# The Fuzzified Natural Transformation between Categorial Functors and Its Selected Categorial Aspects

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Paper Motivation

- ‘How to define a multi-fuzzy natural transformation as based on multi-diagrams?’ and
- ‘It is possible to predict its general form via the appropriate version of Yoneda’s lemma?’.

#### 1.2. Paper Objectives and Organization

- extending an idea of fuzzy natural transformations towards multi-fuzzy natural transformations–as determined by the multi-fuzzy commutativity condition,
- introducing and proving the appropriate version of multi-fuzzy Yoneda’s lemma and
- introducing a sketch of the conceptual tissue’ application area concerning the combinatorial problem of error detection (exploiting the so-called Hamming’s distances).

## 2. The Conceptual Framework and the Leading Problem Formulation

#### 2.1. Terminological Background

**Definition 1**(

**Group of transformations**)

**.**

- If $\alpha ,\beta \in G$, then $\alpha \u2022\beta \in G$ (i.e., G is closed on •),
- There is a unique element in G to be called the neutral one (it is the identity transformation ${i}_{X}\in G$),
- For each transformation α, there is its inverse transformation ${\alpha}^{-1}\in G$ (i.e., each transformation $\alpha \in G$ is reversible).

**Definition 2**(

**Category of transformations**)

**.**

- If $\alpha ,\beta \in K$, then $\beta \u2022\alpha \in K$ (i.e., $\mathcal{K}$ is closed on •),
- There is a unique neutral element in K (the so-called identity transformation ${i}_{X}\in K$),
- For each $\alpha ,\beta ,\gamma \in K$, it holds $\alpha \u2022(\beta \u2022\gamma )=(\alpha \u2022\beta )\u2022\gamma $ (the associativity principle holds for •).

**Example**

**1.**

Categories | Objects | Morphisms |

RelA | sets | binary relations |

Pos | posets | monotone functions |

Grp/Gr | groups | morphisms |

Ring | rings | ring homomorphisms |

Field | fields | monomorphisms |

Mon | monoids | monoid homomorphisms |

Metr | metric spaces | contractions |

Top | topological spaces | continuous maps |

**Definition 3**(

**Small and big category**)

**.**

**big**category.

**Example**

**2.**

**Set, Gr, Metr, Top**are big. Obviously, the category with a singleton $\left\{o\right\}$ consisting of an object o with an identity $i{d}_{o}$ morphism forms a small category.

**Definition**

**4.**

- (a)
- associates an element $F\left(X\right)\in {O}_{L}$ to each element $X\in {O}^{K}$ and
- (b)
- associates to each morphism $f:X\to Y$ from $Mor{p}_{K}$ a new morphism $F\left(f\right):F\left(x\right)\to F\left(Y\right)$ from $Mor{p}_{L}$ such that the following holds:

- $F\left(i{d}_{X}\right)=i{d}_{F\left(X\right)}$, for each object X of K,
- $F(g\u2022f)=F\left(g\right)\u2022F\left(f\right)$, for every morphisms: $f:X\to Y,g:Y\to Z$.

**Example**

**3.**

**Definition**

**5.**

**Example**

**4.**

**Definition**

**6.**

- (a)
- to each object $X\in C$ a morphism ${\eta}_{X}:F\left(X\right)\to G\left(X\right)$ in D is associated (it forms a
**component**on η at X). - (b)
- It holds the commutativity: ${\eta}_{Y}\u2022F\left(f\right)=G\left(f\right)\u2022{\eta}_{X}$.

**Example**

**5.**

**Definition 7.**

**Representable functor**) Let us assume that $\mathcal{C}$ forms a

`(locally) small category`It does not mean that the category is small. Let also $Hom(A,-)$, for each object $A\in \mathcal{C}$, be the so-called hom-functor, which maps object X to the set $Hom(A,X)$ (set of homomorphisms from A to X). Then a functor $F:\mathcal{C}\to \mathit{Set}$ is a representable functor for $Hom(A,-)$ if it is naturally isomorphic to $Hom(A,-)$, for some A of $\mathcal{C}$. The representation of F is given by a pair $(A,\varphi )$, for:

**Example**

**6.**

**Theorem**

**1.**

**Yoneda’s Lemma**) Let assume that $\mathcal{C}$ is a locally small category, let $Hom(c,-)$ be a hom-functor, for each object $c\in \mathcal{C}$ and let F be a functor from $\mathcal{C}\to $

**Sets**. Then the natural transformations η from $Hom(c,-)$ to F is a one-to-one (bijective) correspondence with the set $F\left(c\right)$, i.e.,

**The proof idea:**The core of the proof relies on the demonstration of the fact that the entire transformation $\eta :H(-,c)\to F$ may be fully determined by a single value $\xi ;={\eta}_{c}\left(i{d}_{c}\right)\in F\left(c\right)$, for any object $c\in \mathcal{C}$. For the proof of the fact, a naturality of $\eta $ will be exploited.

#### 2.2. The Leading Problem Formulation

- A
- How to fuzzify (i.e., to violate the commutativity of the diagram for) the natural transformation $\eta =(T,{T}_{con})$ between functors $co{n}_{X}$ and $co{n}_{Y}$,
- B
- What is the effect of defuzzification of the transformation?
- C
- How to manage correctness (lack of errors) of the translation if the translation process will be continued up to k-stage It means that the corresponding multi-diagram contains k single diagrams. will be constructed?
- D
- How to control the errors in exemplary scenario after 3 multi-diagram stages if the pairs of sets (achieved by the upward and the downward functor composition) are as follows:

- $[1,1,1,1,1,1]$ and $[1,2,1,0,0,0]$–after the first stage,
- $[1,1,2,0,0,0]$ and $[1,2,1,0,0,0,0]$–after the second stage and
- $[1,1,0,0,0,0]$ and $[1,0,1,0,1,0]$–after the third stage ?

## 3. The Natural Transformation Based on Fuzzified Commutativity

- $F\left(f\right)\left(\xi \right)/{\eta}_{b}\left(f\right)=\varnothing $,
- $F\left(f\right)\left(\xi \right)/{\eta}_{b}\left(f\right)=A$, and A forms a finite set,
- $F\left(f\right)\left(\xi \right)/{\eta}_{b}\left(f\right)=A$, and A forms a denumerable set,
- $F\left(f\right)\left(\xi \right)/{\eta}_{b}\left(f\right)=A$, and A forms a uncountable set.

**Definition**

**8.**

**The upward fuzzy natural transformation**) The name of this type of transformation is motivated by the fact that the result set achieved by the composition $G\left(f\right)\u2022{\eta}_{X}$ (see: Figure 6 or Figure 7)–has a greater cardinality than the set ${\eta}_{Y}\u2022F\left(f\right)$ achieved via the downward composition..) Let us assume that two categories $\mathcal{C}$ and $\mathcal{D}$ together with the functors F and G are given. Then the upward fuzzy natural transformation η between F and G forms a family of mappings such that the following conditions are satisfied:

- to each object $X\in C$ a mapping ${\eta}_{X}:F\left(X\right)\to G\left(X\right)$ in D is associated (it forms a
**component**on η at X). - the upward fuzzy-commutativity:$${\eta}_{Y}\u2022F\left(f\right)\subset G\left(f\right)\u2022{\eta}_{X}$$holds.

#### Towards a Multi-Fuzzy Natural Transformation

**Definition**

**9.**

**The upward multi-fuzzy natural transformation**As in [20], the name–as previously–was motivated by the observation that the composition $G\left(f\right)\u2022{\eta}_{X}$ by the upward diagram part gives the set of values with a cardinality greater than ${\eta}_{Y}\u2022F\left(f\right)$–obtained by the downward diagram part.) Let us assume that a finite sequence of categories ${C}_{i}$ with two corresponding sequences ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ and ${\left\{{G}_{i}\right\}}_{i=1}^{k-1}$ of functors operating between the categories are given, for a finite k and $i=1,2,\dots ,k$. Then the upward multi-fuzzy natural transformation η from ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ to ${\left\{{G}_{i}\right\}}_{i}^{k-1}$ forms a family of mapping such that the following requirements are satisfied:

- to each object $X\in {C}_{i}$ a mapping ${\eta}_{i}^{X}:{F}_{i}\left(X\right)\to {G}_{i}\left(X\right)$ in ${C}_{i+1}$ is associated (it forms an i-
**component**of η at X), for $i=1,2\dots ,k$. - it holds the upward fuzzy commutativity:$${\eta}_{k}^{X}\u2022({F}_{k}\u2022\dots \u2022{F}_{1}\left(f\right))\subset ({G}_{k}\u2022\dots \u2022{G}_{1}\left(f\right))\u2022{\eta}_{1}^{X}$$(the result set of the ‘upward’ functor composition contains the result set of the ‘downward’ composition of the functors.)

**Definition**

**10.**

**The downward multi-fuzzy natural transformation**.) Let us assume that a finite sequence of categories ${C}_{i}$ together with two corresponding and finite sequences ${\left\{{F}_{i}\right\}}_{i}^{k-1}$ and ${\left\{{G}_{i}\right\}}_{i}^{k-1}$ of functors operating between the categories are given, for $i=1,2\dots ,k$. Then the downward multi-fuzzy natural transformation η from ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ to ${\left\{{G}_{i}\right\}}_{i}^{k-1}$ forms a family of mappings such that the following requirements are satisfied:

- to each object $X\in {C}_{i}$ a mapping ${\eta}_{i}^{X}:{F}_{i}\left(X\right)\to {G}_{i}\left(X\right)$ in ${C}_{i+1}$ is associated (it forms an i-
**component**of η at X), for $i=1,2\dots ,k$, - it holds the downward multi-fuzzy commutativity:$$({G}_{k}\u2022\dots \u2022{G}_{1}\left(f\right))\u2022{\eta}_{1}^{X}\subset {\eta}_{k}^{X}\u2022({F}_{k}\u2022\dots \u2022{F}_{1}\left(f\right))$$(the result set of the ‘downward’ functor composition contains the result set of the ‘upward’ functor composition.).

**Definition**

**11.**

**(The finite-valued upward multi-fuzzy natural transformation**.) Let us assume that a finite sequence of categories ${C}_{i}$ and two corresponding and finite sequences of functors ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ operating between the categories (resp.) are given, for $i=1,2\dots ,k$. Then the finite-valued upward multi-fuzzy natural transformation η from ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ to ${\left\{{G}_{i}\right\}}_{i}^{k-1}$ forms a family of such the mappings that the following requirements are satisfied:

- to each object $X\in {C}_{i}$ a mapping ${\eta}_{i}^{X}:{F}_{i}\left(X\right)\to {G}_{i}\left(X\right)$ in ${C}_{i+1}$ is associated (it forms an i-
**component**of η at X), for $i=1,2\dots ,k$, - it holds the upward multi-fuzzy commutativity:$${\eta}_{k}^{X}\u2022({F}_{k}\u2022\dots \u2022{F}_{1}\left(f\right))\subset ({G}_{k}\u2022\dots \u2022{G}_{1}\left(f\right))\u2022{\eta}_{1}^{X},$$

**Definition 12.**(

**The denumerable-valued upward multi-fuzzy natural transformation.**)

- to each object $X\in {C}_{i}$ a mapping ${\eta}_{i}^{X}:{F}_{i}\left(X\right)\to {G}_{i}\left(X\right)$ in ${C}_{i+1}$ is associated (it forms an i-
**component**of η at X), for $i=1,2\dots ,k$, - it holds the upward multi-fuzzy commutativity:$${\eta}_{k}^{X}\u2022({F}_{k}\u2022\dots \u2022{F}_{1}\left(f\right))\subset ({G}_{k}\u2022\dots \u2022{G}_{1}\left(f\right))\u2022{\eta}_{1}^{X},$$

**Definition**

**13.**

**(The uncountable-valued upward fuzzy natural transformation**.) Let us assume that a finite sequence of categories ${C}_{i}$ together with two corresponding sequences ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ and ${\left\{{G}_{i}\right\}}_{i=1}^{k-1}$ of functors operating between the categories (resp.) are given, for $i=1,2\dots ,k$. Then the upward multi-fuzzy natural transformation η from ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ to ${\left\{{G}_{i}\right\}}_{i}^{k-1}$ forms a family of such mappings that the following requirements are satisfied:

**component**of η at X), for $i=1,2\dots ,k$,- it holds the upward multi-fuzzy commutativity:$${\eta}_{k}^{X}\u2022({F}_{k}\u2022\dots \u2022{F}_{1}\left(f\right))\subset ({G}_{k}\u2022\dots \u2022{G}_{1}\left(f\right))\u2022{\eta}_{1}^{X}$$

## 4. The Natural Transformation with Multi-Fuzzy Commutativity in Terms of Hamming’s Distances

**a**=$\langle {a}_{1}\dots ,{a}_{n}\rangle $ and $\mathbf{b}=\langle {b}_{1}\dots ,{b}_{n}\rangle $ in ${A}^{m}$ remain mutually different.

**Definition**

**14.**

**Hamming’s countable upward fuzzy natural transformation**, see: [20].) Let us assume that two categories C and D together with two functors $F,G$ between them are given. Let us also assume that $\eta :F\to G$ forms an upward fuzzy natural transformation. Then η is Hamming’s countable-valued upward fuzzy natural transformation provided that the following condition:

**Definition**

**15.**

**a**= $\langle {a}_{1}\dots ,{a}_{n}\rangle $, $\mathit{b}=\langle {b}_{1}\dots ,{b}_{n}\rangle \in {\Sigma}^{n}$, for some fixed $n\in \mathbb{N}$. Then

`the Hamming’s distance`between words

**a**and

**b**in ${\Sigma}^{n}$ is a cardinality of the set $\{i:{a}_{i}\ne {b}_{i}\}$. We write is ${d}_{H}(\mathit{a},\mathit{b})$.

**Definition**

**16.**

**a**is the following set

**Theorem**

**2.**

**Definition**

**17.**

**Hamming’s countable downward multi-fuzzy natural transformation**) Let us assume that a finite sequence of categories ${C}_{i}$ together with two corresponding finite sequences ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ and ${\left\{{G}_{i}\right\}}_{i=1}^{k}$ of functors between them are given, for $i=1,2\dots ,k$. Then Hamming’s countable upward multi-fuzzy natural transformation η from ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ from ${\left\{{G}_{i}\right\}}_{i=1}^{k}$ forms a family of such morphisms that the following requirements are satisfied:

- to each object $X\in {C}_{i}$ a morphism ${\eta}_{i}^{X}:{F}_{i}\left(X\right)\to {G}_{i}\left(X\right)$ in ${C}_{i+1}$ is associated (it forms an i-
**component**of η at X), for $i=1,2\dots ,k$, - it holds either:$$ma{x}_{i=1,2\dots ,k-1}\left\{card\left({A}_{i}\right)\right\}={\aleph}_{0}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{or}$$$$ma{x}_{i=1,2\dots ,k-1}\left\{card\left({A}_{i}\right)\right\}<\infty $$for each ${A}_{i}$ in ${\u2a02}_{i=1}^{k-1}{A}_{i}$ Let us observe that we have exactly $k-1$ relative sets as the result of $k-1$ single diagrams if the number of η-components is k., where$$\underset{i=1}{\overset{k-1}{\u2a01}}{A}_{i}={\eta}_{k}^{X}\u2022({F}_{k}\u2022\dots \u2022{F}_{1}\left(f\right))/({G}_{k}\u2022\dots \u2022{G}_{1}\left(f\right))\u2022{\eta}_{1}^{X}$$

## 5. Fuzzy Yoneda’s Lemma

**Definition**

**18.**

**Multi-similarity up to the difference set A**.) Let $K,M$ be two arbitrary sets. It will be said that K is multi-similar toM up to d.s. A if and only if $K/M=A$, where $A={\oplus}_{i}^{k}{A}_{i}$ for some ${A}_{i}$, for a fixed k In practice, we only consider finite simple sums. We will write $K\sim M$ (up to d.s. A). Conversely, we say that M is multi-similar to The sens of the name may be explained by the fact that each finite simple sum might be identified with the sequence of its components. K up to the difference set A if and only if $M/K=A$, and A forms a finite simple sum of some components.

**Theorem**

**3.**

**The multi-fuzzy Yoneda’s Lemma**.) Let assume that ${\mathcal{C}}_{i}^{k}{}_{i=1}$ be a sequence of locally small categories and let ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ be a sequence of functors from ${\mathcal{C}}_{i}\to $

**Sets**and let $Ho{m}_{i}(c,-)$–be the hom-functor for $c\in {\mathcal{C}}_{\mathcal{i}}$. Then the downward fuzzy natural transformations η between ${\left\{Ho{m}_{i}(c,-)\right\}}_{i=1}^{k-1}$ and ${\left\{{F}_{i}\right\}}_{i=1}^{k-1}$ is similar to the set ${F}_{k-1}\u2022\dots {F}_{1}\left(c\right)$ up to d. s. A, and we will denote this fact by:

**An outline of the proof:**The proof runs by induction over the complication degree of the multi-diagram.

## 6. The Problem Solving and Closing Remarks

**A**,

**B**,

**C**, and

**D**.

**Ad. A**. According to the previous results from Section IV–a possible fuzzification of the natural transformation $\eta =(T,{T}_{con})$ should be identified with similarity up to d. s. A for the relative set $co{n}_{Y}\left(T\right)\left(l\right)/{T}_{con}(co{n}_{X}\left(l\right)$, for $l=\left[\right[a,b],[a,c,a,b,c],[c\left]\right]$ (see: Figure 5). Because the finally obtained lists [1,1,1,2,1,1,2,2], [1,1,2,3,2,2,1,1]–in the Hamming’s representation–may be identified with the words

**a**= $({a}_{1},\dots ,{a}_{8})$ and

**b**= $({b}_{1},\dots ,{b}_{8})$ (over $\Sigma =\{1,2,3\}$) by Hamming’s distance ${d}_{H}(\mathbf{a},\mathbf{b})=\{i:{a}_{i}\ne {b}_{i}\}$. Obviously, the lists remain different in 6 places, thus their ${d}_{H}=6$.

**Ad. B**. In our case, a defuzzification process (returning to the non-fuzzy natural transformation) requires to detect all the 6 errors. The necessary and sufficient condition of an error detection (for at most d errors) indicates that the condition (15) $\mathbf{a}\ne \mathbf{b}\to \mathbf{a}\notin {S}_{6}\left(\mathbf{b}\right)=\{\mathbf{c}:{d}_{H}(\mathbf{b},\mathbf{c})\le 6\}$ should be verified. As the lists

**a, b**are mutually different, one only needs to check whether $a\notin {S}_{6}\left(\mathbf{b}\right)$. However, ${d}_{H}\left(\mathbf{a},\mathbf{b}\right)=6$, thus $a\in {S}_{6}\left(\mathbf{b}\right)$, so the fuzzification process is not feasible for that pair. Hence, a pair of lists with at least 7 different values is required.

**Ad. C**It has already been said that the simple sum-based construction of the final relative sets (over a given multi-fuzzy diagram) does not lose any piece of information about the provenance of the set elements. In other words, we are in a position (at each i-stage of the k-multi-diagram analysis) to indicate the intermediate relative set, from which a given element is taken. Simultaneously, each relative set ${A}_{i}$ in the final simple sum, for $i=1,\dots ,k$, may be seen as a result of the error propagation at an i-stage (diagram) of the k-multi-diagram. Thus, the final error propagation A over the whole multi-diagram is controllable by considering the simple sum $A={\u2a01}_{i}^{k}{A}_{i}$.

**Ad. D**Let us initially assume that the initial lists of points (after each stage), i.e., [1,1,1,1,1,1], [1,1,2,0,0,0] and [1,1,0,0,0,0] have been obtained via the downward composition of the appropriate functors. Let us also assume that we are interested in the relative sets as a difference between the ${S}^{{U}_{i}}/{S}^{{D}_{i}}$, where ${S}^{{U}_{i}}$ is an ‘upward composition’ set (after an i stage) and ${S}^{{D}_{i}}$ is a ‘downward composition’ set (after an i stage), for $i=1,2,3$ It is also noteworthy that a kind of a pattern of correctness must be established here in order to solve the case. In our case, the ‘downward composition set’ plays such a role. Meanwhile, the ‘upward composition’ set is considered as the modified one with potential errors to be detected. This establishment plays a role of a feasibility criterion for the task.. In other words, we need to detect the relative sets ${A}_{1},{A}_{2}$ and ${A}_{3}$ after each diagram stage. For clarity we will mark the places where the corresponding values are different, by letters ${a}_{j}^{i}$, where $i\in \{1,2\dots ,6\}$ informs about the place of the error in the order of errors of a given list, and $j\in \{1,2,3\}$ denotes the diagram stage.

- ${A}_{1}=[0,0,0,{a}_{1}^{1},{a}_{2}^{1},0]$,
- ${A}_{2}=[0,{a}_{1}^{2},{a}_{2}^{2},0,0,0]$,
- ${A}_{3}=[0,{a}_{1}^{3},{a}_{2}^{3},0,{a}_{3}^{3},0]$,

**C**–the final error propagation relative set A is given as $A=[0,0,0,{a}_{1}^{1},{a}_{2}^{1},0]\u2a01[0,{a}_{1}^{2},{a}_{2}^{2},0,0,0]\u2a01[0,{a}_{1}^{3},{a}_{2}^{3},0,{a}_{3}^{3},0]$.

## 7. The State-Of-The-Art

**Definition**

**19.**

## 8. Closing Remarks

**Set, GR, Top, Metr**and their mappings do not exhaust the class of all entities, which may be modeled by them. It seems that some structures and the diagram-based relations of a linguistic nature may constitute an attractive application area of the paper constructions and theoretic considerations. It has been shown how the multi-fuzzy diagrams work in the word error detection, this—in the context of lexis of formal languages. Meanwhile, many categorial diagrams may be exploited to represent the appropriate (parts of) derivation trees in the natural language sentences’ syntax analysis. If a single diagram can represent a phrase of a single sentence, the appropriate and modified multi-diagrams may be potentially exploited to represent broader fragments of statements. This idea seems to be a promising area of future exploration.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Eilenberg, S.; MacLane, S. General theory of natural equivalences. Trans. Am. Math. Soc.
**1945**, 28, 247. [Google Scholar] - Lawvere, F.; Rosebrugh, R. Sets for Mathematics; Oxford University Press: Oxford, UK, 2003. [Google Scholar]
- Lawvere, F.; Schnauel, S. Conceptual Mathematics: A First Introduction to Categories; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Awoday, S. Category Theory; Oxford University Press: Oxford, UK, 2010. [Google Scholar]
- Mac Lane, S. Categories for the Working Mathematician; Springer: Berlin/Heidelberg, Germany, 1998. [Google Scholar]
- Spivak, D. Category Theory for the Sciences; Cambridge Press: Cambridge, UK, 2014. [Google Scholar]
- Fagin, R.; Halpern, J.; Moses, Y.; Vardi, M. Reasoning about Knowledge; Cambridge Press: Cambridge, UK, 1995. [Google Scholar]
- Lomuscio, A.; Michaliszyn, J. An Epistemic Halpern-Shoham logic. In Proceedings of the IJCAI-2013, Beijing, China, 3–9 August 2013; pp. 1010–1016. [Google Scholar]
- Rosenfeld, A.; Moderson, J.; Bhutani, K. Fuzzy Group Theory; Springer: Berlin/Heidelberg, Germany, 2006; ISBN 0-691-12993-2. [Google Scholar]
- Moderson, J.; Mathew, S.; Malik, D. Fuzzy Graph Theory with Applications to Human Trafficking; Springer: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
- Fattahi, F. Fuzzy quasi C*-algebra. Fuzzy Inf. Eng.
**2013**, 5, 327–333. [Google Scholar] [CrossRef] - Rodabaugh, S.; Klement, E.-P.; Höhle, U. Applications of Category Theory to Fuzzy Subsets; Kluwer Academic Publisher: Norwell, MA, USA, 1991. [Google Scholar]
- Coulon, J.; Coulon, J.L.; Hoehle, U. Classification of Extremal Subobjects of Algebras over SM-SET. In Applications of Category Theory to Fuzzy Subsets; Kluwer Academic Publisher: Norwell, MA, USA, 1991; pp. 9–31. [Google Scholar]
- Lowen, E.; Lowen, R. A Topological Universe Extension of FTS. In Applications of Category Theory to Fuzzy Subsets; Kluwer Academic Publisher: Norwell, MA, USA, 1991; pp. 153–176. [Google Scholar]
- Mockor, J. Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces. Axioms
**2020**, 9, 63. [Google Scholar] [CrossRef] - Belohlávek, R. Fuzzy closure operators I. J. Math. Anal. Appl.
**2001**, 62, 473–489. [Google Scholar] [CrossRef] [Green Version] - Belohlávek, R. Fuzzy closure operators II. Soft Comput.
**2002**, 7, 53–64. [Google Scholar] [CrossRef] - Jobczyk, K.; Ligȩza, A. The Natural Transformation with Fuzzy Commutativity Condition. In Proceedings of the FUZZ-IEEE, New Orleans, LA, USA, 23–26 June 2019; pp. 1–8. [Google Scholar]
- Jobczyk, K.; Ligȩza, A. The Natural Transformation with Multi-Fuzzy Commutativity Condition. In Proceedings of the FUZZ-IEEE, Glasgow, UK, 19–24 July 2020. to be published. [Google Scholar]
- Jobczyk, K.; Gał czyński, P. The natural transformation with fuzzified commutativity. In Proceedings of the 2019 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), New Orleans, LA, USA, 23–26 June 2019; pp. 1–6. [Google Scholar]
- Lipski, W.; Marek, W. Combinatorial Analysis; PWN: Warsaw, Poland, 1986. [Google Scholar]
- Hoehle, U.; Klement, E.P. Non-Classical Logics and its Applications of Category Theory to Fuzzy Subsets; Springer Science: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Takeuti, G.; Titani, S. Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. Symb. Log.
**1984**, 49, 851–866. [Google Scholar] [CrossRef] - Hajek, P. Metamathematics of Fuzzy Logic; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1998. [Google Scholar]
- Dubois, D.; Gottwald, S.; Hajek, P.; Kacprzyk, J.; Prade, H. Terminological difficulties in fuzzy set theory—The case of ‘Intuitionistic Fuzzy Sets’. Fuzzy Sets Syst.
**2005**, 156, 485–491. [Google Scholar] [CrossRef] - Maruyama, Y. First-Order Typed Fuzzy Logics And Their Categorical Semantics: Linear Completeness And Baaz Translation Via Lawvere Hyperdoctrine Theory. Proc. FUZZ-IEEE
**2020**. (to be published). [Google Scholar] - Rodabaugh, S. Categorial Frameworks for Stone Representation Theorem. In Applications of Category Theory to Fuzzy Subsets; Kluwer Academic Publisher: Norwell, MA, USA, 1991; pp. 177–231. [Google Scholar]
- Hoehle, U. M-valued Sets and Sheaves over Integral Commutative CL-Monoids. In Applications of Category Theory to Fuzzy Subsets; Kluwer Academic Publisher: Norwell, MA, USA, 1991; pp. 33–72. [Google Scholar]
- Chen, J.; Li, S.; Ma, S.; Wang, X. m-Polar fuzy sets: An extension of bipolar fuzzy sets. Sci. World J.
**2014**, 2014, 416530. [Google Scholar] - Al-Masarwah, A.; Ahmad, A.G. m-Polar fuzzy ideals of BCK/BCI-algebras. J. King Saud Univ.
**2019**, 31, 1200–1226. [Google Scholar] [CrossRef] - Singh, P. m-polar fuzzy graph representation of concept lattice. Eng. Appl. Artif. Intell.
**2018**, 67, 52–62. [Google Scholar] [CrossRef] - Sabu, S.; Ramakrishnan, T. Multi-fuzzy sets: An extension of fuzzy sets. Fuzzy Inf. Eng.
**2011**, 1, 35–43. [Google Scholar]

**Figure 1.**A visual illustration of a founding idea of the natural transformation $\eta $ for two functors F and G between categories K and L. The natural transformation is represented here by a pair of components ${\eta}_{0},{\eta}_{1}$. See: [18].

**Figure 2.**The diagrams–presenting the natural transformations with their components marked in green. The left-side diagram presents an initial general situation. The right-side diagrams specify this general situation in two ways. In consequence, two different components of natural transformations guarantee commutativity of the diagrams. In the upward diagram (

**a**)– the natural transformation as a pair $(f,f)$, in the downward diagram (

**b**)–the natural transformation as a pair $(i{d}_{0},i{d}_{0})$ see:[18].

**Figure 3.**A visual and a table-based presentation of a Hom-functor in the category objects $B,C$ and E. ID in the table for B object contains all arrows beginning from B and ‘@’ denotes an argument. Similarly–for points C and E. See: [18].

**Figure 4.**The illustration of the proof idea of Yoneda’s lemma for the natural transformation with the components ${\eta}_{b},{\eta}_{c}$. By $\mathcal{C}(c,c)$ we denote a class of $\mathcal{C}$-morphisms from c object to c itself, by $\mathcal{C}(b,c)$–a class of $\mathcal{C}$-morphisms from c to b. Meanwhile, the upward diagram presents the general scenario, the right one–the particular situation, as: $i{d}_{c}\in \mathcal{C}(c,c)$, $f\in \mathcal{C}(b,c)$, etc. Meanwhile, the lacking functors between $\mathcal{C}(c,c)$ and $\mathcal{C}(b,c)$ are omitted for a clarity of the presentation (see [18].)

**Figure 5.**The illustration of the leading problem. (see: [20]).

**Figure 6.**A multi-diagram with two sequences of functors ${\left\{{F}_{i}\right\}}_{i=1}^{k}$ and ${\left\{{G}_{i}\right\}}_{i=1}^{k}$ as a construction basis for the multi-fuzzy natural transformation. See: [19].

**Figure 7.**The multi-diagram-based natural transformation given as the k-tuple $({\eta}_{0},{\eta}_{1},\dots ,{\eta}_{k-1})$. Each component ${\eta}_{i}$ operates between the Hom-functor $Ho{m}_{i}(c,-)$ and its corresponding functor ${F}_{i}$, for $i=1,\dots ,k-1$. (See: [19].)

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**MDPI and ACS Style**

Jobczyk, K.
The Fuzzified Natural Transformation between Categorial Functors and Its Selected Categorial Aspects. *Symmetry* **2020**, *12*, 1578.
https://doi.org/10.3390/sym12091578

**AMA Style**

Jobczyk K.
The Fuzzified Natural Transformation between Categorial Functors and Its Selected Categorial Aspects. *Symmetry*. 2020; 12(9):1578.
https://doi.org/10.3390/sym12091578

**Chicago/Turabian Style**

Jobczyk, Krystian.
2020. "The Fuzzified Natural Transformation between Categorial Functors and Its Selected Categorial Aspects" *Symmetry* 12, no. 9: 1578.
https://doi.org/10.3390/sym12091578