Double Category Research Articles

TQFTs and quantum computing

Quantum computing is captured in the formalism of the monoidal subcategory of VectC generated by C2 — in particular, quantum circuits are diagrams in VectC — while topological quantum field theories, in the sense of Atiyah, are diagrams in VectC indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call FVectC. We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to FVectC, which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.

A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks

Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.

Cartesian double theories: A double-categorical framework for categorical doctrines

The categorified theories known as “doctrines” specify a category equipped with extra structure, analogous to how ordinary theories specify a set with extra structure. We introduce a new framework for doctrines based on double category theory. A cartesian double theory is defined to be a small double category with finite products and a model of a cartesian double theory to be a finite product-preserving lax functor out of it. Many familiar categorical structures are models of cartesian double theories, including categories, presheaves, monoidal categories, braided and symmetric monoidal categories, 2-groups, multicategories, and cartesian and cocartesian categories. We show that every cartesian double theory has a unital virtual double category of models, with lax maps between models given by cartesian lax natural transformations, bimodules between models given by cartesian modules, and multicells given by multimodulations. In many cases, the virtual double category of models is representable, hence is a genuine double category. Moreover, when restricted to pseudo maps, every cartesian double theory has a virtual equipment of models, hence an equipment of models in the representable case. Compared with 2-monads, double theories have the advantage of being straightforwardly presentable by generators and relations, as we illustrate through a large number of examples.

Physical attributes and performance differences: Server vs. spiker players in men’s doubles sepak takraw

Background Problems: Scientific research exploring the comparative analysis of the physical, physiological, and anthropometric characteristics associated with the server and spiker positions in double-category sepak takraw remains relatively scarce. Research Objectives: This study aims to compare the physical, physiological, and anthropometric characteristics between players in the server and spiker positions in the double category of sepak takraw. Methods: The research method used is an observational study with a quantitative approach. This study consisted of 18 male sepak takraw athletes, with 9 athletes each playing as servers and 9 athletes playing as spikers. The statistical analyses were conducted utilizing the SPSS Version 22 application. Findings and Results: The results of this study showed that there were statistically significant differences in body height, aerobic capacity, and leg strength between server and speaker positions (p < 0.05). Meanwhile, there were no significant differences in the variables age, weight, BMI, fat percentage, speed, agility, and flexibility. Dynamic, attractive game requirements and manoeuvres in the air for players in the spiker position have the potential to contribute to superior VO2max values and increased leg muscle strength compared to players in the server position. Conclusion: The conclusion of this study is that athletes in the spiker position show superior anthropometric and physiological attributes compared to those in the server position. Different understandings of the different characteristics associated with each position in the sport of sepak takraw provide new insights and views for coaches and athletes themselves. In addition, it is hoped that this research will have implications for optimizing training programs that adapt to the demands of each position. Furthermore, these findings serve as a reference for talent search events that assist in the identification and development of athletes with optimal physiological profiles for their respective roles in double-category sepak takraw.

DEVELOPMENT OF WEB-BASED PENCAK SILAT DOUBLE CATEGORY TRAINING MEDIA

Pencak silat rules have changed in terms of the competition system and in terms of scoring. The artistic category in pencak silat consists of single category, double category, team category and solo creative category, each of which has its own characteristics. 49 people participated in filling out the questionnaire from the preliminary research which was divided from 19 coaches and 30 artistic category athletes from the Special Region of Yogyakarta and Central Java province. 83.7% of respondents agreed that the doubles category is the most difficult move to learn. As many as 87.8% of respondents have never encountered digital-based double-category training media. This research aims to produce a training media in the form of a web-based to facilitate the double-category training process that is valid, feasible, and effective to be used by coaches and athletes. This study uses the research and development method with the ADDIE model. Product quality results are tested from 3 aspects: validity, feasibility, and effectiveness. The web product developed is considered valid as it has an average score of 90.8% from material experts and 95% from media expert. The media developed are also considered feasible, with what was obtained in this research being web media with an average score percentage of 89°% from the athletes and an average score of 92% from the coaches. The media developed is declared effective using the paired sample t-test and both media obtain the results that show sig, (2-tailed) of 0.000 < 0.05. So that the development of this training media can be used to practice easily by arts category pencak silat athletes. The hope is that the development of digital-based media can continue to be developed and studied repeatedly academically.<p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/soc/0069/a.php" alt="Hit counter" /></p>

Structured and Decorated Cospans from the Viewpoint of Double Category Theory

Structured and Decorated Cospans from the Viewpoint of Double Category Theory

A Simplicial Category for Higher Correspondences

In this work we propose a realization of Lurie’s prediction that inner fibrations \(p: X \rightarrow A\) are classified by A-indexed diagrams in a “higher category” whose objects are \(\infty \)-categories, morphisms are correspondences between them and higher morphisms are higher correspondences. We will obtain this as a corollary of a more general result which classifies all simplicial maps between ordinary simplicial sets in a similar fashion. Correspondences between simplicial sets (and \(\infty \)-categories) are a generalization of the concept of profunctor (or bimodule) pertaining to categories. While categories, functors and profunctors are organized in a double category, we will exhibit simplicial sets, simplicial maps, and correspondences as part of a simplicial category. This allows us to make precise statements and provide proofs. Our main tool is the language of double categories, which we use in the context of simplicial categories as well.

Structured versus Decorated Cospans

One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor L:A→X, a `structured cospan' is a diagram in X of the form L(a)→x←L(b). We give a new proof that if A and X have finite colimits and L preserves them, there is a symmetric monoidal double category whose objects are those of A and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor F:A→Cat, a `decorated cospan' is a diagram in A of the form a→m←b together with an object of F(m). Generalizing the work of Fong, we show that if A has finite colimits and F:(A,+)→(Cat,×) is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of A and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take X=∫F to be the Grothendieck category of F. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.

Il luogo del sogno e l’allucinazione negativa. Merleau-Ponty e Freud

In his notes for the course entitled “The Problem of Passivity: Sleep, the Unconscious, Memory,” Merleau-Ponty describe the notion of “negative hallucination” as “a perception, but not recognized for what it is.” This essay analyses this figure as it is taken up by Merleau-Ponty in direct dialogue with Freud’s work. To begin, through the double category of the “negative” and the “perceived,” Merleau-Ponty broaches the question of the “place” of dreams by adopting an eccentric position that sheds light on an unexplored pathway of Freudian theory. Moreover, a crucial point can be found in the idea of a negative margin, that appears to implicitly configure a redefinition of the hallucinatory in the direction of chiasm. To highlight this aspect, I review in a preliminary manner some passages of Phenomenology of Perception in which the question of the “place” of negative hallucination emerges in particularly dramatic terms.

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.

Physiological Characteristics of Indonesian Junior Badminton Players: Men's Double Category

The primary aim of this study is to assess the VO2max and anthropometric characteristics of men's double category; Next is to analyze match characteristics of men's double category; and finally, to compare the heart rate (HR) and blood lactate concentration calculated on-court with that obtained under laboratory conditions during an incremental test. 12 junior badminton players from Jaya Raya Badminton Club were enrolled as the subjects for this study. All participants were partnered up in 6 pairs (pair 1 to pair 6). During the first session (laboratory condition), VO2max was measured by using a velocity dependent ramp test (INCS) based on incremental protocols. During the second session (on-court condition), the participants played a simulated match on court. A total of 350 rallies were analyzed. Statistical analyses revealed that double junior men's players hit the shuttlecock more often using drive shots techniques (466 shots). The ANOVA revealed significant differences regarding fB between pair 3 vs pair 6 (p=0.004), pair 4 vs pair 6 (p=0.030), and pair 5 vs pair 6 (p=0.023). For the lactate measures, ANOVA revealed significant main effects of time (p=0.001) and group (p=0.001). Furthermore, the significance of HR average among laboratory and on-court conditions was established with the help of an independent t-test. The study demonstrated that the physiological characteristics of Indonesian men's double category are intermittent activities of high and low intensities, interspersed by short recovery periods. It was also discovered that the adrenergic strategy results in enhanced HR and lactate levels in the laboratory than on-court conditions.

Системный подход в отечественной системологии: транзитивность философского смысла и значения

The article is devoted to the issue of a systems approach in the context of philosophical problems. The necessity of revising system representations and expanding the dynamic component, according to the requirements of the time, is substantiated. Attention is focused on the evolution of the concept of the systems approach in Russian systemology. It is shown that the uniqueness of the evolution of the systems approach in Russia is due to the problem of determining its role and place in the conditions of the dominance of materialistic dialectics in the Soviet period. The aim of the article is to trace the modern formation of the synthesis of the systems approach and dialectics. Through historical analysis, the emergence of basic concepts on which system-dialectical synthesis is carried out today is revealed. Variants of interpretation of the systems approach in Russian science and philosophy are highlighted – its relationship with the philosophical principle of sistemicity, an interdisciplinary scientific field, and the field of private sciences. It is shown that the systems approach, enriched by the dynamic capabilities of the dialectic and synergetic approaches, is qualitatively expanding its application possibilities. The line of research raised by V. N. Sagatovskiy and continued by E. G. Vinogray is the problem of the causes of the formation of systems as a result of resolving systemic contradictions, complemented by the possibility of forming systems based on emerging synergistic relationships. It is proposed to consider the contradiction and synergy as a double category of system-forming factors in the process of formation of the modern concept of the systems approach.

Open Petri nets

AbstractThe reachability semantics for Petri nets can be studied using open Petri nets. For us, an “open” Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category Open(Petri), which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category ${\mathbb O}$ pen(Petri). We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of ${\mathbb O}$ pen(Petri). The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs.

The Gray Monoidal Product of Double Categories

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category {mathbb {A}}, the corresponding internal hom functor sends a double category {mathbb {B}} to the double category whose 0-cells are the double functors {mathbb {A}} rightarrow {mathbb {B}}, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

2-group actions and moduli spaces of higher gauge theory

A framework for higher gauge theory based on a 2-group is presented, by constructing a groupoid of connections on a manifold acted on by a 2-group of gauge transformations, following previous work by the authors where the general notion of the action of a 2-group on a category was defined. The connections are discretized, given by assignments of 2-group data to 1- and 2-cells coming from a given cell structure on the manifold, and likewise the gauge transformations are given by 2-group assignments to 0-cells. The 2-cells of the manifold are endowed with a bigon structure, matching the 2-dimensional algebra of squares which is used for calculating with 2-group data. Showing that the action of the 2-group of gauge transformations on the groupoid of connections is well-defined is the central result. The effect, on the groupoid of connections, of changing the discretization is studied, and partial results and conjectures are presented around this issue. The transformation double category that arises from the action of a 2-group on a category, as defined in previous work by the authors, is described for the case at hand, where it becomes a transformation double groupoid. Finally, examples of the construction are given for simple choices of manifold: the circle, the 2-sphere and the torus.

At the origins of a tenacious narrative: Jacob Thomasius and the history of double truth

ABSTRACTThis article enquires into the origins of the historiographical notion of double truth, a prominent and controversial category in the modern study of medieval philosophy. I believe that these origins are to be found in a short text by Jacob Thomasius from 1663, entitled De duplici & contradictoria veritate, which stands as a very early and highly original example of a history of double truth. I propose a detailed analysis of this document in order to shed light on the mechanisms that transformed duplex veritas from a keyword in Thomasius’s Protestant milieu into a historiographical category. As I show, the De duplici & contradictoria veritate provides a historical legitimation of Thomasius’s own brand of Lutheran Aristotelianism. It does so in a highly ambiguous fashion, namely by bringing together the Lutheran theologian and proponent of double truth Daniel Hofmann with anonymous medieval “Averroists”. I venture an explanation for Thomasius’s line of action by uncovering two of his implicit sources.

Morphisms of Open Games

We define a notion of morphisms between open games, exploiting a surprising connection between lenses in computer science and compositional game theory. This extends the (perhaps more intuitively obvious) definition of globular morphisms as mappings between strategy profiles that preserve best responses, and hence in particular preserve Nash equilibria. We construct a symmetric monoidal double category in which the horizontal 1-cells are open games, vertical 1-morphisms are lenses, and 2-cells are morphisms of open games. States (morphisms out of the monoidal unit game) in the vertical category give a flexible solution concept that includes both Nash and subgame perfect equilibria. Products in the vertical category give an external choice operator that is reminiscent of products in game semantics, and is useful in practical examples. We illustrate the above two features with a simple worked example from microeconomics, the market entry game.

What We Learned from Mediated Embodiment Experiments and Why It Should Matter to Policymakers

Abstract When people embody a virtual or a robotic avatar, their sense of self extends to the body of that avatar. We argue that, as a consequence, if the avatar gets harmed, the person embodied in that avatar suffers the harm in the first person. Potential scenarios involving physical or psychological harm caused to avatars gives rise to legal, moral, and policy implications that need to be considered by policymakers. We maintain that the prevailing distinction in law between “property” and “person” categories compromises the legal protection of the embodied users. We advocate for the inclusion of robotic and virtual avatars in a double category, property–person, as the property and the person mingle in one: the avatar. This hybrid category is critical to protecting users of mediated embodiment experiences both from potential physical or psychological harm and property damage.

Geodesics on path spaces and double category

Let [Formula: see text] be a Riemannian manifold and [Formula: see text] be the space of all smooth paths on [Formula: see text]. We describe geodesics on path space [Formula: see text]. Normal neighborhoods on [Formula: see text] have been discussed. We identify paths on [Formula: see text] under “back-track” equivalence. Under this identification, we show that if [Formula: see text] is complete, then geodesics on the path space yield a double category. This double category has a natural interpretation in terms of the worldsheets generated by freely moving (without any external force) strings.

Algebraic weak factorisation systems I: Accessible AWFS

Algebraic weak factorisation systems (awfs) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad–monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of awfs—drawing on work of previous authors—and complete the theory with two main new results. The first provides a characterisation of awfs and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an awfs by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an awfs, and that the awfs so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of awfs are developed, emphasising particularly those aspects which go beyond the non-algebraic situation.