# Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me.

Question

In the usual setting of open subsets of $\mathbb{R}^n$, differential forms are defined as follows:

• Definition 1 – Given an open subset $U \subset \mathbb{R}^n$, a smooth differential $p$-form on $U$ is a smooth function $\omega : U \mapsto \bigwedge^p(\mathbb{R}^n)^*$ such that $\omega = \sum_I{f_Ie_{i_1}^* \wedge \cdots \wedge e_{i_p}^*}$ for the smooth function $f_I$ on $U$ and the dual basis $\{e_{1}^*,\ldots,e_{n}^*\}$ of the basis $\{e_1,\ldots,e_n\}$ where $I = \{i_1,\ldots,i_p\} \subseteq \{1,\ldots,n\}$ with $i_1 < \cdots < i_p$. The vector space of all $p$-forms on $U$ is denoted $\Omega^p(U)$. The vector space $\Omega^*(U) = \bigoplus_{p \geq 0}{\Omega^p(U)}$ is the set of all differential forms on $U$.

which can be lifted up to differential manifolds

• Definition 2 - Let $M$ be a smooth manifold. The set $\Omega^p(M)$ of smooth differential $p$-forms on $M$ is the set of smooth sections of the bundle $\bigwedge^pT^*M$ and the set $\Omega^*(M)$ of all smooth differential forms on $M$ is the set of smooth sections of the bundle $\bigwedge T^*M$.

Definition 2 is all well and good when one is dealing with "standard" differential manifolds $M$, i.e., $M$ is a second countable Hausdorff space that is locally homeomorphic to $\mathbb{R}^n$. However, is it possible to define differential forms, along with other related concepts, on manifolds that are not homeomorphic to $\mathbb{R}^n$, but rather homeomorphic to a more abstract topological vector space? More specifically, can this be done on fuzzy topological vector spaces (see the background section/papers below for clarification) and hence on fuzzy differential manifolds? If so, how?

My intuition is that differential forms can be defined for fuzzy differential manifolds and hence fuzzy topological vector spaces. In fact, it looks like it, along with the supporting notions of sections, fiber bundles, and alternating multilinear maps, can be defined without explicitly worrying much about the "fuzzy baggage" associated with fuzzy manifolds/fuzzy spaces.

• Definition 3 - Let $X$ be a differentiable fuzzy manifold (see definition 17 below), $TX = \bigsqcup_{x\in X}T_xX$ the associated tangent bundle (see definition 18 below for the tangent space), and $T^*X$ the dual bundle to $TX$. The set $\Omega^p(X)$ of smooth differential $p$-forms on $X$ is the set of smooth sections of the bundle $\bigwedge^pT^*X$.

The same appears to be true for the local representation of a $p$-form

• Definition 4 - Let $X$ be a differentiable fuzzy manifold $(A,\phi)$ and fuzzy chart of $X$ at $x$ (see definition 16 below), where $x$ lies in the support of the fuzzy set $A$. A $p$-form $\omega$ can be expressed in terms of its coordinate functions with respect to the basis $\{e_{i_1} \wedge \cdots \wedge e_{i_p}\}$ of $T^*X$, $i_1 < \cdots < i_p$, as $\omega = \sum_I{f_{i_1,\ldots,i_p}e_{i_1}\wedge \cdots \wedge e_{i_p}}$, where $f_{i_1,\ldots,i_p}$ is a function on $A$.

Background

In the paper "$C^1$ fuzzy manifolds", Fuzzy Sets and Systems, vol. 54, pp. 99-106, 1993 (PDF link), Ferraro and Foster introduced one possible realization of a "fuzzy" manifold: a differentiable manifold that is locally homeomorphic to open fuzzy sets living in a certain kind of fuzzy topological vector space. More formally,

Fuzzy Topological Space

• Definition 5 - Let $X$ be a set and $I$ the unit interval $[0,1]$. A fuzzy set $A$ is characterized by a membership function $\mu_A$ which associates each point $x \in X$ with a membership grade $\mu_A(x) \in I$.

• Definition 6 - Let $k_c$ be the fuzzy set in $X$ with the membership function $\mu_{k_c}(x) = c$, $c \in I$, $x \in X$. The fuzzy set $k_1$ corresponds to the set $X$ and the fuzzy set $k_0$ to the empty set.

• Definition 7 - Let $X$ be a set. A fuzzy topology on $X$ is a family $\mathscr{F}$ of fuzzy sets in $X$ which satisfy the following conditions: (i) $k_0,k_1 \in \mathscr{F}$, (ii) if $A,B \in \mathscr{F}$ then $A \cap B \in \mathscr{F}$, (iii) if $A_j \in \mathscr{F}$ then $\bigcup_j{A_j} = \sup_j \mu_{A_j}(x) \in \mathscr{F}$, for finite $j$ and $x \in X$; we call $(X,\mathscr{F})$ a fuzzy topological space.

• Definition 8 - An open fuzzy set $A \in X$ is one in $\mathscr{F}$. A closed fuzzy set is one whose compliment, i.e., $\mu_{A^c}(x) = 1 - \mu_A(x)$, $\forall x \in X$, is in $\mathscr{F}$.

Fuzzy Topological Vector Space

• Definition 9 - A fuzzy point in $X$ is a fuzzy set with membership function $\mu_{y_\lambda}(x)$, $x \in X$, such that $\mu_{y_\lambda}(x) = \lambda$ for $x = y$ and $0$ otherwise.

• Definition 10 - Let $(X_1,\mathscr{F}_1)$ and $(X_2,\mathscr{F}_2)$ be two fuzzy topological spaces. A mapping $f$ of $(X_1,\mathscr{F}_1)$ onto $(X_2,\mathscr{F}_2)$ is said to be fuzzy continuous if for each open fuzzy set $B \in \mathscr{F}_2$ the inverse image $f^{-1}(B)$ is in $\mathscr{F}_1$. Conversely, $f$ is fuzzy open if for each $A \in \mathscr{F}_1$ the image $f(A)$ is in $\mathscr{F}_2$. If $f$ is a bijective mapping that is both fuzzy continuous and fuzzy open, it is called a fuzzy homeomorphism.

• Definition 11 - Let $\{A_j\}$ be a finite family of fuzzy sets in a vector space $E$ over the field $K$ of real numbers. The sum $A = \sum_j A_j$ of the family $\{A_j\}$ is the fuzzy set in $E$ that has membership function $\mu_A(x) = \sup_{\sum_j A_j}\min_j \mu_{A_j}(x_j)$, $x \in E$. The scalar product $\alpha A$ of $\alpha \in K$ and $A \in E$ is the fuzzy set in $E$ with membership function $\mu_{\alpha A}(x)$, $x \in E$, given by: $\mu_{\alpha A}(x) = \mu_{A}(x/\alpha)$ for $\alpha \neq 0$ and $\mu_{\alpha A}(x) = \mu_{0_\lambda}(x)$ for $\alpha = 0$ where $0_\lambda$ is a fuzzy point at $0$ in $E$ with $\lambda = \sup_{y \in E}\mu_A(y)$.

• Definition 12 - A fuzzy topological vector space is a vector space $E$ over the field $K$ of real numbers, $E$ equipped with the fuzzy topology $\mathscr{F}$ and $K$ with the usual topology $\mathscr{H}$, such that the two mappings: (i) $(x,y) \mapsto x+y$ of $(E,\mathscr{F}) \times (E,\mathscr{F})$ into $(E,\mathscr{F})$ and (ii) $(\alpha,x) \mapsto \alpha x$ of $(K,\mathscr{H}) \times (E,\mathscr{F})$ into $(E,\mathscr{F})$ are fuzzy continuous.

Fuzzy Manifold and Tangent Space

• Definition 13 - Let $\mathscr{E}_1$ and $\mathscr{E}_2$ be fuzzy topological vector spaces. Let $o(t)$ denote a function of a real variable $t$ such that $\lim_{t\to 0}o(t)/t = 0$. A mapping $\beta$ is said to be tangent to 0 if given a neighborhood $W$ of $0_\delta$ (a fuzzy point), $0 < \delta \leq 1$, in $\mathscr{E}_1$ there exists a neighborhood $V$ of $0_\lambda$ (a fuzzy point), for every $\lambda$, $0 < \lambda < \delta$, in $\mathscr{E}_2$ such that $\beta(tV) \subset o(t)W$.

• Definition 14 - Let $\mathscr{E}_1$ and $\mathscr{E}_2$ be fuzzy topological vector spaces endowed with a $T_1$ fuzzy topology (a $T_1$ fuzzy topological space is one where every fuzzy point is a closed fuzzy set). Let $f : \mathscr{E}_1 \mapsto \mathscr{E}_2$ be a fuzzy continuous mapping. $f$ is said to be fuzzy differentiable at a point $x \in \mathscr{E}_1$ if there exists a linear fuzzy continuous mapping $u$ of $\mathscr{E}_1$ into $\mathscr{E}_2$ such that $f(x+y) = f(x)+u(y)+\beta(y)$, $y \in \mathscr{E}_1$ where $\beta$ is tangent to $0$. The mapping $u$ is called the fuzzy derivative $f$ at $x$ and $f$ is called fuzzy differentiable if it is fuzzy differentiable at every point of $\mathscr{E}_1$.

• Comment 1 - One can also define the fuzzy partial derivative in a manner analogous to Lang in $\S I.3$ of: S. Lang, Introduction to Differentiable Manifolds, New York City, NY, USA: Springer, 2002; a definition of fuzzy integration also follows from $\S I.4$, ibid.

• Definition 15 - Let $\mathscr{E}_1$ and $\mathscr{E}_2$ be fuzzy topological vector spaces. A bijection $f$ of $\mathscr{E}_1$ into $\mathscr{E}_2$ is said to be a fuzzy diffeomorphism of class $C^1$ if it and its inverse $f^{-1}$ are fuzzy differentiable and $f'$ and $(f^{-1})'$ are fuzzy continuous.

• Definition 16 - Let $X$ be a set. A fuzzy atlas of class $C^1$ on $X$ is a collection of pairs $(A_j,\phi_j)$ which satisfies the following conditions: (i) each $A_j$ is a fuzzy set on $X$ and $\sup_j \mu_{A_j}(x)$, $\forall x \in X$ (ii) each $\phi_j$ is a bijection defined on the support of $A_j$, i.e., $\{x \in X,\mu_{A_j}(x) > 0\}$, that maps $A_j$ onto an open fuzzy set $\phi_j(A_j)$ in some fuzzy vector space $\mathscr{E}_j$ and for each $l$ in the index set $\phi_j(A_j \cap A_l)$ is an open fuzzy set $\mathscr{E}_j$ (iii) the mapping $\phi_l \circ \phi_j^{-1}$ which maps $\phi_j(A_j \cap A_l)$ onto $\phi_l(A_j \cap A_l)$ is a $C^1$ fuzzy diffeomorphism for each pair of indices $j,l$.

• Definition 17 - Let $(X,\mathscr{F})$ be a fuzzy topological space. Suppose there exists an open fuzzy set $A$ on $X$ and a fuzzy continuous mapping $\phi$ defined on the support of $A$, mapping $A$ onto an open fuzzy set $V$ in a fuzzy topological vector space $\mathscr{E}$. $(A,\phi)$ is said to be compatible with the fuzzy atlas $\{(A_j,\phi_j)\}$ if each mapping $\phi_j \circ \phi^{-1}$ of $\phi(A \cap A_j)$ into $\phi_j(A \cap A_j)$ is a fuzzy diffeomorphism of class $C^1$. Two fuzzy $C^1$ atlases are compatible if each fuzzy chart in one atlas is compatible with each fuzzy chart of another atlas; this relation of compatibility between $C^1$ fuzzy atlases is an equivalence relation. An equivalence class of $C^1$ fuzzy atlases on $X$ is said to define a $C^1$ fuzzy manifold on $X$.

• Comment 2 - As an aside, it appears one can modify Hirsch's arguments in Lemma 2.8 and Theorem 2.9 (M. W. Hirsch, Differential Topology, New York City, NY, USA: Springer, 1976) to show that every $C^1$ fuzzy manifold is $C^1$ fuzzy diffeomorphic to a $C^\infty$ fuzzy manifold.

• Definition 18 - Let $X$ be a fuzzy manifold. Consider the triples $(A_1,\phi_1,v_\lambda)$ and $(A_2,\phi_2,w_\lambda)$; here $(A_1,\phi_1)$ is a fuzzy chart at $x$ in $X$ with $v_\lambda$ being a fuzzy point of the fuzzy vector space in which $\phi(A_1)$ lies and $(A_2,\phi_2)$ and $w_\lambda$ are defined similarly. These pair of triples are related, denoted by $(A_1,\phi_1,v_\lambda) \sim (A_2,\phi_2,w_\lambda)$, if the fuzzy derivative of $\phi_2 \circ \phi_1^{-1}$ maps $v_\lambda$ into $w_\lambda$, i.e., $(\phi_2 \circ \phi_1^{-1})'(\phi_1(x))v_\lambda = w_\lambda$; this relation $(A_1,\phi_1,v_\lambda) \sim (A_2,\phi_2,w_\lambda)$ is an equivalence relation. An equivalence class of triples $(A_1,\phi_1,v_\lambda)$ is called a tangent vector of the fuzzy manifold $X$ at $x$ and the tangent space $T_x(X)$ at $x$ is the set of all tangent vectors at $x$.

• Comment 3 - As usual, $T_x(X)$ can be given the structure of a vector space.

For further discussion about the above terms, e.g., fuzzy derivatives, it is advisable to consult: M. Ferraro and D. Foster "Differentiation of fuzzy continuous mappings on fuzzy topological vector spaces", Journal of Mathematical Analysis and Applications, vol. 121, pp. 589-601, 1987 (PDF link).

I think your intuition is right: you already have the structure in place to talk about fuzzy tangent bundles (Definition 18). So you can talk about (fuzzy) sections of bundles, hence fuzzy vector fields, fuzzy forms and all that. So, if your question is "does the notion of fuzzy $C^1$ manifold you've defined lead to a natural notion of fuzzy differential forms?" then the answer is certainly yes.