Category Theory Research Articles

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.

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Expanding the modelling capabilities of systemic interactions utilizing applied category theory

AbstractCurrent systems analysis approaches often lack the necessary clarity to capture details relating to the interactions of systems, defining arbitrary system boundaries, and expanding boundaries to capture interactions of interest. This paper proposes a new approach for identifying and defining system boundaries based on system‐level properties that can be defined in any system. The research proposes a unification of methodological understandings between systems analysis and applied category theory to bring further mathematical rigor to abstract system boundaries and interactions. This unification is demonstrated by the expansion of systemic interactions between an individual and an organization that employs them, periodically integrating applied category theory concepts to expand the scope of modelled information. We conclude with a call to action to further define the applications of applied category theory to systems analysis and to apply the rigorous practices that exist in the field to further the advancement of methods for modelling the exploration of systemic interactions.

A Study on the Semantics of the Sichuan Dialect Word Ba Based on the Cognitive Perspective

The Sichuan dialect word ba is widely used and has a high frequency of occurrence within the dialect area. It is characterized by its rich semantic meanings and diverse parts of speech, making it one of the most representative common words in Sichuan dialect. However, existing dialect dictionaries have issues such as incomplete definitions, unclear interpretations, and uncertain grammatical properties, which call for further careful revisions. Cognitive linguistics, one of the current research hotspots in linguistics, can fundamentally explain the intrinsic mechanisms and external manifestations of lexical semantic formation. This paper, based on the cognitive perspective and within the macro-framework of Prototype Category Theory, employs the theories of metaphor, metonymy, and image schema to conduct a micro-level analysis and clarification of the original meaning and extended meanings of the Sichuan dialect word ba. It attempts to provide cognitive reasoning for interpreting the meanings of dialect words in dictionaries, as well as to offer new perspectives and cases for exploring the semantics of dialect words.

A Fundamental Duality in the Exact Sciences: The Application to Quantum Mechanics

There is a fundamental subsets–partitions duality that runs through the exact sciences. In more concrete terms, it is the duality between elements of a subset and the distinctions of a partition. In more abstract terms, it is the reverse-the-arrows of category theory that provides a major architectonic of mathematics. The paper first develops the duality between the Boolean logic of subsets and the logic of partitions. Then, probability theory and information theory (as based on logical entropy) are shown to start with the quantitative versions of subsets and partitions. Some basic universal mapping properties in the category of Sets are developed that precede the abstract duality of category theory. But by far the main application is to the clarification and interpretation of quantum mechanics. Since classical mechanics illustrates the Boolean worldview of full distinctness, it is natural that quantum mechanics would be based on the indefiniteness of its characteristic superposition states, which is modeled at the set level by partitions (or equivalence relations). This approach to interpreting quantum mechanics is not a jury-rigged or ad hoc attempt at the interpretation of quantum mechanics but is a natural application of the fundamental duality running throughout the exact sciences.

On preradicals, persistence, and the flow of information

We use Category Theory notions to explore the conceptual axiom that, in a general sense, information flows through a neural network following a path of “least resistance”. To this end, we consider particular endofunctors, called preradicals, whereby we describe persistence in small shape diagrams defined in R-Mod. Specifically, we show that the α preradicals naturally describe persistence in commutative G-modules, for G a directed acyclic graph. Then, we use this results to generalize the notion of persistence to any diagram labeled by a quiver Q. These results, in turn, set the theoretical foundation for our formal framework that explores the notion of “paths of least resistance”. Lastly, we provide a notion of entropy for preradicals in R-Mod, and prove that it respects the order and operations between preradicals.