Categorical Formulation Research Articles

Categorical formulation of the concepts of «investment attractiveness» and «investment climate»

Methods. To achieve the research goal, the following methods were applied: abstraction and logical analysis for examining the evolution of views on the essence of the concepts of «investment attractiveness» and «investment climate» and improving their categorical formulation; comparative analysis for comparing the essential characteristics of the categories of «investment attractiveness» and «investment climate», which allowed identifying their differences and interrelations; factor and functional analysis for isolating factors affecting investment attractiveness and establishing the interrelation between them. Results. Theoretical and methodological approaches to the study of the essence of the concepts «investment attractiveness» and «investment climate» have been considered, revealing a lack of a consistent understanding of these terms among scholars. Attention is focused on the necessity of viewing investment attractiveness as an integrated system, combining economic, political, and sociocultural groups of factors. It is demonstrated that such an approach allows for the analysis of the interrelations between different system components and determining their impact on the overall investment attractiveness of the national economy. Moreover, it enables a deeper understanding of the interaction and influence of various system components on each other, which facilitates the development of more effective investment attraction strategies. Novelty. The investment climate is defined as a complex category that covers a wide range of conditions – from political stability to economic attractiveness. The latter is presented as a dynamic, evolutionary process that reflects the country's ability to meet the needs and expectations of investors at various stages of their activity, and at the same time, as an indicator of the country's potential in the long term. Practical value. The research results can be applied to optimize economic policy aimed at attracting investments. They may also be useful for understanding cross-cultural differences in investment concepts and their application as an educational resource.

FIRST-ORDER HOMOTOPICAL LOGIC

Abstract We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from [8] that any Grothendieck fibration of the kind being considered can automatically be upgraded to a two-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into two-dimensional fibrations.

Correspondence modules and persistence sheaves: a unifying perspective on one-parameter persistent homology

We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.

A planar algebraic description of conditional expectations

Let [Formula: see text] be a unital inclusion of arbitrary von Neumann algebras. We give a 2-[Formula: see text]-categorical/planar algebraic description of normal faithful conditional expectations [Formula: see text] with finite index and their duals [Formula: see text] by means of the solutions of the conjugate equations for the inclusion morphism [Formula: see text] and its conjugate morphism [Formula: see text]. In particular, the theory of index for conditional expectations admits a 2-[Formula: see text]-categorical formulation in full generality. Moreover, we show that a pair [Formula: see text] as above can be described by a Q-system, and vice versa. These results are due to Longo in the subfactor/simple tensor unit case [R. Longo, Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial, Comm. Math. Phys. 130(1990) 285–309; A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994) 133–150].

A MALL geometry of interaction based on indexed linear logic

AbstractWe construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation.

Питання дотримання норм позиційного чергування у//в у навчальній літературі 60-х років ХХ ст

Formal adheres to the norm of positive alteration of у/в does not always correspond to an individual emotional and aesthetic perception of sonority. As a result today modern language practice often demonstrates the need to require a valid “Ukrainian spelling” in the relevant issue. It is necessary to investigate whether enough attention is paid to the categorical formulation of the rules of euphony.

Adjoint functor theorems for ∞‐categories

Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an $\infty$-categorical generalization of Freyd's classical General Adjoint Functor Theorem. As an application of this result, we recover Lurie's adjoint functor theorems for presentable $\infty$-categories. We also discuss the comparison between adjunctions of $\infty$-categories and homotopy adjunctions, and give a treatment of Brown representability for $\infty$-categories based on Heller's purely categorical formulation of the classical Brown representability theorem.

Categorical formulation of mathematical design theories applied to system design process analysis

Given hierarchical system specifications, embodiment and integration processes of subsystems are crucial in designing complex engineered systems. To complement the current mathematical design theories that do not sufficiently support these processes, this study applies general design theory to formulate these processes based on the mathematical structure of design knowledge studied in the literature using category theory. Specifically, the composition of embodiment operations and the relations among subsystems in integration operations on the category of sets and topological spaces are elucidated. Furthermore, the formulation is applied to analysis of a system design process of a production line in model-based systems engineering.

Fermionic modular categories and the 16-fold way

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin topological quantum field theories at low energy. We formulate a 16-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of PSU(2)4m+2 with an eye towards a classification of the low-rank cases.

Semantics of higher-order quantum computation via geometry of interaction

While much of the current study on quantum computation employs low-level formalisms such as quantum circuits, several high-level languages/calculi have been recently proposed aiming at structured quantum programming. The current work contributes to the semantical study of such languages by providing interaction-based semantics of a functional quantum programming language; the latter is, much like Selinger and Valiron's, based on linear lambda calculus and equipped with features like the ! modality and recursion. The proposed denotational model is the first one that supports the full features of a quantum functional programming language; we prove adequacy of our semantics. The construction of our model is by a series of existing techniques taken from the semantics of classical computation as well as from process theory. The most notable among them is Girard's Geometry of Interaction (GoI), categorically formulated by Abramsky, Haghverdi and Scott. The mathematical genericity of these techniques—largely due to their categorical formulation—is exploited for our move from classical to quantum.

Planning for Desirable Land Uses in Periurban Landscapes: Application of a Spatial Concept for Territorial Sustainability

With the aim to test the categorical formulation of a spatial concept on the functional, desirable characteristics of land-use distributions inside periurban land- scapes, in the research we envisaged and developed an original quantitative method, deployed in nine study-cases worldwide. Increasingly facing a generic set of pressures from suburbanization and unregulated land-use change, unwanted spatial outcomes in those landscapes often arise after trespassing critical thresholds of ecosystems' processes inadvertently: Overriding the minimum-area requirements, for example, of crucial ecosystems providing basic ecological services to whole urban regions. Spatial concepts can be powerful tools for strategic planning of territorial systems, helping to avoid shortsighted plans and policies. From the narrative of a spatial con- cept supportive of strategic planning and regulatory policies for territorial sustain- ability (the Aggregates-with-Outliers -AWO), we developed an original synthetic, quantitative method fostering statistical analysis between diverse locales, years, and scales -as well desirable scenarios. From the joint maximization of size variance and land-use diversity (JMV+D mod- el), we evidenced deficits and potentials on the composition of heterogeneous peri- urban landscapes, from the universal perspective of the concept. It is considered a tool (among the many required) for the strategic planning of territorial sustainability in those landscapes, especially required of a flexible definition on the desirable spa- tial outcomes of plans and policies.

$\mathfrak{g}$-quasi-Frobenius Lie algebras

A Lie version of Turaev’s $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $\mathfrak{g}$-quasi-Frobenius Lie algebra for $\mathfrak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\mathfrak{q},\beta )$ together with a left $\mathfrak{g}$-module structure which acts on $\mathfrak{q}$ via derivations and for which $\beta $ is $\mathfrak{g}$-invariant. Geometrically, $\mathfrak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group $G$ which acts via symplectic Lie group automorphisms. In addition to geometry, $\mathfrak{g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\mathfrak{g}$-quasi Frobenius Lie algebras correspond to quasi Frobenius Lie objects in $\mathbf{Rep}(\mathfrak{g})$. If $\mathfrak{g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in [16] suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf{Rep}(D(\mathfrak{g}))$, where $D(\mathfrak{g})$ is the associated (semiclassical) Drinfeld double. We show that if $\mathfrak{g}$ is a quasitriangular Lie bialgebra, then every $\mathfrak{g}$-quasi-Frobenius Lie algebra has an induced $D(\mathfrak{g})$-action which gives it the structure of a $D(\mathfrak{g})$-quasi-Frobenius Lie algebra.

Freyd categories are Enriched Lawvere Theories

Lawvere theories provide a categorical formulation of the algebraic theories from universal algebra. Freyd categories are categorical models of first-order effectful programming languages.The notion of sound limit doctrine has been used to classify accessible categories. We provide a definition of Lawvere theory that is enriched in a closed category that is locally presentable with respect to a sound limit doctrine.For the doctrine of finite limits, we recover Power's enriched Lawvere theories. For the empty limit doctrine, our Lawvere theories are Freyd categories, and for the doctrine of finite products, our Lawvere theories are distributive Freyd categories. In this sense, computational effects are algebraic.

‘Why do these people’s opinions matter?’ Positioning known referents as unnameable others

The way we refer to third parties in talk is one means through which relationships between speaker, recipients and referents are made relevant. A range of referring expressions is available and any number of expressions might correctly refer to a referent. One guide to selection is the preference for achieving recognition and the default practice is, where possible, to use a name. This conversation analytic article describes a practice that does not fit the default pattern. In this practice, speakers select a broad social category (typically gendered, e.g. guy, woman, but not always, e.g. people) when a recognitional form could (and perhaps, ought to) have been used. Despite the designed selection of a categorical form, the referent(s) remains recognitional. For example, in one extract, a mother in conversation with her teenage daughter refers to a collective made up of her former husband and his girlfriend as ‘these people’. The daughter has no difficulty working out who ‘these people’ are and recognizes it as a reference to her father and stepmother. I show that this designedly categorical formulation often contributes to hostile action by distancing the referent(s) from parties to the interaction – making the referent(s) unnameable and not connected to the speaker and recipient. The role of demonstrative pronouns – this, that, these – are discussed in relation to constructing social distance between speakers, recipients and referents.

Categorical foundations of topology with applications to quantaloid enriched topological spaces

Based on ordered monads this paper uncovers the categorical basis of topology in terms of a categorical formulation of neighborhood axioms. Here dense subobjects, lower separation axioms and regularity receive a purely categorical representation. In the case of appropriate submonads of the double presheaf monad this theory is applied to quantaloid-enriched topological spaces which form a common framework for many valued topology as well as for non-commutative topology. As an illumination of this situation two examples are given: the first one is chosen from probability theory and has the following characteristics: Weak convergence of τ-smooth probability measures is topological. The Hausdorff separation axiom is valid. Dirac measures form a dense subset. The second example is related to operator theory and explains the topologization of spectra of non-commutative C*-algebras.

Grasping a Concept as an Image or as a Word – A Categorical Formulation of Visual and Verbal Thinking Processes

Aim: Categorical formulations for thought processes and communications are provided. Study Design: Category theory and Sheaf theory. Place and Duration of Study: June 2010 to April 2013 in the Department of Mathematics, California Polytechnic State University and the Institute of Economic Research, Kyoto University. Methodology: In order to capture the changing state of an individual over a time period, we introduce a state controlling variable in terms of objects (called generalized time) in a site (i.e., a category with a Grothendieck topology). We also capture communications (information flows) as observation morphisms between individuals. The interplay among the change of the state of an individual, the communication between individuals, and the thought processes of those individuals is unified as the commutativity of the fundamental diagram in Section 3 of [1]. Results: We provide formulations for communication diagrams (information flows) between the language dependency type and the image dependency type which are denoted as Type (L) and Type (I), respectively. Conclusion:We have used categorical notions to formulate to express Type (I) and Type (L) of thought patterns. Then we have formulated the information flows in terms of the induced morphisms between Type (I) and Type (L). Research Article Kato and Nishimura; JSRR, Article no. JSRR.2013.016 683

Rosen's modelling relations via categorical adjunctions

Rosen's modelling relations constitute a conceptual schema for the understanding of the bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems used in this study refers to information structures constructed as algebraic rings of observable attributes of natural systems, in which the notion of observable signifies a physical attribute that, in principle, can be measured. Due to the fact that modelling relations are bidirectional by construction, they admit a precise categorical formulation in terms of the category-theoretic syntactic language of adjoint functors, representing the inverse processes of information encoding/decoding via adjunctions. As an application, we construct a topological modelling schema of complex systems. The crucial distinguishing requirement between simple and complex systems in this schema is reflected with respect to their rings of observables by the property of global commutativity. The global information structure representing the behaviour of a complex system is modelled functorially in terms of its spectrum functor. An exact modelling relation is obtained by means of a complex encoding/decoding adjunction restricted to an equivalence between the category of complex information structures and the category of sheaves over a base category of partial or local information carriers equipped with an appropriate topology.

Symplectic microgeometry II: generating functions

We adapt the notion of generating functions for lagrangian submanifolds to symplectic microgeometry. We show that a symplectic micromorphism always admits a global generating function. As an application, we describe hamiltonian flows as special symplectic micromorphisms whose local generating functions are the solutions of Hamilton-Jacobi equations. We obtain a purely categorical formulation of the temporal evolution in classical mechanics.

More on upper bicompletion-true functorial quasi-uniformities

Let T : QU 0 → Top 0 denote the usual forgetful functor from the category of quasi-uniform T 0 -spaces to that of the topological T 0 -spaces. We regard the bicompletion reflector as a (pointed) endofunctor K : QU 0 → QU 0 . For any section F : Top 0 → QU 0 of T we consider the (pointed) endofunctor R = T K F : Top 0 → Top 0 . The T-section F is called upper bicompletion-true (briefly, upper K-true) if the quasi-uniform space KFX is finer than FRX for every X in Top 0 . An important known characterisation is that F is upper K-true iff the canonical embedding X → R X is an epimorphism in Top 0 for every X in Top 0 . We show that this result admits a simple, purely categorical formulation and proof, independent of the setting of quasi-uniform and topological spaces. We thus mention a few other settings where the result is applicable. Returning then to the setting T : QU 0 → Top 0 , we prove: Any T-section F is upper K-true iff for all X the bitopology of KFX equals that of FRX; and iff the join topology of KFX equals the strong topology (also called the b- or Skula topology) of RX.