Mapping and conservation law

Question

What is a general map from a point sitting in one dimensional space, to a set of points sitting in two dimensional space, to a set of points sitting in three dimensional space, to a set of points embedded in a torus sitting in four dimensional space? This is what I have but I'm not sure if it's correct notation. The parentheses are to show that those mappings inside the parentheses occur before the mapping to the torus.

The motivation for this comes from physics and viewing these points as fixed points in space throughout all transformations, which lends itself to a conservation of momentum law for these points, which can be thought of as particles. The last embedding onto the torus is to provide a ring structure to the solution space. Physicists are not necessarily known as tidy mathematicians so I thought I'd seek help from the pros.

$ F : ((\Bbb R \to \Bbb R^2) \to \Bbb R^3) \hookrightarrow \Bbb T^4 $.

Thanks.

Answer 1

You need more brackets to make the notation unambiguous. Suppose we write something like

$F: \big ( (\mathbb R \to \mathbb R^2) \to \mathbb R^3\big) \to \mathbb T^4 $.

That could be reasonably interpreted to mean $F$ takes as input a function $f:(\mathbb R \to \mathbb R^2) \to \mathbb R^3 $ and each $F(f)$ is an element of $\mathbb T^4$. But what does $f:(\mathbb R \to \mathbb R^2) \to \mathbb R^3 $ mean? It means $f$ takes as input a function $\alpha: \mathbb R \to \mathbb R^2$ and each $f(\alpha)$ is an element of $\mathbb R^3$.

Exercise: What does $G: \big ( \mathbb R \to (\mathbb R^2 \to \mathbb R^3) \big) \to \mathbb T^4 $ mean?

A slightly more common notation would be to write something like $\hom(X,Y)$, or $\text{Mor} (X,Y)$ or $\text{fun}(X,Y)$ or $F(X,Y)$ for the set of all maps from $X$ to $Y$, or $C(X,Y)$ for the set of all continuous maps. In that case the above example is written:

$F \in C \, (C \,(C\,(\mathbb R, \mathbb R^2)\,, \mathbb R^3)\,, \mathbb T^4)$.

Personally I'd stick to the earlier notation. It is easier on the eyes and also resembles function definition syntax from some programming languages.

Answer 2

If I understand correctly you have first a mapping $\mathbb R \stackrel{f}{\longrightarrow} \mathscr P(\mathbb R^2),$ where $\mathscr P$ denotes power set, and then a chain of mappings $\mathbb R^2 \stackrel{g}{\longrightarrow} \mathbb R^3 \stackrel{\iota} {\hookrightarrow} \mathbb T^4$ acting on each element in $f(t)$ for $t \in \mathbb R.$