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These are comments on David McNally's 2024 David Gordon Memorial Lecture. In this response, I point to the immanent critique present in Professor McNally's remarks. I then show the potential usefulness of immanent critique for an engagement with neoclassical economics, and I consider the consequences for Marxist economics of characterizing the critique of political economy as a critical theory of economic categories. JEL Classification : B14, B51, B24 [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]

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau–Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau–Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent configuration of rational curves together with some extra data at the nodal points. In this context, we introduce a version of Lagrangian Floer theory and the Fukaya category for trivalent graphs, and show that homological mirror symmetry holds, namely, that the Fukaya category of a trivalent configuration of rational curves is equivalent to the derived category of a non-Archimedean generalized Tate curve. To illustrate the concrete nature of this equivalence, we show how explicit formulas for theta functions and for the canonical map of the curve arise naturally under mirror symmetry. [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]

The document "A.C. Paseau, ed. Philosophy of Mathematics: Critical Concepts in Philosophy" is a collection of historical and contemporary readings on the philosophy of mathematics. It covers various topics such as logicism, intuitionism, formalism, and the indispensability argument. The collection offers a comprehensive overview of the philosophical foundations of mathematics and is a valuable resource for researchers in the field. In the article "Scientific Platonism" by A.C. Paseau, the concept of scientific platonism is explored, discussing the idea that mathematical objects exist independently of human thought and are discovered rather than invented. The article presents arguments for and against scientific platonism, contributing to the ongoing debate on this philosophical perspective. This article provides valuable insights for library patrons interested in understanding the nature of mathematical knowledge and the different philosophical viewpoints surrounding it. [Extracted from the article]

Modular and distributed coding theories of category selectivity along the human ventral visual stream have long existed in tension. Here, we present a reconciling framework--contrastive coding--based on a series of analyses relating category selectivity within biological and artificial neural networks. We discover that, in models trained with contrastive self-supervised objectives over a rich natural image diet, category-selective tuning naturally emerges for faces, bodies, scenes, and words. Further, lesions of these model units lead to selective, dissociable recognition deficits, highlighting their distinct functional roles in information processing. Finally, these pre-identified units can predict neural responses in all corresponding face-, scene-, body-, and word-selective regions of human visual cortex, under a highly constrained sparse positive encoding procedure. The success of this single model indicates that brain-like functional specialization can emerge without category-specific learning pressures, as the system learns to untangle rich image content. Contrastive coding, therefore, provides a unifying account of object category emergence and representation in the human brain. [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]

In arithmetic geometry, we study problems of numbers by transforming them into problems of shapes called algebraic varieties (geometric objects). Cohomology theories extract the information of algebraic varieties as linear data. Various cohomology theories have been developed to study different aspects of algebraic varieties. However, it is widely believed that there is a universal theory called the motive theory, which unifies these cohomology theories. This article gives an overview of the motive theory and presents attempts by the author and his collaborators to generalize it. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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]

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]

The article discusses the historical significance and enduring appeal of chalkboards in mathematics. It mentions the legend of Archimedes appealing to his attacker not to destroy his drawings on his abacus, and how the modern blackboard evolved from smaller slate boards. The article highlights the benefits of chalkboards, such as their simplicity, accessibility, and ability to facilitate the creative process. It also describes a book by Jessica Wynne, which features photographs of chalkboards used by mathematicians from around the world, along with their personal descriptions of the illustrations. The book is seen as a tribute to the timeless medium of chalkboards and a reflection of the mathematical culture of our time. [Extracted from the article]

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]

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]

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]

The polygroup theory is a natural generalization of the group theory. In a group the composition of two elements is an element, while in a polygroup the composition of two elements is a set. Polygroups have been applied in many area, such as geometry, lattices, combinatorics, and color scheme. Also, Crossed modules and its applications play very important roles in category theory, homology and cohomology of groups, homotopy theory, algebra, k-theory, etc. In this paper, we have definition of a polyfunctor and transformation for polygroups. Also, we introduce the concept of the symmetric crossed module to the symmetric crossed polymodules. Our results extend the classical results of crossed modules to crossed polymodules of polygroups. [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]

Let R be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of R-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes. [ABSTRACT FROM AUTHOR]

The multilevel hidden Markov model (MHMM) is a promising method to investigate intense longitudinal data obtained within the social and behavioral sciences. The MHMM quantifies information on the latent dynamics of behavior over time. In addition, heterogeneity between individuals is accommodated with the inclusion of individual-specific random effects, facilitating the study of individual differences in dynamics. However, the performance of the MHMM has not been sufficiently explored. We performed an extensive simulation to assess the effect of the number of dependent variables (1–8), number of individuals (5–90), and number of observations per individual (100–1600) on the estimation performance of a Bayesian MHMM with categorical data including various levels of state distinctiveness and separation. We found that using multivariate data generally alleviates the sample size needed and improves the stability of the results. Moreover, including variables only consisting of random noise was generally not detrimental to model performance. Regarding the estimation of group-level parameters, the number of individuals and observations largely compensate for each other. However, only the former drives the estimation of between-individual variability. We conclude with guidelines on the sample size necessary based on the level of state distinctiveness and separation and study objectives of the researcher. [ABSTRACT FROM AUTHOR]

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In 1957, M. Krasner described a complete valued field (K , v) as the inverse limit of a system of certain structures, called hyperfields, associated with (K , v) . We put this result in purely category-theoretic terms by translating it into a limit construction in certain slice categories of the category of valued hyperfields and their homomorphisms. We replace the original metric-dependent arguments employed by Krasner with a clean and elegant transition to certain slice categories. [ABSTRACT FROM AUTHOR]

This research aimed to establish relations between the Brazilian national historiography, worked by Jessé Souza, with the historiography of the Brazilian science education research field found in the specialized literature. In his works, the theoretical reference of this research, Jessé Souza, works with the historical interpretations of Brazil - interpretations, in the plural, because, in the work of Jessé Souza, two interpretations of the history of Brazil are exposed. The first interpretation, called patrimonialist, seeks our origins in the remote past of Portuguese society and identifies, as the origin of our ills, a supposed backwardness of the country and, mainly, of the State. The second interpretation is proposed by Jessé Souza based on empirical works and other national interpreters such as Gilberto Freyre and Florestan Fernandes. This second interpretation identifies our origins in slavery and perceives exclusion and social inequality as our greatest evils. In order to dialogue with national history, texts available in the specialized literature that deal with the history of the Brazilian science education research field were selected. Through an approach based on the historiography of the problem, and Marc Bloch's regressive method and after numerous joint readings of the texts selected for analysis and the theoretical framework, categories were created that relate the history of the Brazilian science education research field and the historical interpretations of Brazil thought on the Jesse Souza's work. In this article, two of the created categories are presented. The first category connects the theory of modernization, closely linked to the patrimonialist interpretation of Brazil, with financial aid from international and North American agencies for the production of didactic materials and teacher training, recurrently mentioned in the history of Brazilian science education research field. The second category connects the yearning for national modernization, fostered by the patrimonialist historiographical line, with the research and teaching of science in Brazil. Reflections based on these categories point out that the history of the Brazilian science education research field presented in the analyzed literature has connections with the patrimonialist interpretation of Brazil that belittles the slavery aspects of our history and covers up our brutal and current social inequality. Finally, it is left as a suggestion, for the Brazilian science education research field a greater intellectual involvement with the issues of inequality and social exclusion in Brazil. [ABSTRACT FROM AUTHOR]

Using a pedagogic experiment at an architectural school in Tallinn as an empirical and conceptual starting point, this article explores the significance of autopoiesis in contemporary urban design. We suggest that organic processes—in this case the use of vegetable peels as a novel substrate—have been widely deployed in architectural discourse as a form of biomimicry. At a theoretical level these conceptual moves mark part of a wider set of dialogues between the arts and the sciences that rest on a form of degraded or even "phantom" modernism. The article draws on various insights, including the recent work of Fredric Jameson and Sianne Ngai, to explore the changing relationship between aesthetic categories and critical theory in the urban arena. We argue that aesthetic motifs derived from nature, including various forms of organicist architecture, are being effectively recycled under the aegis of late capital. [ABSTRACT FROM AUTHOR]