arXiv subjects

David Jaz Myers

Publications and source records attributed to David Jaz Myers.

At least 19 recordsLinked to original sources

arXiv2026

Twisted double functors and loosely discrete opfibrations

Various situations in the theory and applications of double categories, ranging from a loose Yoneda theory and loose compact closure to double-operadic systems theory, require a notion of double copresheaf in which the action is by loose morphisms rather than tight ones. In this paper, we develop and compare several models for loose copresheaves on double categories. First, we introduce a new notion of morphism between double categories, called twisted double functors, which send tight morphisms to loose morphisms and vice versa, and use these to define twisted copresheaves. We exhibit numerous examples of twisted double functors, starting with the twisted Hom functor and the twisted representables on a double category. Corresponding to this functorial notion of loose copresheaf is a fibrational one, an internal version of a discrete opfibration that we call a loosely discrete opfibration. We prove that twisted copresheaves and cloven loosely discrete opfibrations are equivalent via an elements construction. Finally, we compare twisted bimodules with double categories over the walking loose arrow, or double barrels, via a collage construction.

math.CT↗

arXiv2026

Classifying strict discrete opfibrations with lax morphisms

We study discrete opfibration classifiers in enhanced 2-categories and show how, under suitable hypotheses, such classifiers can be endowed with the structure of a (lax or pseudo-)T-algebra and classify strict discrete opfibrations in 2-categories of (lax or pseudo-)T-algebras and lax morphisms. This leads to a notion of discrete opfibration classifier in the enhanced setting, in which `small' (e.g. strict) discrete opfibrations are classified by `loose' (e.g. lax) maps. We identify conditions on an enhanced 2-monad T and on a discrete opfibration classifier ensuring that this lifting to algebras is possible. These conditions hold in a broad range of examples, including double categories, monoidal and symmetric monoidal categories, orthogonal factorization systems, and, more generally, structures encoded by opfamilial 2-monads. In particular, this recovers and explains the role of Span(Set) as a classifier for strict double discrete opfibrations via lax double functors. We also characterize when representable copresheaves are pseudo rather than merely lax in terms of `cartesianness at the representing object', for an abstract notion of cartesianness we introduce.

math.CT↗

arXiv2026

Compositionality of Lyapunov functions via assume-guarantee reasoning

Assume-guarantee reasoning is a technique for compositional model checking in which system specifications are checked under certain assumptions on system parameters or inputs, and provide guarantees on observations of system state. We present a categorical framework for assume-guarantee reasoning for safety problems by viewing systems as lenses, following our earlier work on the compositionality of generalized Moore machines. Generalized Moore machines include ordinary Moore machines, partially observable Markov (decision) processes, and systems of parameterized ODEs (control systems); our framework gives assume-guarantee reasoning specially adapted to each of these cases. In particular, we give a novel formulation of assume-guarantee reasoning for (local) input-to-state stability ((L)ISS) Lyapunov functions on systems of parameterized ODEs. Our framework is categorically natural and straightforwardly compositional. A flavor of generalized Moore machine is determined by a tangency: a fibration with a section. We show that symmetric monoidal loose right modules of assume-guarantee certified generalized Moore machines over symmetric monoidal double categories of certified wiring diagrams can be constructed 2-functorially from fibrations internal to the 2-category of tangencies.

cs.LO↗

arXiv2026

Clock systems for stochastic and non-deterministic categorical systems theories

One of the characteristic features of categorical systems theory is that the behavior of systems can be characterized by certain morphisms into them. In other words, behaviors form a representable covariant functor to Set. And more generally, in the compositional setting, behaviors form a representable double functor to Span. Clock systems are convenient because behavior functors represented by clock systems are automatically well-behaved. It was previously not known whether stochastic and non-deterministic systems theories have clock systems. In this paper, we show that indeed they do have clock systems. Moreover, the clock systems for non-deterministic systems point to generalized notions of behavior for non-linear time.

math.CT↗

arXiv2025

Comparing loose bimodules and double barrels using pseudo-models of enhanced sketches

(Pseudo) double categories have two sorts of morphisms: tight ones which compose strictly, and loose ones which compose up to coherent isomorphism. In this paper, we consider bimodules between double categories in the loose direction. We provide two formulation of this concept -- first as pseudo-bimodules between pseudo-categories in the 2-category of categories, and second as double barrels generalizing Joyal's definition of bimodules between categories as functors into the walking arrow -- and prove these two formulations equivalent. In order to prove this equivalence, we define a notion of \emph{pseudo-model} of an enhanced sketch, which may be of independent interest. We then consider some double category theory unlocked by the theory of loose bimodules: loose adjunctions, and loose limits.

math.CT↗

arXiv2025

Proceedings Seventh International Conference on Applied Category Theory 2024

Proceedings of the Seventh International Conference on Applied Category Theory, held at the University of Oxford on 17 - 21 June 2024. The contributions to ACT 2024 ranged from pure to applied and included contributions in a wide range of disciplines in science and engineering. ACT 2024 included talks in classical mechanics, quantum physics, probability theory, linguistics, decision theory, machine learning, epidemiology, thermodynamics, engineering, and logic.

cs.LO↗

arXiv2025

Towards a double operadic theory of systems

We present a unified framework for categorical systems theory which packages a collection of open systems, their interactions, and their maps into a symmetric monoidal loose right module of systems over a symmetric monoidal double category of interfaces and interactions. As examples, we give detailed descriptions of (1) the module of open Petri nets over undirected wiring diagrams and (2) the module of deterministic Moore machines over lenses. We define several pseudo-functorial constructions of modules of systems in the form of doctrines of systems theories. In particular, we introduce doctrines for port-plugging systems, variable sharing systems, and generalized Moore machines, each of which generalizes existing work in categorical systems theory. Finally, we observe how diagrammatic interaction patterns are free processes in particular doctrines.

math.CT↗

arXiv2025

Modal Fracture of Higher Groups

In this paper, we examine the modal aspects of higher groups in Shulman's Cohesive Homotopy Type Theory. We show that every higher group sits within a modal fracture hexagon which renders it into its discrete, infinitesimal, and contractible components. This gives an unstable and synthetic construction of Schreiber's differential cohomology hexagon. As an example of this modal fracture hexagon, we recover the character diagram characterizing ordinary differential cohomology by its relation to its underlying integral cohomology and differential form data, although there is a subtle obstruction to generalizing the usual hexagon to higher types. Assuming the existence of a long exact sequence of differential form classifiers, we construct the classifiers for circle k-gerbes with connection and describe their modal fracture hexagon.

math.CT↗

arXiv2024

Contextads as Wreaths; Kleisli, Para, and Span Constructions as Wreath Products

We introduce contextads and the Ctx construction, unifying various structures and constructions in category theory dealing with context and contextful arrows -- comonads and their Kleisli construction, actegories and their Para construction, adequate triples and their Span construction. Contextads are defined in terms of Lack--Street wreaths, suitably categorified for pseudomonads in a tricategory of spans in a 2-category with display maps. The associated wreath product provides the Ctx construction, and by its universal property we conclude trifunctoriality. This abstract approach lets us work up to structure, and thus swiftly prove that, under very mild assumptions, a contextad equipped colaxly with a 2-algebraic structure produces a similarly structured double category of contextful arrows. We also explore the role contextads might play qua dependently graded comonads in organizing contextful computation in functional programming. We show that many side-effects monads can be dually captured by dependently graded comonads, and gesture towards a general result on the `transposability' of parametric right adjoint monads to dependently graded comonads.

math.CT↗

arXiv2023

Topological Quantum Gates in Homotopy Type Theory

Despite the evident necessity of topological protection for realizing scalable quantum computers, the conceptual underpinnings of topological quantum logic gates had arguably remained shaky, both regarding their physical realization as well as their information-theoretic nature. Building on recent results on defect branes in string/M-theory and on their holographically dual anyonic defects in condensed matter theory, here we explain how the specification of realistic topological quantum gates, operating by anyon defect braiding in topologically ordered quantum materials, has a surprisingly slick formulation in parameterized point-set topology, which is so fundamental that it lends itself to certification in modern homotopically typed programming languages, such as cubical Agda. We propose that this remarkable confluence of concepts may jointly kickstart the development of topological quantum programming proper as well as of real-world application of homotopy type theory, both of which have arguably been falling behind their high expectations; in any case, it provides a powerful paradigm for simulating and verifying topological quantum computing architectures with high-level certification languages aware of the actual physical principles of realistic topological quantum hardware. In a companion article, we will explain how further passage to "dependent linear" homotopy data types naturally extends this scheme to a full-blown quantum programming/certification language in which our topological quantum gates may be compiled to verified quantum circuits, complete with quantum measurement gates and classical control.

quant-ph↗

arXiv2023

Commuting Cohesions

Shulman's spatial type theory internalizes the modalities of Lawvere's axiomatic cohesion in a homotopy type theory, enabling many of the constructions from Schreiber's modal approach to differential cohomology to be carried out synthetically. In spatial type theory, every type carries a spatial cohesion among its points and every function is continuous with respect to this. But in mathematical practice, objects may be spatial in more than one way at the same time; for example, a simplicial space has both topological and simplicial structures. In this paper, we put forward a type theory with "commuting focuses" which allows for types to carry multiple kinds of spatial structure. The theory is a relatively painless extension of spatial type theory, and enables us to give a synthetic account of simplicial, differential, equivariant, and other cohesions carried by the same types. We demonstrate the theory by showing that the homotopy type of any differential stack may be computed from a discrete simplicial set derived from the Čech nerve of any good cover. We also give other examples of commuting cohesions, such as differential equivariant types and supergeometric types, laying the groundwork for a synthetic account of Schreiber and Sati's proper orbifold cohomology.

math.CT↗

arXiv2022

Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory

Informally, an orbifold is a smooth space whose points may have finitely many internal symmetries. Formally, however, the notion of orbifold has been presented in a number of different guises -- from Satake's V-manifolds to Moerdijk and Pronk's proper étale groupoids -- which do not on their face resemble the informal definition. The reason for this divergence between formalism and intuition is that the points of spaces cannot have internal symmetries in traditional, set-level foundations. The extra data of these symmetries must be carried around and accounted for throughout the theory. More drastically, maps between orbifolds presented in the usual ways cannot be defined pointwise. In this paper, we will put forward a definition of orbifold in synthetic differential cohesive homotopy type theory: an orbifold is a microlinear type for which the type of identifications between any two points is properly finite. In homotopy type theory, a point of a type may have internal symmetries, and we will be able to construct examples of orbifolds by defining their type of points directly. Moreover, the mapping space between two orbifolds is merely the type of functions. We will justify this synthetic definition by proving, internally, that every proper étale groupoid is an orbifold. In this way, we will show that the synthetic theory faithfully extends the usual theory of orbifolds. Along the way, we will investigate the microlinearity of higher types such as étale groupoids, showing that the methods of synthetic differential geometry generalize gracefully to higher analogues of smooth spaces. We will also investigate the relationship between the Dubuc-Penon and open-cover definitions of compactness in synthetic differential geometry, and show that any discrete, Dubuc-Penon compact subset of a second-countable manifold is subfinitely enumerable.

math.AT↗

arXiv2022

Good Fibrations through the Modal Prism

Homotopy type theory is a formal language for doing abstract homotopy theory -- the study of identifications. But in unmodified homotopy type theory, there is no way to say that these identifications come from identifying the path-connected points of a space. In other words, we can do abstract homotopy theory, but not algebraic topology. Shulman's Real Cohesive HoTT remedies this issue by introducing a system of modalities that relate the spatial structure of types to their homotopical structure. In this paper, we develop a theory of modal fibrations for a general modality, and apply it in particular to the shape modality of Real Cohesion. We then give examples of modal fibrations in Real Cohesive HoTT, and develop the theory of covering spaces.

math.CT↗

arXiv2021

Double Categories of Open Dynamical Systems (Extended Abstract)

A (closed) dynamical system is a notion of how things can be, together with a notion of how they may change given how they are. The idea and mathematics of closed dynamical systems has proven incredibly useful in those sciences that can isolate their object of study from its environment. But many changing situations in the world cannot be meaningfully isolated from their environment - a cell will die if it is removed from everything beyond its walls. To study systems that interact with their environment, and to design such systems in a modular way, we need a robust theory of open dynamical systems. In this extended abstract, we put forward a general definition of open dynamical system. We define two general sorts of morphisms between these systems: covariant morphisms which include trajectories, steady states, and periodic orbits; and contravariant morphisms which allow for plugging variables of some systems into parameters of other systems. We define an indexed double category of open dynamical systems indexed by their interface and use a double Grothendieck construction to construct a double category of open dynamical systems. In our main theorem, we construct covariantly representable indexed double functors from the indexed double category of dynamical systems to an indexed double category of spans. This shows that all covariantly representable structures of dynamical systems - including trajectories, steady states, and periodic orbits - compose according to the laws of matrix arithmetic.

math.CT↗

arXiv2021

Behavioral Mereology: A Modal Logic for Passing Constraints

Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos, providing maximal generality; however, by using only its internal logic we can hide the details and readers may assume a completely elementary set-theoretic discussion. We consider the relationship between various parts of a whole in terms of how behavioral constraints are passed between them, and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetric accessibility.

cs.LO↗

arXiv2021

Cartesian Factorization Systems and Grothendieck Fibrations

Every Grothendieck fibration gives rise to a vertical/cartesian orthogonal factorization system on its domain. We define a cartesian factorization system to be an orthogonal factorization in which the left class satisfies 2-of-3 and is closed under pullback along the right class. We endeavor to show that this definition abstracts crucial features of the vertical/cartesian factorization system associated to a Grothendieck fibration, and give comparisons between various 2-categories of factorization systems and Grothendieck fibrations to demonstrate this relationship. We then give a construction which corresponds to the fiberwise opposite of a Grothendieck fibration on the level of cartesian factorization systems. Apart from the final double categorical results, this paper is entirely review of previously established material. It should be read as an expository note.

math.CT↗

arXiv2020

Dirichlet Polynomials form a Topos

One can think of power series or polynomials in one variable, such as $P(x)=2x^3+x+5$, as functors from the category $\mathsf{Set}$ of sets to itself; these are known as polynomial functors. Denote by $\mathsf{Poly}_{\mathsf{Set}}$ the category of polynomial functors on $\mathsf{Set}$ and natural transformations between them. The constants $0,1$ and operations $+,\times$ that occur in $P(x)$ are actually the initial and terminal objects and the coproduct and product in $\mathsf{Poly}_{\mathsf{Set}}$. Just as the polynomial functors on $\mathsf{Set}$ are the copresheaves that can be written as sums of representables, one can express any Dirichlet series, e.g.\ $\sum_{n=0}^\infty n^x$, as a coproduct of representable presheaves. A Dirichlet polynomial is a finite Dirichlet series, that is, a finite sum of representables $n^x$. We discuss how both polynomial functors and their Dirichlet analogues can be understood in terms of bundles, and go on to prove that the category of Dirichlet polynomials is an elementary topos.

math.CT↗

arXiv2020

Behavioral Mereology (Proofs and Properties)

math.LO↗