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Compositionality is at the core of programming languages research and has become an important goal toward scalable verification of large systems. Despite that, there is no compositional account of linearizability, the gold standard of correctness for concurrent objects. In this article, we develop a compositional semantics for linearizable concurrent objects. We start by showcasing a common issue, which is independent of linearizability, in the construction of compositional models of concurrent computation: interaction with the neutral element for composition can lead to emergent behaviors, a hindrance to compositionality. Category theory provides a solution for the issue in the form of the Karoubi envelope. Surprisingly, and this is the main discovery of our work, this abstract construction is deeply related to linearizability and leads to a novel formulation of it. Notably, this new formulation neither relies on atomicity nor directly upon happens-before ordering and is only possible because of compositionality, revealing that linearizability and compositionality are intrinsically related to each other. We use this new, and compositional, understanding of linearizability to revisit much of the theory of linearizability, providing novel, simple, algebraic proofs of the locality property and of an analogue of the equivalence with observational refinement. We show our techniques can be used in practice by connecting our semantics with a simple program logic that is nonetheless sound concerning this generalized linearizability. [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]

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]

Unlike traditional investing, where decisions follow a clear financial calculus, it is unclear how and why funders support hybrid ventures. To address this question, we analyze the varied priority that investors place on social impact versus financial returns and draw on categories theory to argue that different priority orderings associate with different perceptions of how hybridity aligns with different investment goals. Results show that funders who prioritize financial goals react positively when they perceive a venture exhibits greater hybridity, whereas funders who prioritize social impact do not. Our findings contribute to research on impact investing, hybrid organizations, and categories theory. [ABSTRACT FROM AUTHOR]

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Diverse explanations or theories of consciousness are arrayed on a roughly physicalist-to-nonphysicalist landscape of essences and mechanisms. Categories: Materialism Theories (philosophical, neurobiological, electromagnetic field, computational and informational, homeostatic and affective, embodied and enactive, relational, representational, language, phylogenetic evolution); Non-Reductive Physicalism; Quantum Theories; Integrated Information Theory; Panpsychisms; Monisms; Dualisms; Idealisms; Anomalous and Altered States Theories; Challenge Theories. There are many subcategories, especially for Materialism Theories. Each explanation is self-described by its adherents, critique is minimal and only for clarification, and there is no attempt to adjudicate among theories. The implications of consciousness explanations or theories are assessed with respect to four questions: meaning/purpose/value (if any); AI consciousness; virtual immortality; and survival beyond death. A Landscape of Consciousness, I suggest, offers perspective. [ABSTRACT FROM AUTHOR]

Collapsibility is a practical and useful technique for dimension reduction in multidimensional contingency tables. In this paper, we consider marginal log‐linear models for studying collapsibility and related aspects in such tables. These models generalize ordinary log‐linear and multivariate logistic models, besides several others. First, we obtain some characteristic properties of marginal log‐linear parameters. Then we define collapsibility and strict collapsibility of these parameters in a general sense. Several necessary and sufficient conditions for collapsibility and strict collapsibility are derived based on simple functions of only the cell probabilities, which are easily verifiable. These include results for an arbitrary set of marginal log‐linear parameters having some common effects. The connections of strict collapsibility to various forms of independence of the variables are explored. We analyze some real‐life datasets to illustrate the above results on collapsibility and strict collapsibility. Finally, we obtain a result relating parameters with the same effect, but different margins for an arbitrary table, and demonstrate smoothness of marginal log‐linear models under collapsibility conditions. [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]

Aristotle is believed to have introduced the focal meaning of friendship in Eudemian Ethics VII.2 and then to have formulated it more generally in Metaphysics Γ.2. Bonitz's unjustifiable emendation of the text underscores these interpretations. This paper therefore reads the MSS and supposes that the EE passage introduces a wider focal meaning based on the theory presented in the Categories, one that does not imply referential relationships between the primary and secondary terms. This wider focal meaning can help explain disputed terms in practical philosophy, such as "friendship" and "like-mindedness," without invoking the Form of the Good. [ABSTRACT FROM AUTHOR]

Early literacy acquisition is influenced by children’s social and cultural background. Several empirical studies have shown that by considering literacy activities at home and providing cultural resources, the correlation between the social and cultural background and the children’s literacy achievement can be significantly reduced. This suggests a crucial importance of parents’ beliefs and their role as active agents in the acquisition of literacy skills at home. The aim of the present study is therefore to provide an overview of the range of relevant parental attitudes towards joint literacy activities. In an elicitation study based on the theory of planned behavior, the statements of N  =  25 German parents of pre-preschool and preschool children were recorded. All deductive categories based on the theory of planned behavior were confirmed, furthermore the category system could be differentiated inductively. Thus the parental beliefs about joint literacy activities were comprehensively mapped. It is discussed how the explorative results of this study can be used in further studies, for example, to generate a questionnaire based on the TPB to assess parental beliefs for joint literacy activities. [ABSTRACT FROM AUTHOR]

Lesson planning makes possible organizing a teaching sequence coherent with the expected knowledge. The purpose of this work is to characterize the knowledge that a secondary mathematics teacher puts into play when designs class lessons about quadrilaterals. We present a case study in which the categories of the Mathematics Teacher Specialized Knowledge are used to show the knowledge he/she uses in the designing of the aforementioned lesson plans. The outcomes show the relevance of the decisions taken by the teacher during teacher planning related to the didactical strategies and resources. [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]

We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high enough cardinality will have a single model in any high enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation. [ABSTRACT FROM AUTHOR]

In globular higher category theory the small category Θ0 of finite rooted trees plays an important role: for example the objects of Θ0 are the arities of the operations inside the free globular ω-operad B0 of Batanin, which B0 -algebras are models of globular weak ∞-categories; also this globular Θ0 is an important tool to build the coherator Θ∞W0 of Grothendieck which Sets-models are globular weak ∞-groupoids. Cubical higher category needs similarly its Θ0. In this work we describe, combinatorially, the small category Θ0 which objects are cubical pasting diagrams and which morphisms are morphisms of cubical sets. [ABSTRACT FROM AUTHOR]

We develop a theory of Cauchy sequences and intertwinings for morphisms of Cuntz semigroups, which generalizes all past approaches to study metric-like properties of the invariant. Further, the techniques presented here can be applied to all known refinements of the Cuntz semigroup, including those that may be used in new classification results. As a particular application, we introduce a Fraïssé theory for abstract Cuntz semigroups akin to the theory of Fraïssé categories developed by Kubiś. We also show that any (Cuntz) Fraïssé category has a unique Fraïssé limit which is both universal and homogeneous. Several examples of such categories and their Fraïssé limits are given throughout the paper. [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]

Enriched motivic A -spaces are introduced and studied in this paper, where A is an additive category of correspondences. They are linear counterparts of motivic Γ-spaces. It is shown that rational special enriched motivic Cor ˜ -spaces recover connective motivic bispectra with rational coefficients, where Cor ˜ is the category of Milnor–Witt correspondences. [ABSTRACT FROM AUTHOR]

Abstract algebraic concepts such as category are considered cornerstones on which logical consistency relies in any sophisticated study of natural phenomena. However, to the best of our knowledge, in molecular/genetic biology, their application is still severely limited because they capture neither the dynamics nor provide a visual form. The Petri net (PN) has often been used to illustrate visually parallel, asynchronous dynamic events in small data systems. A prototypal hybrid model combining both category theory and extended PNs may instead be indispensable for that purpose. This hybrid model incorporates 1) token-like elements of a group, 2) object-like places of a category, 3) square poles (rather than pentagon poles) that enable unique identifications of single-strand DNA sequences from the shape of its polygonal line, 4) creation/annihilation morphisms that generate/erase tokens, 5) Cartesian products 'Z5×Z2×...' that enable conversions between DNA and RNA sequences, 6) somatic recombinations (VDJ recombinations) for antibodies displayed concretely in category-theoretic form, 7) 'identity protein Δ' translated from a triplet of identity bases 'EEE' as an advanced concept from our previous display of the canonical central dogma, 8) illustrations of an incidence-matrix-like matrix A that includes operators as coordinates, and 9) basic topics concerning the canonical central dogma being displayed concretely using concepts of conventional category theory such as 'adjoint', 'adjoint functor', 'natural transformation', 'Yoneda's lemma' and 'Kan extension'. These ideas provide more advanced tools that expand our previous model concerning nucleic-acid-base sequences. Despite the nascent nature of our methodology, our hybrid model has potential in a variety of applications, illustrated using molecular/genetic sequences, in particular providing a simple dynamic/visual representation. With further improvements, this approach may prove effective in reducing the need for large data-storing systems. [ABSTRACT FROM AUTHOR]

Practical approaches to the notions of subsets and extension sets are compared, coming from broadly set-theoretic and category-theoretic traditions of mathematics. I argue that the set-theoretic approach is the most practical for 'looking down' or 'in' at subsets and the category-theoretic approach is the most practical for 'looking up' or 'out' at extensions, and suggest some guiding principles for using these approaches without recourse to either category theory or axiomatic set theory. [ABSTRACT FROM AUTHOR]

Theories of category learning have typically focused on how the underlying category structure affects the category representations acquired by learners. However, there is limited research as to how other factors affect what representations are learned and utilized and how representations might change across the time course of learning. We used a novel "5/5" categorization task developed from the well-studied 5/4 task with the addition of one more stimulus to clarify an ambiguity in the 5/4 prototypes. We used multiple methods including computational modeling to identify whether participants categorized on the basis of exemplar or prototype representations. We found that, overall, for the stimuli we used (schematic robot-like stimuli), learning was best characterized by the use of prototypes. Most importantly, we found that relative use of prototype and exemplar strategies changed across learning, with use of exemplar representations decreasing and prototype representations increasing across blocks. [ABSTRACT FROM AUTHOR]

Let T be a triangulated category with a proper class ξ of triangles. We introduce the notions of left Frobenius pairs, left (n -)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to ξ in T. We show how to construct left cotorsion pais from left n -cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. Some applications are given in the Gorenstein homological theory of triangulated categories. [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]

We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai's factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [ Finitary isomorphism of m-dependent processes , Amer. Math. Soc., Birkhauser, Providence, RI, 1992, pp. 373–376], which was first suggested by D. Rudolph [ A characterization of those processes finitarily isomorphic to a Bernoulli shift , Birkhäuser, Boston, Mass., 1981, pp. 1–64]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai's factor theorem for ergodic processes. Another consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [Math. Inst. Hautes Études Sci. 128 (2018), 1–119]. [ABSTRACT FROM AUTHOR]

Numerous real-world applications apply categorical data clustering to find hidden patterns in the data. The K-modes-based algorithm is a popular algorithm for solving common issues in categorical data, from outlier and noise sensitivity to local optima, utilizing metaheuristic methods. Many studies have focused on increasing clustering performance, with new methods now outperforming the traditional K-modes algorithm. It is important to investigate this evolution to help scholars understand how the existing algorithms overcome the common issues of categorical data. Using a research-area-based bibliometric analysis, this study retrieved articles from the Web of Science (WoS) Core Collection published between 2014 and 2023. This study presents a deep analysis of 64 articles to develop a new taxonomy of categorical data clustering algorithms. This study also discusses the potential challenges and opportunities in possible alternative solutions to categorical data clustering. [ABSTRACT FROM AUTHOR]

Business model innovation faces multiple tests of legitimacy. Most extant research in this area has been conducted from institutional and strategic perspectives while paying insufficient attention to the perspective of evaluators. Based on the institutionalization of China's online car-hailing industry from 2012 to 2018, this paper analyzes the legitimacy judgment of the stakeholders from the perspective of evaluator categorization and explores the legitimation mechanism of business model innovation. It finds that evaluators judge the legitimacy of business models based on category cognition. Therefore, to achieve the bridging, spillover, and accumulation effects of legitimacy, the legitimation strategy of online car-hailing platforms should dynamically adapt to different evaluators, judgment models, and categorization standards. Ultimately, as quantitative changes lead to qualitative changes, the legitimation of innovative business models is achieved in this way. In this process, stakeholders categorize and evaluate online car-hailing based on prototypes and value goals, and establish a two-way interactive mechanism, which is from behavior guided by cognition to cognition given feedback by behavior. This paper combines the legitimacy judgment with category theory to explain how individual cognition drives the emergence of new categories and identifies a series of legitimacy strategies based on categorization, thus providing theoretical support and practical inspiration for exploring the legitimation of business model innovation. [ABSTRACT FROM AUTHOR]

The theory of extensive categories determines in particular the notion of separable MV-algebra (equivalently, of separable unital lattice-ordered Abelian group). We establish the following structure theorem: An MV-algebra is separable if, and only if, it is a finite product of algebras of rational numbers—i.e., of subalgebras of the MV-algebra [ 0 , 1 ] ∩ Q. Beyond its intrinsic algebraic interest, this research is motivated by the long-term programme of developing the algebraic geometry of the opposite of the category of MV-algebras, in analogy with the classical case of commutative K -algebras over a field K. [ABSTRACT FROM AUTHOR]

The Language of Thought (LoT) hypothesis proposes that some collections of mental states and processes are symbol systems to explain language-like systematic properties of thought. Recent proponents of this hypothesis point to additional LoT-like properties in non-linguistic domains to claim that LoT remains the "best game in town" in terms of explanatory coverage. Nonetheless, LoT assumes but does not explain why/how symbolic representations connect to other (non-symbolic) formats. The perspective presented here is supposed to bridge this gap as a duality in a category theoretical sense: (perceptual) data are projected onto a base (conceptual) space in one direction, and in the opposite direction, these data are referenced by that space. Accordingly, perception is dual to conception. These constructions follow from a universal mapping principle affording an explanation for why/how symbolic and non-symbolic formats are connected: as the "best" possible transformation between the two forms-- so the slogan, LoT is universal. This view also sheds some light on the apparent pervasiveness of logic-like capacities across age-groups and species, and these constructions constitute special types of categories called toposes (topoi), and every topos has an interpretation in first-order logic. [ABSTRACT FROM AUTHOR]

We use Category Theory to construct a 'bridge' relating directed graphs with undirected graphs, such that the notion of direction is preserved. Specifically, we provide an isomorphism between the category of simple directed graphs and a category we call 'prime graphs category'; this has as objects labeled undirected bipartite graphs (which we call prime graphs), and as morphisms undirected graph morphisms that preserve the labeling (which we call prime graph morphisms). This theoretical bridge allows us to extend undirected graph techniques to directed graphs by converting the directed graphs into prime graphs. To give a proof of concept, we show that our construction preserves topological features when applied to the problems of network alignment and spectral graph clustering. [ABSTRACT FROM AUTHOR]

Idioms are important elements of language and culture, forming the essence of language and the core of vocabulary. Animal idioms naturally have different connotations in the Chinese and Western cultures, which can pose a tremendous challenge for translators. Thus, the reasons for translators' choices in this context are significant. To better understand this subject, this study investigates the cognitive category shifts in the translations of animal idioms in Journey to the West. Through the theory of cognitive category shifts, the paper finds that translators' choices are influenced by cognitive concepts derived from their living environments, cultural backgrounds, and knowledge reserves. [ABSTRACT FROM AUTHOR]

We express the fundamental properties of commutative polygroups (also known as canonical hypergroups) in category-theoretic terms, over the category Set formed by sets and functions. For this, we employ regularity as well as the monoidal structure induced on the category Rel of sets and relations by cartesian products. We highlight how our approach can be generalised to any regular category. In addition, we consider the theory of partial multirings and find fully faithful functors between certain slice or coslice categories of the category of partial multirings and other categories formed by well-known mathematical structures and their morphisms. [ABSTRACT FROM AUTHOR]

Category theory finds that markets partition producers into categories and producers who do not fit one specific category—or who span multiple categories—perform worse than their single-category peers. The dominant thread of this work argues that this miscategorization penalty arises when cognitive limits of categorization cause individual members of the market's audience to exclude or denigrate ill-fitting producers. I present a null model of markets in which a miscategorization penalty appears without being caused by a market audience: drawing on cognitive science and research on rugged landscapes, the model shows that producer herding behavior generates a spurious correlation between market outcomes and miscategorizations. The model further predicts the dynamics of categorical emergence and change over time. I establish these results in a simulation and discuss strategies by which this landscape model can be empirically distinguished or integrated with the standard account of an audience-driven penalty. Supplemental Material: The online appendix is available at https://doi.org/10.1287/orsc.2020.13770. [ABSTRACT FROM AUTHOR]

In this paper, we propose to consider the dependence structure of the trade/no trade categorical sequence of individual illiquid stocks returns. The framework considered here is wide as constant and time-varying zero returns probability are allowed. The ability of our approach in highlighting illiquid stock's features is underlined for a variety of situations. More specifically, we show that long-run effects for the trade/no trade categorical sequence may be spuriously detected in presence of a non-constant zero returns probability. Monte Carlo experiments, and the analysis of stocks taken from the Chilean financial market, illustrate the usefulness of the tools developed in the paper. [ABSTRACT FROM AUTHOR]

In a recent work [On factorization and vector bundles of conformal blocks from vertex algebras, preprint (2019), arXiv:1909.04683], Damiolini et al. showed that for a C 2 -cofinite rational vertex operator algebra , sheaves of conformal blocks are locally free and satisfy the factorization property. In this paper, we prove that if is C 2 -cofinite, the sewing of conformal blocks is convergent. This proves a conjecture proposed by Zhu [Global vertex operators on Riemann surfaces, Commun. Math. Phys. 165(3) (1994) 485–531] and Huang [Some open problems in mathematical two-dimensional conformal field theory, preprint (2016), arXiv:1606.04493], generalizing previous results on the convergence of products, iterates, and traces (but not pseudo-traces) of vertex operators and intertwining operators in [Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9(1) (1996) 237–302; Y. Z. Huang, A theory of tensor products for module categories for a vertex operator algebra, IV, J. Pure Appl. Algebra 100(1–3) (1995) 173–216; Y. Z. Huang, Differential equations, duality and modular invariance, Commun. Contemp. Math. 7(05) (2005) 649–706; Y. Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011), arXiv:1110.1929] and on the convergence of projective factors related to the central charge of the Virasoro algebra in [Y. Z. Huang, Two-Dimensional Conformal Geometry and Vertex Operator Algebras, Vol. 148 (Springer Science & Business Media, 1997)]. [ABSTRACT FROM AUTHOR]

Action of a pomonoid on partially ordered sets (S -posets) has beautiful aspects in practical subjects such as automata theory, projection algebra and theoretical computer science which makes it always capture the interest of mathematicians. On the other hand, the study of different kinds of weakly injectivity (which category theory inherited from homological and commutative algebra) is an interesting subject for mathematicians. One of the important kinds of weakly injectivity is quasi-injectivity. In this paper, we study quasi-injectivity in the category of S -posets with respect to special kind of order embeddings, namely, down-closed embeddings (dc-quasi-injectivity). [ABSTRACT FROM AUTHOR]

In this paper, we unite concepts from Husserlian phenomenology, the active inference framework in theoretical biology, and category theory in mathematics to develop a comprehensive framework for understanding social action premised on shared goals. We begin with an overview of Husserlian phenomenology, focusing on aspects of inner time-consciousness, namely, retention, primal impression, and protention. We then review active inference as a formal approach to modeling agent behavior based on variational (approximate Bayesian) inference. Expanding upon Husserl's model of time consciousness, we consider collective goal-directed behavior, emphasizing shared protentions among agents and their connection to the shared generative models of active inference. This integrated framework aims to formalize shared goals in terms of shared protentions, and thereby shed light on the emergence of group intentionality. Building on this foundation, we incorporate mathematical tools from category theory, in particular, sheaf and topos theory, to furnish a mathematical image of individual and group interactions within a stochastic environment. Specifically, we employ morphisms between polynomial representations of individual agent models, allowing predictions not only of their own behaviors but also those of other agents and environmental responses. Sheaf and topos theory facilitates the construction of coherent agent worldviews and provides a way of representing consensus or shared understanding. We explore the emergence of shared protentions, bridging the phenomenology of temporal structure, multi-agent active inference systems, and category theory. Shared protentions are highlighted as pivotal for coordination and achieving common objectives. We conclude by acknowledging the intricacies stemming from stochastic systems and uncertainties in realizing shared goals. [ABSTRACT FROM AUTHOR]

Simulation studies are needed to investigate how many score categories are sufficient to treat ordered categorical data as continuous, particularly for bifactor models. The current simulation study aims to address such needs by investigating the performance of estimation methods in the bifactor models with ordered categorical data. Results support the application of categorical estimators to the ordered categorical data rather than the continuous estimators when sample size is large (750). Otherwise, an applied researcher may have to use the continuous estimators due to the model non-convergence. In this circumstance, the number of response categories needs to be at least 6 to avoid the rejection of correctly specified bifactor models by the chi-square test and estimate the model parameters accurately. The robust maximum likelihood (MLR) may be chosen among two continuous estimators due to its smaller type I error rate for the chi-square test than the ML. Practical implications of study findings are discussed. [ABSTRACT FROM AUTHOR]

In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I'll offer a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation. [ABSTRACT FROM AUTHOR]

Gauging is a powerful operation on symmetries in quantum field theory (QFT), as it connects distinct theories and also reveals hidden structures in a given theory. We initiate a systematic investigation of gauging discrete generalized symmetries in two-dimensional QFT. Such symmetries are described by topological defect lines (TDLs) which obey fusion rules that are non-invertible in general. Despite this seemingly exotic feature, all well-known properties in gauging invertible symmetries carry over to this general setting, which greatly enhances both the scope and the power of gauging. This is established by formulating generalized gauging in terms of topological interfaces between QFTs, which explains the physical picture for the mathematical concept of algebra objects and associated module categories over fusion categories that encapsulate the algebraic properties of generalized symmetries and their gaugings. This perspective also provides simple physical derivations of well-known mathematical theorems in category theory from basic axiomatic properties of QFT in the presence of such interfaces. We discuss a bootstrap-type analysis to classify such topological interfaces and thus the possible generalized gaugings and demonstrate the procedure in concrete examples of fusion categories. Moreover we present a number of examples to illustrate generalized gauging and its properties in concrete conformal field theories (CFTs). In particular, we identify the generalized orbifold groupoid that captures the structure of fusion between topological interfaces (equivalently sequential gaugings) as well as a plethora of new self-dualities in CFTs under generalized gaugings. [ABSTRACT FROM AUTHOR]

We consider the locally self-injective property of the product FI m of category FI of finite sets and injections. Explicitly, we prove that the external tensor product commutes with the coinduction functor, and hence preserves injective modules. As corollaries, every projective FI m -module over a field of characteristic 0 is injective, and the Serre quotient of the category of finitely generated FI m -modules by the category of finitely generated torsion FI m -modules is equivalent to the category of finite-dimensional FI m -modules. [ABSTRACT FROM AUTHOR]

In this article I model John Cage's pragmatics of silence using the mathematics of category theory with the framework of ontological logs. I use this approach in order to represent knowledge within Cage's so-called silent compositions – 4 $ ^{\prime } $ ′ 33 $ ^{\prime \prime } $ ′′ , 0 $ ^{\prime } $ ′ 00 $ ^{\prime \prime } $ ′′ (4 $ ^{\prime} $ ′ 33 $ ^{\prime \prime } $ ′′ No. 2), and One $ ^{3} $ 3 = 4 $ ^{\prime } $ ′ 33 $ ^{\prime \prime } $ ′′ (0 $ ^{\prime } $ ′ 00 $ ^{\prime \prime } $ ′′ ) +. I first generate a categorical schema $ \mathcal {F} $ F and an accompanying database that describes an instance of 4 $ ^{\prime } $ ′ 33 $ ^{\prime \prime } $ ′′ from its premiere in 1952 which is translated via two functors into the categorical schema $ \mathcal {Z} $ Z (0 $ ^{\prime } $ ′ 00 $ ^{\prime \prime } $ ′′ ) and $ \mathcal {O} $ O (One $ ^{3} $ 3 ). A pushout of $ \mathcal {Z} $ Z and $ \mathcal {O} $ O along $ \mathcal {F} $ F allows for the presentation of the categorical schema $ \mathcal {S} $ S and its accompanying ontological log as an enfolding of Cage's pragmatics of silence. I then introduce the score Listening to John Cage listening in order to conceptualize the category of instances $ \mathcal {S} $ S - $ \mathbf {Set} $ S e t , and further utilize it to investigate the language of $ \mathcal {S} $ S through its fibre order. I conclude by reporting on persistent spatio-temporal structures of Cagean silence embodied in $ \mathcal {S} $ S as meta-work. [ABSTRACT FROM AUTHOR]

The last five decades have witnessed sociologists formulating various scales to measure and assess the degree of alienation of workers. Critical Marxists, however, argue that de-ideologisation and valueneutrality cannot be seen as desirable properties of a reconceptualization of the Marxian notion of alienation. Most Marxist scholars are not in favour of a comparative-quantitative analysis of Marx's theory of alienation. Nevertheless, Sen situates Marx's theory in the category of those which carry out "realization-focused comparison" (as opposed to "transcendental institutionalism"), by comparing societies that actually exist or may evolve. This paper articulates the need for an operationalization of the concept of alienation in empirical terms and calls for a meaningful dialogue between the capability approach to meaningful work and the emerging and significant body of literature on alienation and capabilities. It argues that alienation, translated to the capability vocabulary as "impairments in responsible agency to attain the capabilities one has reason to value" may also be mapped onto failed social relationships. Even when we do not limit the concept of alienation to the system-anti-system binary, we need to understand it in the context of the failures of economic institutions existing in the contemporary world. [ABSTRACT FROM AUTHOR]

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example. [ABSTRACT FROM AUTHOR]

Currently, most of the ultra-high-dimensional feature screening methods for categorical data are based on the correlation between covariates and response variables, using some statistics as the screening index to screen important covariates. Thus, with the increasing number of data types and model availability limitations, there may be a potential problem with the existence of a class of unimportant covariates that are also highly correlated with the response variable due to their high correlation with the other covariates. To address this issue, in this paper, we establish a model-free feature screening procedure for binary categorical response variables from the perspective of the contribution of features to classification. The idea is to introduce the Jensen-Shannon divergence to measure the difference between the conditional probability distributions of the covariates when the response variables take on different values. The larger the value of the Jensen-Shannon divergence, the stronger the covariate's contribution to the classification of the response variable, and the more important the covariate is. We propose two kinds of model-free ultra-high-dimensional feature screening methods for binary response data. Meanwhile, the methods are suitable for continuous or categorical covariates. When the numbers of covariate categories are the same, the feature screening is based on traditional Jensen-Shannon divergence. When the numbers of covariate categories are different, the Jensen-Shannon divergence is adjusted using the logarithmic factor of the number of categories. We theoretically prove that the proposed methods have sure screening and ranking consistency properties, and through simulations and real data analysis, we demonstrate that, in feature screening, the approaches proposed in this paper have the advantages of effectiveness, stability, and less computing time compared with an existing method. [ABSTRACT FROM AUTHOR]

Multiple correspondence analysis (MCA) has started to gain popularity within sociology as a method of mapping 'fields' and 'social spaces' in the style of Pierre Bourdieu, its capacity to document multidimensional geometric relationships within data being a snug fit for the relational mode of thought he championed. There is a risk, however, of over-relying on MCA when the data suggest alternative methods and, as a result, drawing unsound conclusions. As a case in point, I take a recent analysis of political attitudes in the UK using MCA that drew bold inferences about the relationship with social class and reanalyse the same data with categorical principal components analysis (CatPCA). The results suggest the opposite conclusion to what was originally argued. I thus urge greater methodological flexibility and openness among those wishing to chart fields and social spaces and, more specifically, I make a case for CatPCA as a tool of geometric data analysis. [ABSTRACT FROM AUTHOR]

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander–Reiten theory for extriangulated categories. This unifies Auslander–Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander–Reiten–Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category \underline {\mathscr {C}} of an extriangulated category \mathscr {C} is a \tau-category (see O. Iyama [Algebr. Represent. Theory 8 (2005), pp. 297–321]) if \mathscr {C} has enough projectives, almost split extensions and source morphisms. This gives various consequences on \underline {\mathscr {C}}, including Igusa–Todorov's Radical Layers Theorem (see K. Igusa and G. Todorov [J. Algebra 89 (1984), pp. 105–147]), Auslander–Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of \underline {\mathscr {C}} via the complete mesh category of the Auslander–Reiten species of \underline {\mathscr {C}}. Finally we prove that any locally finite symmetrizable \tau-quiver (=valued translation quiver) is an Auslander–Reiten quiver of some extriangulated category with sink morphisms and source morphisms. [ABSTRACT FROM AUTHOR]

A new marker reflecting the pathophysiology of chronic kidney disease (CKD) has been desired for its therapy. In this study, we developed a virtual space where data in medical words and those of actual CKD patients were unified by natural language processing and category theory. A virtual space of medical words was constructed from the CKD-related literature (n = 165,271) using Word2Vec, in which 106,612 words composed a network. The network satisfied vector calculations, and retained the meanings of medical words. The data of CKD patients of a cohort study for 3 years (n = 26,433) were transformed into the network as medical-word vectors. We let the relationship between vectors of patient data and the outcome (dialysis or death) be a marker (inner product). Then, the inner product accurately predicted the outcomes: C-statistics of 0.911 (95% CI 0.897, 0.924). Cox proportional hazards models showed that the risk of the outcomes in the high-inner-product group was 21.92 (95% CI 14.77, 32.51) times higher than that in the low-inner-product group. This study showed that CKD patients can be treated as a network of medical words that reflect the pathophysiological condition of CKD and the risks of CKD progression and mortality. [ABSTRACT FROM AUTHOR]

Mallios and Zafiris emphasize that adjoint functors, or adjunctions, are not only "ubiquitous" in category theory but also characterize the naturality of their approach to physical geometry. Hence, in this paper, the history and theory of adjoint functors is investigated. Where do adjoint functors come from mathematically, and how did the concept develop historically? [ABSTRACT FROM AUTHOR]

We developed a model to represent the time evolution phenomena of life through physics constraints. To do this, we took into account that living organisms are open systems that exchange messages through intracellular communication, intercellular communication and sensory systems, and introduced the concept of a message force field. As a result, we showed that the maximum entropy generation principle is valid in time evolution. Then, in order to explain life phenomena based on this principle, we modelled the living system as a nonlinear oscillator coupled by a message and derived the governing equations. The governing equations consist of two laws: one states that the systems are synchronized when the variation of the natural frequencies between them is small or the coupling strength through the message is sufficiently large, and the other states that the synchronization is broken by the proliferation of biological systems. Next, to simulate the phenomena using data obtained from observations of the temporal evolution of life, we developed an inference model that combines physics constraints and a discrete surrogate model using category theory, and simulated the phenomenon of early embryogenesis using this inference model. The results show that symmetry creation and breaking based on message force fields can be widely used to model life phenomena. [ABSTRACT FROM AUTHOR]

Differential cohomology is a topic that has been attracting considerable interest. Many interesting applications in mathematics and physics have been known, including the description of WZW terms, string structures, the study of conformal immersions, and classifications of Ramond–Ramond fields, to list a few. Additionally, it is an interesting application of the theory of infinity categories. In this paper, we give an expository account of differential cohomology and the classification of higher line bundles (also known as S 1 -banded gerbes) with a connection.We begin with how Čech cohomology is used to classify principal bundles and define their characteristic classes, introduce differential cohomology à la Cheeger and Simons, and introduce S 1 -banded gerbes with a connection. [ABSTRACT FROM AUTHOR]