Abstract
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.
Highlights
A category theory may be viewed as the newest approach to formalization of mathematical structures by means of a general algebra-based conceptualization
An exhaustive characterization of category theory encounters different difficulties of a methodological nature. It forms the aftermath of a dichotomy between a purely theoretic provenience of category theory and a programming-oriented application area outside its theoretic parent environment
For monoids and grupoids the form of the natural transformation is predictable from a single value of the so-called representable functor for an initial object of these structures
Summary
A category theory may be viewed as the newest approach to formalization of mathematical structures by means of a general algebra-based conceptualization. An exhaustive characterization of category theory encounters different difficulties of a methodological nature. It forms the aftermath of a dichotomy between a purely theoretic provenience of category theory (with its conceptual sources located in homology and universal algebra) and a programming-oriented application area outside its theoretic parent environment. The main formal tool–often adopted to this task –is the so-called Yoneda’s lemma It enables predicting a general form of the natural transformations in a relatively smart way and it often simplifies the whole reasoning. For monoids and grupoids (broader: for algebraic structures of a cyclic nature They may be generated by a single element.) the form of the natural transformation is predictable from a single value of the so-called representable functor (introduced in detail in Section 2) for an initial object of these structures
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
