample source-\'etale topological categories; these are generalisations of the Boolean inverse monoids and ample \'etale topological groupoids used to encode self-similar structures such as Cuntz and Cuntz--Krieger $C^\ast$-algebras, Leavitt path algebras and the $C^\ast$-algebras associated to self-similar group actions. We explain and illustrate these links, and begin the programme of understanding how topological and algebraic properties of such groupoids can be understood from the logical perspective of the associated varieties.

math.OA↗
arXiv2022

Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences

We extend the arithmetic product of species of structures and symmetric sequences studied by Maia and Mendez and by Dwyer and Hess to coloured symmetric sequences and show that it determines a normal oplax monoidal structure on the bicategory of coloured symmetric sequences. In order to do this, we establish general results on extending monoidal structures to Kleisli bicategories. Our approach uses monoidal double categories, which help us to attack the difficult problem of verifying the coherence conditions for a monoidal bicategory in an efficient way.

math.CT↗
arXiv2021

Functorial aggregation

We study polynomial comonads and polynomial bicomodules. Polynomial comonads amount to categories. Polynomial bicomodules between categories amount to parametric right adjoint functors between corresponding copresheaf categories. These may themselves be understood as generalized polynomial functors. They are also called data migration functors because of applications in categorical database theory. We investigate several universal constructions in the framed bicategory of categories, retrofunctors, and parametric right adjoints. We then use the theory we develop to model database aggregation alongside querying, all within this rich ecosystem.

math.CT↗
arXiv2021

Stream processors and comodels

In 2009, Hancock, Pattinson and Ghani gave a coalgebraic characterisation of stream processors $A^\mathbb{N} \to B^\mathbb{N}$ drawing on ideas of Brouwerian constructivism. Their stream processors have an intensional character; in this paper, we give a corresponding coalgebraic characterisation of extensional stream processors, i.e., the set of continuous functions $A^\mathbb{N} \to B^\mathbb{N}$. Our account sites both our result and that of op. cit. within the apparatus of comodels for algebraic effects originating with Power-Shkaravska. Within this apparatus, the distinction between intensional and extensional equivalence for stream processors arises in the same way as the the distinction between bisimulation and trace equivalence for labelled transition systems and probabilistic generative systems.

cs.LO↗
arXiv2020

The costructure-cosemantics adjunction for comodels for computational effects

It is well established that equational algebraic theories, and the monads they generate, can be used to encode computational effects. An important insight of Power and Shkaravska is that comodels of an algebraic theory T -- i.e., models in the opposite category Set^op -- provide a suitable environment for evaluating the computational effects encoded by T. As already noted by Power and Shkaravska, taking comodels yields a functor from accessible monads to accessible comonads on Set. In this paper, we show that this functor is part of an adjunction -- the "costructure-cosemantics adjunction" of the title -- and undertake a thorough investigation of its properties. We show that, on the one hand, the cosemantics functor takes its image in what we term the presheaf comonads induced by small categories; and that, on the other, costructure takes its image in the presheaf monads induced by small categories. In particular, the cosemantics comonad of an accessible monad will be induced by an explicitly-described category called its behaviour category that encodes the static and dynamic properties of the comodels. Similarly, the costructure monad of an accessible comonad will be induced by a behaviour category encoding static and dynamic properties of the comonad coalgebras. We tie these results together by showing that the costructure-cosemantics adjunction is idempotent, with fixpoints to either side given precisely by the presheaf monads and comonads. Along the way, we illustrate the value of our results with numerous examples drawn from computation and mathematics.

cs.LO↗
arXiv2020

Bousfield localisation and colocalisation of one-dimensional model structures

We give an account of Bousfield localisation and colocalisation for one-dimensional model categories---ones enriched over the model category of $0$-types. A distinguishing feature of our treatment is that it builds localisations and colocalisations using only the constructions of projective and injective transfer of model structures along right and left adjoint functors, and without any reference to Smith's theorem.

math.CT↗
arXiv2020

The Vietoris monad and weak distributive laws

The Vietoris monad on the category of compact Hausdorff spaces is a topological analogue of the power-set monad on the category of sets. Exploiting Manes' characterisation of the compact Hausdorff spaces as algebras for the ultrafilter monad on sets, we give precise form to the above analogy by exhibiting the Vietoris monad as induced by a weak distributive law, in the sense of Böhm, of the power-set monad over the ultrafilter monad.

math.CT↗
arXiv2020

An embedding theorem for tangent categories

Tangent categories were introduced by Rosicky as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category---one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem for tangent categories due to Leung to exhibit tangent categories as a certain kind of enriched category.

math.CT↗
arXiv2020

Monads and theories

Given a locally presentable enriched category $\mathcal{E}$ together with a small dense full subcategory $\mathcal A$ of arities, we study the relationship between monads on $\mathcal E$ and identity-on-objects functors out of $\mathcal A$, which we call $\mathcal A$-pretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised as the $\mathcal A$-nervous monads---those for which the conclusions of Weber's nerve theorem hold---and the $\mathcal A$-theories, which we introduce here. The resulting equivalence between $\mathcal A$-nervous monads and $\mathcal A$-theories is best possible in a precise sense, and extends almost all previously known monad--theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak $ω$-groupoids. Besides establishing our general correspondence and illustrating its reach, we study good properties of $\mathcal A$-nervous monads and $\mathcal A$-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Melliès and Weber.

math.CT↗
arXiv2020

Ultrafilters, finite coproducts and locally connected classifying toposes

We prove a single category-theoretic result encapsulating the notions of ultrafilters, ultrapower, ultraproduct, tensor product of ultrafilters, the Rudin--Kiesler partial ordering on ultrafilters, and Blass's category of ultrafilters UF. The result in its most basic form states that the category FC(Set,Set) of finite-coproduct-preserving endofunctors of Set is equivalent to the presheaf category [UF,Set]. Using this result, and some of its evident generalisations, we re-find in a natural manner the important model-theoretic realisation relation between n-types and n-tuples of model elements; and draw connections with Makkai and Lurie's work on conceptual completeness for first-order logic via ultracategories. As a further application of our main result, we use it to describe a first-order analogue of Jónsson and Tarski's canonical extension. Canonical extension is an algebraic formulation of the link between Lindenbaum--Tarski and Kripke semantics for intuitionistic and modal logic, and extending it to first-order logic has precedent in the topos of types construction studied by Joyal, Reyes, Makkai, Pitts, Coumans and others. Here, we study the closely related, but distinct, construction of the locally connected classifying topos of a first-order theory. The existence of this is known from work of Funk, but the description is inexplicit; ours, by contrast, is quite concrete.

math.CT↗
arXiv2020

Generalising the étale groupoid--complete pseudogroup correspondence

We prove a generalisation of the correspondence, due to Resende and Lawson--Lenz, between étale groupoids---which are topological groupoids whose source map is a local homeomorphisms---and complete pseudogroups---which are inverse monoids equipped with a particularly nice representation on a topological space. Our generalisation improves on the existing functorial correspondence in four ways. Firstly, we enlarge the classes of maps appearing to each side. Secondly, we generalise on one side from inverse monoids to inverse categories, and on the other side, from étale groupoids to what we call partite étale groupoids. Thirdly, we generalise from étale groupoids to source-étale categories, and on the other side, from inverse monoids to restriction monoids. Fourthly, and most far-reachingly, we generalise from topological étale groupoids to étale groupoids internal to any join restriction category C with local glueings; and on the other side, from complete pseudogroups to ``complete C-pseudogroups'', i.e., inverse monoids with a nice representation on an object of C. Taken together, our results yield an equivalence, for a join restriction category C with local glueings, between join restriction categories with a well-behaved functor to C, and partite source-étale internal categories in C. In fact, we obtain this by cutting down a larger adjunction between arbitrary restriction categories over C, and partite internal categories in C. Beyond proving this main result, numerous applications are given, which reconstruct and extend existing correspondences in the literature, and provide general formulations of completion processes.

math.CT↗
arXiv2020

Cartesian differential categories as skew enriched categories

We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids -- or in a straightforward generalisation, the category of modules over a commutative rig $k$. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad $Q$. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlach\'anyi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad $Q$ involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal $k$-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category -- thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

math.CT↗