scholarly journals The Fuzzified Natural Transformation between Categorial Functors and Its Selected Categorial Aspects

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1578
Author(s):  
Krystian Jobczyk
Keyword(s):  
Category Theory ◽  
Coding Theory ◽  
Natural Transformation ◽  
Unique Mapping ◽  
Natural Transformations

The natural transformation constitutes one of the most important entity of category theory and it introduces a piece of sophisticated dynamism to the categorial structures. Each natural transformation forms a unique mapping between the so-called functors, which live between categories. In the most simple contexts, natural transformations may be recognized by commutativity of diagrams, which determine them. In fact, the natural transformation does not form any single mapping, but a pair of two components, which–together with the commutativity condition itself–introduces a kind of a symmetry to the functor diagrams. Meanwhile, the general form of the natural transformation may be predicted by means the so-called Yoneda’s lemma in each scenario based on two-valued logic. Meanwhile, the situation may be radically different if we deal with multi-diagrams (instead of the single ones) and if we exchange the two-valued scenario for a multi-valued or fuzzy one. Due to this background–the paper introduces a new concept of multi-fuzzy natural transformation. Its definition exploits the notion of fuzzy natural transformation. Moreover, a multi-fuzzy Yoneda’s lemma is formulated and proved. Finally, some references of these constructions to coding theory are elucidated in last parts of the paper.

scholarly journals A Category Theoretic Approach to Metaphor Comprehension: Theory of Indeterminate Natural Transformation

2020 ◽  
Author(s):  
Miho Fuyama ◽  
Hayato Saigo ◽  
Tatsuji Takahashi
Keyword(s):  
Category Theory ◽  
Natural Transformation ◽  
Mathematical Structure ◽  
Theoretic Approach ◽  
Metaphor Comprehension ◽  
Central Notion ◽  
Meaning Creation ◽  
Natural Transformations ◽  
Categories And Functors ◽  
The Relationship

We propose the theory of indeterminate natural transformation (TINT) to investigate the dynamical creation of meaning as an association relationship between images, focusing on metaphor comprehension as an example. TINT models meaning creation as a type of stochastic process based on mathematical structure and defined by association relationships, such as morphisms in category theory, to represent the indeterminate nature of structure-structure interactions between the systems of image meanings. Such interactions are formulated in terms of the so-called coslice categories and functors as structure-preserving correspondences between them. The relationship between such functors is “indeterminate natural transformation”, the central notion in TINT, which models the creation of meanings in a precise manner. For instance, metaphor comprehension is modeled by the construction of indeterminate natural transformations from a canonically defined functor, which we call the base-of-metaphor functor.

Category theory

2020 ◽  
pp. 19-38
Author(s):  
Ash Asudeh ◽  
Gianluca Giorgolo
Keyword(s):  
Natural Language ◽  
Category Theory ◽  
Categorial Grammar ◽  
Natural Language Semantics ◽  
Fundamental Notion ◽  
Language Semantics ◽  
Natural Transformations ◽  
Glue Semantics

This chapter aims to introduce sufficient category theory to enable a formal understanding of the rest of the book. It first introduces the fundamental notion of a category. It then introduces functors, which are maps between categories. Next it introduces natural transformations, which are natural ways of mapping between functors. The stage is then set to at last introduces monads, which are defined in terms of functors and natural transformations. The last part of the chapter provides a compositional calculus with monads for natural language semantics (in other words, a logic for working with monads) and then relates the compositional calculus to Glue Semantics and to a very simple categorial grammar for parsing. The chapter ends with some exercises to aid understanding.

A Calculus for (Meta)Models and Transformations

2014 ◽  
Vol 24 (05) ◽  
pp. 715-730 ◽  
Author(s):  
Laurent Thiry ◽  
Michel Hassenforder
Keyword(s):  
Category Theory ◽  
Point Of View ◽  
Modeling Languages ◽  
Formal Representation ◽  
Equational Reasoning ◽  
Natural Transformations ◽  
Formal Point

This paper proposes a formal representation of modeling languages based on category theory. These languages are generally described by "metamodels", i.e. structures composed by classes and relations, and related by "transformations". Thus, this paper studies how the key categorical concepts such as functors and relations between functors (called natural transformations) can be used for equational reasoning about modeling artifacts (models, metamodels, transformations). As a result, this paper proposes a formal point of view of models usable to specify/prove equivalence between models or transformations (with an application to refactoring).

Morita context functors

1988 ◽  
Vol 103 (3) ◽  
pp. 399-408 ◽  
Author(s):  
W. K. Nicholson ◽  
J. F. Watters
Keyword(s):  
Natural Transformation ◽  
Morita Context ◽  
Natural Transformations ◽  
Tensor Functor

AbstractGiven a Morita context (R, V, W, S), there are functors W⊗() and hom (V, ) from R-mod to; S-mod and a natural transformation λ from the first to the second. This has an epi-mono factorization and the intermediate functor we denote by ()° with natural transformations and . The tensor functor is exact if and only if WR is flat, whilst the hom functor is exact if and only if RV is projective. We begin by determining conditions under which ()° is exact; this is Theorem 1.

scholarly journals Natural Transformations in Statistics

2015 ◽  
Vol 6 ◽  
pp. 1-5 ◽  
Author(s):  
Gennadii V. Kondratiev
Keyword(s):  
Statistical Analysis ◽  
Empirical Data ◽  
Category Theory ◽  
Probabilistic Approach ◽  
Model Error ◽  
Natural Approach ◽  
Natural Transformations

The old idea of internal uniform regularity of empirical data is discussed within the framework of category theory. A new concept and technique of statistical analysis is being introduced. It is independent on and fully compatible with the classical probabilistic approach. The absence of the model in the natural approach to statistics eliminates the model error and allows to use it in all areas with poor models. The existing error is fully determined by incompleteness of the data. It is always uniformly small by the construction of the data extension.

scholarly journals Coherence for monoidal endofunctors

2010 ◽  
Vol 20 (4) ◽  
pp. 523-543 ◽  
Author(s):  
KOSTA DOŠEN ◽  
ZORAN PETRIĆ
Keyword(s):  
Monoidal Category ◽  
Natural Transformation ◽  
Monoidal Categories ◽  
Monoidal Structure ◽  
Natural Transformations ◽  
Finite Products ◽  
Relational Graphs

The goal of this paper is to prove coherence results with respect to relational graphs for monoidal endofunctors, that is, endofunctors of a monoidal category that preserve the monoidal structure up to a natural transformation that need not be an isomorphism. These results are proved first in the absence of symmetry in the monoidal structure, and then with this symmetry. In the later parts of the paper, the coherence results are extended to monoidal endofunctors in monoidal categories that have diagonal or codiagonal natural transformations, or where the monoidal structure is given by finite products or coproducts. Monoidal endofunctors are interesting because they stand behind monoidal monads and comonads, for which coherence will be proved in a sequel to this paper.

The theory of semi-functors

1993 ◽  
Vol 3 (1) ◽  
pp. 93-128 ◽  
Author(s):  
Raymond Hoofman
Keyword(s):  
General Theory ◽  
Natural Transformation ◽  
Lambda Calculus ◽  
Adjoint Functor ◽  
Typed Lambda Calculus ◽  
The World ◽  
Natural Transformations ◽  
Theoretical Characterization

The notion ofsemi-functorwas introduced in Hayashi (1985) in order to make possible a category-theoretical characterization of models of the non-extensional typed lambda calculus. Motivated by the further use of semi-functors in Martini (1987), Jacobs (1991) and Hoofman (1992a), (1992b) and (1992c), we consider the general theory of semi-functors in this paper. It turns out that the notion ofsemi natural transformationplays an important part in this theory, and that various categorical notions involving semi-functors can be viewed as 2-categorical notions in the 2-category of categories, semi-functors and semi natural transformations. In particular, we find that the notion ofnormal semi-adjunctionas defined in Hayashi (1985) is the canonical generalization of the notion of adjunction to the world of semi-functors. Further topics covered in this paper are the relation between semi-functors and splittings, the Karoubi envelope construction, semi-comonads, and a semi-adjoint functor theorem.

scholarly journals Modified Traces and the Nakayama Functor

2021 ◽  
Author(s):  
Taiki Shibata ◽  
Kenichi Shimizu
Keyword(s):  
Existence And Uniqueness ◽  
Natural Transformation ◽  
Tensor Category ◽  
Abelian Category ◽  
Module Category ◽  
Module Structure ◽  
Exact Functor ◽  
Abelian Categories ◽  
Natural Transformations ◽  
Uniqueness Criteria

AbstractWe organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category ${\mathscr{M}}$ M , we introduce the notion of a Σ-twisted trace on the class $\text {Proj}({\mathscr{M}})$ Proj ( M ) of projective objects of ${\mathscr{M}}$ M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on $\text {Proj}({\mathscr{M}})$ Proj ( M ) and the set of natural transformations from Σ to the Nakayama functor of ${\mathscr{M}}$ M . Non-degeneracy and compatibility with the module structure (when ${\mathscr{M}}$ M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

scholarly journals A triple in CAT

1980 ◽  
Vol 23 (3) ◽  
pp. 261-268
Author(s):  
Charles Wells
Keyword(s):  
Natural Transformation ◽  
Natural Transformations ◽  
Image Position ◽  
The Right

A triple (or monad) in a category K is a triple = (T, μ, η) where, T: K → K is a functor and μ: TT →, T, η: 1k → T are natural transformations for which (1.1) and (1.2) commute:In these diagrams the component of a natural transformation α at an object x is denoted xα. Thus for example (kη)T is the value of the functor T applied to the component of η at k, whereas (kT)η is the component of η at the object kT. I write functions and functors on the right and composition from left to right.

Basics

2019 ◽  
pp. 1-28
Author(s):  
Chris Heunen ◽  
Jamie Vicary
Keyword(s):  
Quantum Computing ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Category Theory ◽  
Quantum Teleportation ◽  
Tensor Products ◽  
Vector Spaces ◽  
Density Matrices ◽  
Natural Transformations ◽  
Basic Ideas

This book assumes familiarity with some basic ideas from category theory, linear algebra and quantum computing. This self-contained chapter gives a quick summary of the essential aspects of these areas, including categories, functors, natural transformations, vector spaces, Hilbert spaces, tensor products, density matrices, measurement and quantum teleportation.

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