Category Theory Research Articles

Category theory in consciousness science: going beyond the correlational project

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

What is a farm? Mental framing and reframing as tools in communication between agricultural advisors and farmers

ABSTRACT Purpose To explore how advisors communicate with farmers and the mental frames that they use in their communication. Method A case study in Norway observing meetings between farmers and advisors at 17 farm visits and 43 separate interviews with 26 different informants (advisors and farmers) related to these visits. Empirical analysis is based on theory of framing and categories of frames. Findings There is variation among advisors in how they communicate with the farmers at farm visits, and how they communicate is related to the mental frames that the advisors apply. The frames identified are those of a machine, the brain, an organism, culture, and family. Some advisors are more skilled than others at switching between these various mental frames – reframing. Practical implications The mental frames can be further developed and applied as tools for advisory services to improve communication according to farmers’ needs and thus make advisory services more relevant for farmers. Theoretical implications Mental frames are applied empirically in an advisory context, and the relevance of mental framing is confirmed. Originality Mental frames have not previously been applied in an agricultural advisory context.

A METHOD FOR SYNTHESIZING TOPOS OF HASHES IN THE CATEGORICAL cN-SCHEME OF THE MARKOV ALGORITHM. Part 1

In organizational systems, economic interactions are studied. The purpose of this study is to synthesize a problem-oriented management subsystem for combating money laundering and terrorist financing. To achieve this goal, the problem of synthesizing a constructive way of processing such interactions is solved. The method is based on the development of the N-scheme of the Markov algorithm with an output that cannot be considered normal according to Markov. This algorithm scheme allows us to consider a quadruple consisting of words in the ordered and extended Markov alphabet A2, morphism numbers N, source numbers i, and the alphabet M. From the N-scheme of the Markov algorithm, its categorical version is synthesized. The need for this scheme of the Markov algorithm is due to the solution of the problem of folding Markov occurrences of words into each other of large length for their more compact representation. The idea of organizing a convolution for the entire topos is tested separately. The term “topos of hashes” is introduced into scientific circulation. This topos is considered as an alternative way to represent blocks of chains of Markov occurrences. The hash topos is formed from hash trees known in blockchain technology. This idea follows from previous studies of classifiers in topos, as well as the possibility of splitting chains of Markov occurrences into blocks. The convolution is considered using gamma and hash functions. In the course of the study, it was possible to theoretically show the connection between Markov algorithms and blockchain technology, taking into account the fact that each chain of occurrences of words in each other is considered as a category, and these chains are formed in different sources, otherwise worlds, and the way they are synthesized may have signs of a creative process, then a connection was found between the theories of categories, algorithms, blockchain technology and intuitionistic models by S. Kripke and L. E. Ya. Brouwer.

An Overview of Cognitive Strategies in SLA of English in 21st Century

Cognitive linguistics scrutinizes the close relationship between the human brain, mind and SLA. Since the beginning of the 21st century, many new paradigms and approaches have emerged in this domain of SLA. Based on English as L2, this paper delves into the review in cognitive dimension from three main aspects: cognitive linguistics represented by constructional grammar, empirical cognitive science represented by neural-brain science and corpus research, as well as category and metaphor-related theories in cognitive semantics. These cognitive perspectives and strategies are interconnected and complementary, constituting a general and comprehensive sketch of new paradigms in cognitive SLA. This paper summarizes and analyzes how cognitive linguistics and cognitive science in the 21st century can provide coherent and appropriate theoretical guidance and methodological reference for the study of SLA, and meanwhile criticize and reflect on the shortcomings of the current research field and offers suggestions for future research directions.

Abelian envelopes of exact categories and highest weight categories

We define admissible and weakly admissible subcategories in exact categories and prove that the former induce semi-orthogonal decompositions on the derived categories. We develop the theory of thin exact categories, an exact-category analogue of triangulated categories generated by exceptional sequences. The right and left abelian envelopes of exact categories are introduced, an example being the category of coherent sheaves on a scheme as the right envelope of the category of vector bundles. The existence of right (left) abelian envelopes is proven for exact categories with projectively (injectively) generating subcategories with weak (co)kernels. We show that highest weight categories are precisely the right/left envelopes of thin categories. Ringel duality on highest weight categories is interpreted as a duality between the right and left abelian envelopes of a thin exact category. A duality for thin exact categories compatible with Ringel duality is introduced by means of derived categories and Serre functor on them.

Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings

We describe connections between concepts arising in Poisson geometry and the theory of Fukaya categories. The key concept is that of a symplectic groupoid, which is an integration of a Poisson manifold. The Fukaya category of a symplectic groupoid is monoidal, and it acts on the Fukaya categories of the symplectic leaves of the Poisson structure. Conversely, we consider a wide range of known monoidal structures on Fukaya categories and observe that they all arise from symplectic groupoids. We also use the picture developed to resolve a conundrum in Floer theory: why are some Lagrangian Floer cohomology rings commutative?

Physical computing: a category theoretic perspective on physical computation and system compositionality

This paper introduces a category theory-based framework to redefine physical computing in light of advancements in quantum computing and non-standard computing systems. By integrating classical definitions within this broader perspective, the paper rigorously recontextualizes what constitutes physical computing devices and processes. It demonstrates how the compositional nature and relational structures of physical computing systems can be coherently formalized using category theory. This approach not only encapsulates recent formalisms in physical computing but also offers a structured method to explore the dynamic interactions within these systems.

Relative sectional category revisited

The concept of relative sectional category expands upon classical sectional category theory by incorporating the pullback of a fibration along a map. Our paper aims not only to explore this extension but also to thoroughly investigate its properties. We seek to uncover how the relative sectional category unifies several homotopic numerical invariants found in recent literature. These include the topological complexity of maps according to Murillo–Wu or Scott, relative topological complexity as defined by Farber, and homotopic distance for continuous maps in the sense of Macías-Virgós and Mosquera-Lois, among others.

Reply to Kottler

ABSTRACT The author appreciates Kottler’s suspension of her own theoretical framework in order more fully to enter and grasp his framework. He finds his theory of needed relationships/complex selfobjects and hers of twinship/alterego selfobject experience to be compatible if it is recognized that the two sets of constructs represent two different categories of analytic theorizing: what the author terms “holistic” vs. “categorical” constructs. Holistic theories and constructs pertain to the analytic relationship as a whole whereas categorical theories and constructs describe particular dimensions of that relationship. He argues that the major concepts of his model—the needed relationship, fittedness, specificity of recognition and specificity of connection—are all holistic constructs whereas twinship and alterego transference phenomena are categorical in that they refer to particular dimensions of certain treatments. The author’s concept of complex selfobject experience is also a categorical construct but “moves in the direction of” a holistic construct. Finally, the author is in full agreement with Kottler that our theories and ways of working need to take into account and address the realities of otherness and difference between ourselves and our patients in such a way that our patients feel met, understood and fully welcome in the analytic space.

On the Differentiation of Integrals in Measure Spaces Along Filters: II

Let X be a complete measure space of finite measure. The Lebesgue transform of an integrable function f on X encodes the collection of all the mean-values of f on all measurable subsets of X of positive measure. In the problem of the differentiation of integrals, one seeks to recapture f from its Lebesgue transform. In previous work we showed that, in all known results, f may be recaptured from its Lebesgue transform by means of a limiting process associated to an appropriate family of filters defined on the collection A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\\mathrm{{\\mathcal {A}}}\\,}}$$\\end{document} of all measurable subsets of X of positive measure. The first result of the present work is that the existence of such a limiting process is equivalent to the existence of a Von Neumann-Maharam lifting of X. In the second result of this work we provide an independent argument that shows that the recourse to filters is a necessary consequence of the requirement that the process of recapturing f from its mean-values is associated to a natural transformation, in the sense of category theory. This result essentially follows from the Yoneda lemma. As far as we know, this is the first instance of a significant interaction between category theory and the problem of the differentiation of integrals. In the Appendix we have proved, in a precise sense, that natural transformations fall within the general concept of homomorphism. As far as we know, this is a novel conclusion: Although it is often said that natural transformations are homomorphisms of functors, this statement appears to be presented as a mere analogy, not in a precise technical sense. In order to achieve this result, we had to bring to the foreground a notion that is implicit in the subject but has remained hidden in the background, i.e., that of partial magma.

Categorifying equivariant monoids

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action.

Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency

In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ > 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.

Of Stones and Horses: Reading the Gōngsūn Lóngži in Terms of Concrete Universals

Abstract: The present article proposes a new take on the realist interpretation of the Gōngsūn Lóngzǐ, which theorizes the existence of a theory of universals in the text. Building on the “two-level ontology” hypothesis, it argues that the Gōngsūn Lóngz ǐ elaborates a complex category theory that should rather be interpreted in terms of concrete rather than abstract universals. Through the analysis of some of the most famous arguments debated in the text, such as the “white horse” and the “hard and white stone,” among others, the article delineates a distinction between what seems to be two different types of categories, “incidental qualities” and “structural qualities.” Through this analysis, the article aims to show how this interpretation can help disambiguate the meaning of certain obscure passages, and to identify an underlying line of reasoning that runs like a thread through the text, despite its complex textual history.

Multiple Positive Solutions for Quasilinear Elliptic Problems in Expanding Domains

In this paper we prove the existence of multiple positive solutions for a quasilinear elliptic problem with unbalanced growth in expanding domains by using variational methods and the Lusternik–Schnirelmann category theory. Based on the properties of the category, we introduce suitable maps between the expanding domains and the critical levels of the energy functional related to the problem, which allow us to estimate the number of positive solutions by the shape of the domain.

The Universality of Functions in the Sciences at Large and in Computing

Universality of a concept here means wide conceptual and practical usefulness in mathematics and applications. The function concept owes its universality to simplicity, generality, and powerful algebraic properties. Advantages proven in the sciences at large significantly benefit computing science as well. Universality critically depends on the definitional choices. The first half of this article shows that a “function” in the sense prevalent throughout the sciences, namely, as fully specified by its domain and its values , entails the characteristics that most contribute to universality. This link is clarified by some less well-understood aspects, including the role of function types as partial specifications, the ramifications of having composition defined for any pair of functions, and unification by capturing various notions not commonly seen as functions. Simple but representative examples are given in diverse areas, mostly computing. When a codomain appears at all in basic textbooks, it mostly involves a self-contradicting definition, corrected by the labeled function variant. Either way, it severely reduces universality, especially for composition. Yet, the axiomatization of category theory common in theoretical computing science offers no other choice. The second half explores how waiving one axiom generalizes category theory to include a wider variety of concepts, primarily the conventional function variant. It is also shown how this can be done unobtrusively for typical categorical notions, such as products, coproducts, functors, natural transformations, adjunctions, Galois connections, and auxiliary concepts, illustrated by example definitions and technical comments. Allowing the familiar function variant renders category theory more appealing to a wider group of scientists. A lesson for mathematics in general is Rogaway’s maxim: “Your definitional choices should be justified!”.

On the Structure of Set-Theoretic Polygon Equations

Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the $(N-1)$-simplex equation can be regarded as a realization of the higher Bruhat order $B(N,N-2)$, the $N$-gon equation is a realization of the higher Tamari order $T(N,N-2)$. The latter and its dual $\tilde T(N,N-2)$, associated with which is the dual $N$-gon equation, have been shown to arise as suborders of $B(N,N-2)$ via a ''three-color decomposition''. There are two different reductions of $T(N,N-2)$ and $\tilde T(N,N-2)$, to ${T(N-1,N-3)}$, respectively $\tilde T(N-1,N-3)$. In this work, we explore the corresponding reductions of (dual) polygon equations, which lead to relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation.

Female CHRO appointments: A crack in the glass ceiling?

Executive succession in conjunction with a gender shift is a key factor for enhancing gender diversity in senior management positions. Although an extensive strategic leadership literature has examined CEO turnover and succession, research is lacking on the succession of top management team (TMT) members or non-CEO executives at the individual level. By focusing on a specific executive position—the chief human resources officer (CHRO)—this study examines how executive succession in conjunction with a gender shift occurs. Although women are underrepresented in TMTs, we observe an increase in the number of women appointed to the role of CHRO. By utilizing social role and social categorization theories, we describe the dynamics of the gender shift in the CHRO position. We find that CHRO succession with a gender shift is not prevalent and that male-to-male or female-to-female successions are more common. However, our results suggest that CEOs’ board positions in a firm with a female CHRO and the industry-level diffusion of female CHROs tend to be negatively associated with male-to-male CHRO successions. Our results highlight how such factors may mitigate the effect of social role perceptions.

Prototypical English every days’ Phrases Utilized via College

The researcher , in the current study , studies one of the human categorization theories which is known as ‘ prototype theory’ . He sets a questionnaire to fourth class students at Kirkuk university by using the prototype theory . A good number of every days’ expressions is used as data . The students are required to choose one of the tabulated expressions and give the reason of choosing that expression as well . The reasons ( factors) are the taxonomy of the used model . Two analysis methods namely qualitative and quantitative are followed to arrive at the desired results . The aims of the study are ( 1) showing what can influence the typical member . (2) Showing how people are distinct in creating the prototype. Its hypotheses are (1) each individual has his / her own prototype . (2) The social factors like ( age , religion , one’s background knowledge , etc …) have roles in the formation of the prototype . The study is concluded with points like the context has a well-built relation with the process of creating the prototypical category member .

Homotopy in Exact Categories

In this monograph we develop various aspects of the homotopy theory of exact categories. We introduce different notions of compactness and generation in exact categories, and use these to study model structures on categories of chain complexes C h ∗ ( E ) Ch_{*}(\mathcal {E}) which are induced by cotorsion pairs on E \mathcal {E} . As a special case we show that under very general conditions the categories C h + ( E ) Ch_{+}(\mathcal {E}) , C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) , and C h ( E ) Ch(\mathcal {E}) are equipped with the projective model structure, and that a generalisation of the Dold-Kan correspondence holds. We also establish conditions under which categories of filtered objects in exact categories are equipped with natural model structures. When E \mathcal {E} is monoidal we also examine when these model structures are monoidal and conclude by studying some homotopical algebra in such categories. In particular we provide conditions under which C h ( E ) Ch(\mathcal {E}) and C h ≥ 0 ( E ) Ch_{\ge 0}(\mathcal {E}) are homotopical algebra contexts, thus making them suitable settings for derived geometry.

Parametrised Presentability Over Orbital Categories

In this paper, we develop the notion of presentability in the parametrised homotopy theory framework of Barwick et al. (Parametrized higher category theory and higher algebra: a general introduction, 2016) over orbital categories. We formulate and prove a characterisation of parametrised presentable categories in terms of its associated straightening. From this we deduce a parametrised adjoint functor theorem from the unparametrised version, prove various localisation results, and we record the interactions of the notion of presentability here with multiplicative matters. Such a theory is of interest for example in equivariant homotopy theory, and we will apply it in Hilman (Parametrised noncommutative motives and cubical descent in equivariant algebraic K-theory, 2022) to construct the category of parametrised noncommutative motives for equivariant algebraic K-theory.