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The category of directed graphs is isomorphic to a particular category whose objects are labeled undirected bipartite graphs and whose morphisms are undirected graph morphisms that respect the labeling. Based on this isomorphism, we begin by showing that the class of all directed graphs is a Graph Isomorphism Complete class. Afterwards, by extending this categorical framework to weighted prime graphs, we prove that the categories of multidirected graphs with and without self-loops are each isomorphic to a particular category of weighted prime graphs. Consequently, we prove that these classes of multidirected graphs are also Graph Isomorphism Complete. [ABSTRACT FROM AUTHOR]

Biologists, philosophers, and mathematicians building upon Robert Rosen's non-algorithmic theories of life using Relational Biology and Category Theory have continued to develop his theory and modeling approaches. There has been general agreement that the impredicative, self-referential, and complex nature of living systems negates an algorithmic approach. Rosen's main goal was to answer, "What is Life?". Many believe he provided the best but minimum answer using a cellular, metabolism–repair or (M, R)-system as a category-theoretic model. It has been challenging, however, to incorporate his theory to develop a fully non-algorithmic methodology that retains the essence of his thinking while creating more operational models of living systems that can be used to explore other facets of life and answer different questions. Living systems do more than the minimum in the real world beyond the confines of definition alone. For example, ecologists ask how living systems inherently mitigate existential risk from climate change and biodiversity loss through their complex self-organization. Loop Analysis, a signed graph technique, is discussed as a hybrid algorithmic/non-algorithmic methodology in Relational Biology. This methodology can be used at the ecosystem level with standard non-algorithmic field data as per McAllister's description of the algorithmic incompressibility of empirical data of this type. An example is described showing how the North Atlantic Carbon Pump, an important planetary life support system, is situated in the plankton community and functions as a mutualistic ecosystem chimera. It captures carbon from the atmosphere as an extended (M, R)-system and processes it until it is sequestered in the marine sediments. This is an important process to alleviate climate change in magnitude equal to or larger than the sequestration of carbon on land with forests. It is suggested that the ecosystem level should replace the cellular and organismic levels as the main system unit in biology and evolution since all life exists and evolves with full functional potential in ecosystem networks and not laboratory test tubes. The plankton ecosystem is the largest after the total biosphere and consists of evolutionary links and relationships that have existed for eons of time. If there was ever a genuine robust, highly self-organized ecosystem, it would be planktonic. Severing the links in these thermodynamically open networks by focusing on lower levels of the biological hierarchy loses the critical organization of how life exists on this planet. There is no theory to regain this crucial 'omitted' ecological relational causality at the cell or organismal levels. At the end of the paper, some future directions are outlined. [ABSTRACT FROM AUTHOR]

In this paper, we propose a formalism for knowledge manipulation, based on conceptual graphs, which is supported by category theory, and which can be the basis for the development of knowledge-based systems. The model we propose is an extension of the model based on conceptual graphs. In this model, the application of an inference rule between logical formulas represented by two conceptual graphs is reduced to the identification of an arrow in a category. To this end, we introduce several new notions such as: the category of conceptual graphs, the category of classes of conceptual graphs, the conceptual category of a model and the conceptual category of inference of a model. [ABSTRACT FROM AUTHOR]

This paper addresses understanding and categorizing language by using Markov categories to establish a mathematical framework for semantic concepts. This framework enables us to measure the semantic similarity between linguistic expressions within a given text. Furthermore, this approach enables the measurement and control of uncertainty in language categorization and the creation of metrics for evaluating semantic similarity. We provide use cases to demonstrate how the proposed methods can be applied and computed, focusing on their interpretability and the universality of categorical constructions. This work contributes to the field by offering a novel perspective on semantic similarity and uncertainty metrics in language processing, generating criteria to automate their computation. [ABSTRACT FROM AUTHOR]

An algebraic interpretation of multigraph networks is introduced in relation to conscious experience, brain and body. These multigraphs have the ability to merge by an associative binary operator ⊙ , accounting for biological composition. We also study a mathematical formulation of splitting layers, resulting in a formal analysis of the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal constraints that conscious interactions may impose. An important result is the prediction of structural topological changes after conscious interactions. These results may inspire further use of formal mathematics to describe and predict new features of conscious experience while aligning well with formal tries to mathematize phenomenology, phenomenological tradition and applications to artificial consciousness. [ABSTRACT FROM AUTHOR]

With a view to the question of a possible revival of systematic transcendental philosophy, the article follows the problem developments from outgoing neo-Kantianism to post-neo-Kantianism to the present day. Neo-Kantianism decisively determined this development, since, following the ‘analytical method’ of the prolegomena, it based its principletheoretical claim on the ‘factum of science’ and other facta of culture. Since the fact of science and culture at the beginning of the 20th century has changed. However, this ‘validity objectivism’ rapidly lost credibility. Although he was also criticized within neo-Kantianism (Emil Lask, Paul Natorp), these isolated approaches and calls for a renewal and deepening of the reflection of principles remained ineffective, since one sought salvation rather in the forced orientation to individual sciences or in new Orthodoxies and academic ‘research’. For a continuity of transcendental philosophical questions, but by no means for a revival of the systematic transcendental philosophy, the works of vom Verf. under the title Kritische Dialektik und Transcendentalontologie , post-neo-Kantian approaches of the interwar and post-war period by Richard Hönigswald and Wolfgang Cramer, Bruno Bauch and Hans Wagner, as well as Robert Reininger and Erich Heintel, documented a systematically oriented succession in the works of Werner Flach and Harald Holz to the present day, and - under conditions of philosophy of consciousness - in the works of Dieter Henrich and Karl-Otto Apel, an at least there was a thematic continuation. However, a “revival of systematic transcendental philosophy” would require more radical approaches than have been delivered in the last 100 years: it would simply require a return to the ‘synthetic method' of critique of reason , which “bases nothing as a given, except reason itself”.

AbstractInteractive data visualization has become a staple of modern data presentation. Yet, despite its growing popularity, we still lack a general framework for turning raw data into summary statistics that can be displayed by interactive graphics. This gap may stem from a subtle yet profound issue: while we would often like to treat graphics, statistics, and interaction in our plots as independent, they are in fact deeply connected. This article examines this interdependence in light of two fundamental concepts from category theory: groups and monoids. We argue that the knowledge of these algebraic structures can help us design sensible interactive graphics. Specifically, if we want our graphics to support interactive features which split our data into parts and then combine these parts back together (such as linked selection), then the statistics underlying our plots need to possess certain properties. By grounding our thinking in these algebraic concepts, we may be able to build more flexible and expressive interactive data visualization systems. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

This paper examines the use of algorithms and non-algorithmic models in mathematics and science, especially in biology, during the past century by summarizing the gradual development of a conceptual rationale for non-algorithmic models in biology. First, beginning a century ago, mathematicians found it impossible to constrain mathematics in an algorithmic straitjacket via öö's Incompleteness Theorems, so how would it be possible in biology? By the 1930s, biology was resolutely imitating classical physics, with biologists enforcing a reductionist agenda to expunge function, purpose, teleology, and vitalism from biology. Interestingly, physicists and mathematicians often understood better than biologists that mathematical representations of living systems required different approaches than those of dead matter. Nicolas Rashevsky, the Father of Mathematical Biology, and Robert Rosen, his student, pointed out that the complex systems of life cannot be reduced to machines or mechanisms as per the Newtonian paradigm. Robert Rosen concluded that living systems are not amenable to algorithmic models that are primarily syntactical. Life requires semantics for its description. Rashevsky and Rosen pioneered Relational Biology, initially using Graph Theory to model living systems. Later, Rosen created a metabolic–repair model (M, R)-system using Category Theory to encode the basic entailments of life itself. Although reductionism still dominates in current biology, several subsequent authors have built upon the Rashevsky–Rosen intellectual foundation and have explained, extended, and explored its ramifications. Algorithmic formulations have become increasingly inadequate for investigating and modeling living systems. Biology is shifting from a science of simple systems to complex ones. This transition will only be successful once mathematics fully depicts what it means to be alive. This paper is a call to mathematicians from biologists asking for help in doing this. [ABSTRACT FROM AUTHOR]

We study existence and multiplicity of nontrivial solutions of the following problem { − Δ p u + (− Δ p) s u = λ | u | q − 2 u + | u | p ⁎ − 2 u in Ω , u = 0 on R N ∖ Ω , where Ω ⊂ R N is a bounded open set with smooth boundary, dimension N ≥ 2 , parameter λ > 0 , exponents 0 < s < 1 < p < N , while q ∈ (1 , p ⁎) with p ⁎ = N p N − p. The problem is driven by an operator of mixed order obtained by the sum of the classical p -Laplacian and of the fractional p -Laplacian. We analyze three different scenarios depending on exponent q. For this, we combine variational methods with some topological techniques, such as the Krasnoselskii genus and the Lusternik–Schnirelman category theories. [ABSTRACT FROM AUTHOR]

This work focuses on establishing the relationship between bipolar fuzzy automata, reverse bipolar fuzzy automata, and double bipolar fuzzy automata. It also introduces the concepts of bipolar fuzzy subsystems, reverse bipolar fuzzy subsystems, and double bipolar fuzzy subsystems, and explores various properties associated with these subsystems. Furthermore, the paper aims to introduce the categorical aspects of bipolar fuzzy automata and reverse bipolar fuzzy automata, along with their functorial relationship. [ABSTRACT FROM AUTHOR]

Cyclic proof systems permit derivations that are finite graphs in contrast to conventional derivation trees. The soundness of such proofs is ensured by imposing a soundness condition on derivations. The most common such condition is the global trace condition (GTC), a condition on the infinite paths through the derivation graph. To give a uniform treatment of such cyclic proof systems, Brotherston proposed an abstract notion of trace. We extend Brotherston's approach into a category theoretical rendition of cyclic derivations, advancing the framework in two ways: first, we introduce activation algebras which allow for a more natural formalisation of trace conditions in extant cyclic proof systems. Second, accounting for the composition of trace information allows us to derive novel results about cyclic proofs, such as introducing a Ramsey-style trace condition. Furthermore, we connect our notion of trace to automata theory and prove that verifying the GTC for abstract cyclic proofs with certain trace conditions is PSPACE-complete. [ABSTRACT FROM AUTHOR]

To analyze traditional jazz music in a way that attends to its idiom and performance practices, I develop two formalizations of jazz leadsheets, while also considering the relationship between sound and algebraic theory it initiates. First, I represent leadsheets as digraphs of chordal movement and articulate a digraph category. Second, I represent leadsheets as strings of chords and develop quotient semigroups on these strings using equivalence relations based on jazz performance practice. I produce a metric for leadsheets from these quotients. Then I apply the two formalizations to a set of twenty jazz standards and consider how the results compare with a listener's experience. I also include two original leadsheets composed to test the formalizations. Though they are grounded in jazz conventions, I conclude these formalizations often do not agree with a listener's perception; I consider what this implies about their construction and the general project of music algebra. [ABSTRACT FROM AUTHOR]

Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, P -affine schemes can be defined as objects of the opposite of the category of algebras over an operad P . An example is the opposite of the category of associative algebras. The category of operadic schemes of an operad carries a canonical tangent structure. This paper aims to initiate the study of the geometry of operadic affine schemes via this tangent category. For example, we expect the tangent structure over the opposite of the category of associative algebras to describe algebraic non-commutative geometry. In order to initiate such a program, the first step is to classify differential bundles, which are the analogs of vector bundles for differential geometry. In this paper, we prove that the tangent category of affine schemes of the enveloping operad P (A) over a P -affine scheme A is precisely the slice tangent category over A of P -affine schemes. We are going to employ this result to show that differential bundles over a P -affine scheme A are precisely A-modules in the operadic sense. [ABSTRACT FROM AUTHOR]

In the present paper, we outline and expound the fundamental and novel qualitative-cum-philosophical premises, principles, ideas, concepts, constructions and results that originate from our ongoing research project of applying the new conceptual panoply and the novel technical machinery of Abstract Differential Geometry (ADG) to various persistently outstanding issues in Quantum Gravity (QG) (Mallios and Raptis Int. J. Theor. Phys. 40, 1885, 2004, Int. J. Theor. Phys. 41, 1857, 2002, Int. J. Theor. Phys. 42, 1479, 2003; Raptis Int. J. Theor. Phys. 39, 1233, 2000, Int. J. Theor. Phys. 39, 1703, 2000, Int. J. Theor. Phys. 45, 79, 2006, Int. J. Theor. Phys. 46, 688, 2007, Int. J. Theor. Phys. 45, 1495, 2006, Int. J. Theor. Phys. 46, 1137, 2007, Int. J. Theor. Phys. 46, 3009, 2007; Raptis and Zapatrin Int. J. Theor. Phys. 39, 1, 2000, Class. Quant. Grav. 20, 4187, 2001). This paper may be regarded as a sequel to the paper (Raptis Int. J. Theor. Phys. 46, 3009, 2007) in the aftermath of the paper (Raptis 2024), which is currently in press. At the end of the paper, we discuss the potential philosophical repercussions of two possible future research routes that the main stream of our applications of ADG to QG may bifurcate towards in view of three independent, but overlapping, research papers that are currently under development (Raptis 2024a, 2024b, 2024c). [ABSTRACT FROM AUTHOR]

Recent research has suggested that category theory can provide useful insights into the field of machine learning (ML). One example is improving the connection between an ML problem and the design of a corresponding ML algorithm. A tool from category theory called a Kan extension is used to derive the design of an unsupervised anomaly detection algorithm for a commonly used benchmark, the Occupancy dataset. Achieving an accuracy of 93.5% and an ROCAUC of 0.98, the performance of this algorithm is compared to state-of-the-art anomaly detection algorithms tested on the Occupancy dataset. These initial results demonstrate that category theory can offer new perspectives with which to attack problems, particularly in making more direct connections between the solutions and the problem's structure. [ABSTRACT FROM AUTHOR]

This paper proposes mathematical foundations for music theory and composition. While mathematical methods have proven effective in music theory, a deficiency in rigorous mathematical foundations often leads to ad-hoc constructions and a reliance on intuitive notions with inexplicit definitions. The proposal introduces a comprehensive environment for encoding musical phenomena, as well as a theory of musical parameter. Drawing from the methodologies of (Mazzola, Guerino. 2002. The Topos of Music: Geometric Logic of Concepts, Theory, and Performance. Basel: Birkhäuser Verlag), the proposed framework aims to extend his theory of forms to accommodate a wider class of musical structures. Additionally, I advocate for a framework characterized by “comprehensive comprehension” (accommodating a broad class of musical objects), “explicit encoding” (capturing essential features of objects), and “limited ontological commitments” (minimizing posited ontological primitives). These three aspects aim to ensure the framework's generality, explicit representation of structure, and economic efficiency in theoretical constructions. Minimizing ontological commitments also has the benefit of facilitating easier comparisons between entities. This approach is motivated by the desire for a highly versatile framework that enables systematic and standardized construction methods without constraining the intended domain of discourse. To achieve these goals, the paper introduces a mathematical theory of structure grounded in topos theory. By offering a unified and systematic approach, this work contributes to establishing a more rigorous and standardized basis for mathematical music theory. This, in turn, fosters a deeper understanding of musical phenomena and enables more robust theoretical constructions and compositional applications. [ABSTRACT FROM AUTHOR]

This paper assembles a unifying framework encompassing a wide variety of mathematical instruments used to compare different theories. The main theme will be the idea that theory comparison techniques are most easily grasped and organized through the lens of category theory. The paper develops a table of different equivalence relations between theories and then answers many of the questions about how those equivalence relations are themselves related to each other. We show that Morita equivalence fits into this framework and provide answers to questions left open in Barrett and Halvorson [4]. We conclude by setting up a diagram of known relationships and leave open some questions for future work. [ABSTRACT FROM AUTHOR]

The study of the interactions among intentional agents, with rationality being the main source of intentional behavior, requires mathematical tools capable of capturing systemic effects. Here, we choose an alternative toolbox based on Category Theory. We examine potential level-agnostic formalisms, presenting three categories: PR , G , and an encompassing one, I. The latter allows for representing dynamic rearrangements of the interactions among different agents. Systems represented in I capture the dynamic interactions among the interfaces of their sub-agents, changing the connections among them based on their internal states. We illustrate the expressive power of this formalism in four different instances, providing practitioners with a toolbox for representing cases of interest and facilitating their modular analysis. [ABSTRACT FROM AUTHOR]

We discuss the potential of applying category theory to the study of consciousness. We first review a recent proposal from the neurosciences of consciousness to illustrate the “correlational project”, using the integrated information theory of consciousness as an example. We then discuss some technical preliminaries related to categories and in particular to the notion of a functor, which carries the bulk of conceptual weight in many current discussions. We then look at possible payoffs of this project—getting to grips with the hard problem, theory integration, and exploiting explanatory dualities—and discuss possible avenues for further research, stressing the need to better develop the categorical representation of consciousness, in particular its phenomenological structure. A better understanding of consciousness cannot be achieved by merely studying the physical brain. By contrast, the categorical treatment even suggests application beyond the domain of neuroscience, for example in computer science and artificial intelligence research, while also emphasizing the primacy of (phenomenal) experience. [ABSTRACT FROM AUTHOR]

Laser powder bed fusion (LPBF) is a promising metal additive manufacturing technology for producing functional components. However, there are still a lot of obstacles to overcome before this technology is considered mature and trustworthy for wider industrial applications. One of the biggest obstacles is the difficulty in ensuring the repeatability of process and the reproducibility of products. To tackle this challenge, a prerequisite is to represent and communicate the data from the part realisation process in an unambiguous and rigorous manner. In this paper, a semantically enriched LPBF part data model is developed using a category theory-based modelling approach. Firstly, a set of objects and morphisms are created to construct categories for design, process planning, part build, post-processing, and qualification. Twenty functors are then established to communicate these categories. Finally, an application of the developed model is illustrated via the realisation of an LPBF truncheon. [ABSTRACT FROM AUTHOR]

Let T。the category of unpointed spaces. We introduce some results on category theory. We construct and proof some propositions and examples for the category of T., which we have used them to prove our main theorem: Suppose that Y be a topological space. If we define two functors on T。by P(X) = X x Y and M(Z) = map (Y, Z) then P and M are adjoint functors. [ABSTRACT FROM AUTHOR]

Where Being and Event was a book of meta-ontology, demonstrating how things be, Alain Badiou tells us that Logics of Worlds is an objective phenomenology demonstrating how (and what) things do. As such, it should also be considered a book about power: how objects function in their worlds, how change is possible, and what the event does to worlds and the objects in them. Attending to this, this article shows two conflicting forms of power emerge in Logics of Worlds, the power of appearance and the power of change. In this article, the author argues their incommensurability suggests an overall failure of a consistent account of power from Badiou. [ABSTRACT FROM AUTHOR]

In every matured theory, there is a need to investigate possible relationships between considered objects. To address this issue, it is natural to relate a category with given model of computing. Thanks to such approach, many properties are unified and simplified. In this paper, we investigate how category theory can be used to give a faithful semantics for reaction systems. In particular, we propose and discuss possible approaches to the problem of defining morphisms between reaction systems. We provide the definition of morphism that keeps the behaviour of the original reaction system. Especially, some equivalences of reaction systems are reflected in terms of morphisms. For this purpose we expressed isomorphisms and sections in term of transition systems. Moreover, the accelerating morphism defined in the last section gives a new approach for including time in reaction systems. [ABSTRACT FROM AUTHOR]