TITLE: Meaning of $\mathbb{R}^0$, $\mathbb{R}^{1/3}$, and $\mathbb{R}^{-2}$.
QUESTION [7 upvotes]: In mathematics, we take $\mathbb{R}^n$, where $n$ is a fixed positive integer, to represent the Cartesian product
$$
\overbrace{\mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R}}^{n \ \text{times}} = \left\{ (x_1, x_2, \dots, x_n) : x_1, x_2, \dots, x_n \in \mathbb{R} \right\}.
$$
My question: do $\mathbb{R}^0$, $\mathbb{R}^{1/3}$, or $\mathbb{R}^{-2}$ have any meaning, mathematically?

REPLY [2 votes]: My two cents: to a certain extent your question is not silly, and can be partially answered.
Let's start unbiased: you have this big box where you put all vector spaces $\mathbb R^n$ and linear maps between them. If you think up to isomorphism it's a category, the category $\bf Vect$ of finite dimensional real vector spaces (in functorial salons people say that it is the skeleton of the category of real vector spaces, but don't worry about this). 
Now, notice that $\mathbb R^n\oplus \mathbb R^m = \mathbb R^{n+m}$ and $\mathbb R^n\otimes \mathbb R^m = \mathbb R^{nm}$, and suchlike relations, so direct sum and multiplication "behave like" addition and multiplication of natural numbers.
In fact this is not so silly: (iso classes of) vector spaces categorify natural numbers.
Now, there is a machinery you feed with a commutative, cancellative monoid (like $\mathbb N$) and which returns you an abelian group (like $\mathbb Z$): it is the Grothendieck group of the monoid. Basically you formally add inverses to those elements which do not have one, and define a compatible group operation extending the old monoid operation.
Now.
What if I was able to do "the same thing" in the upper floor, "formally inverting" vector spaces? In the end, "grouping" the monoid $\mathbb N$ you used some sort of adjunction $G\colon {\bf Mon\leftrightarrows Grp}\colon U$: it is reasonable to expect that this extends (internalizes?) to a 2-adjunction
$$
\underline G\colon {\bf Mon(Cat) \leftrightarrows Grp(Cat)}\colon \underline U
$$
sending a monoidal category (like, for example, the categoy of vector spaces) into its 2-Grothendieck group. Google in fact gives a reference about something like this: an old Joyal paper called "Traced Monoidal Categories".
In principle, you should be able to build a new category (which can have really nothing to share with the old $\bf Vect$ you started with, but which we call $\overline{\bf Vect}$), which is a categorical group, i.e. a monoidal category "where every object $M$ has a multiplicative inverse $\bar M$", such that $M\otimes \bar M\cong \bar M\otimes M\cong I$, if $I$ is the identity object ($\mathbb R^1$, or $\mathbb R^0$, in the case of $\bf Vect$, according to the case you consider $\oplus$- or $\otimes$-monoidal structure).
Now, I've absolutely no clue about the shape of your $\overline{\bf Vect}$, or how does it behave under "natural" constructions, or whether it has some "nice" properties.