There are many ways to create a category from (for example) the real numbers. I could have the numbers as objects with a morphism from a to b if $a\ge b$, for example, or I could have a monoid where morphisms are numbers and composition is addition, or where composition is multiplication.
Each of these captures some important feature of the real numbers, but none of them capture all of them. So my first question is, is there a category that deserves to be called "the category of real numbers" (or complex numbers, rationals, quaternions etc.), In the same sort of way that Set can be called "the category of sets"? If that doesn't exist, what might be a better way to think about numbers in the context of category theory?
In a previous version of this question I used measure theory as an example, but it suffered from my idea being a bit unclear, so here is a more straightforward example. Consider the function that maps finite sets to their cardinality. This seems somehow like an order-preserving map from the finite sets to the natural numbers, and indeed, it is a functor from the set of all finite sets orders by inclusion to $\mathbb{Z}$, considered as an ordered set.
However, it preserves more structure than just order, because it maps products in the category of finite sets ordered by inclusion to multiplication in $\mathbb{Z}$, and it maps coproducts to addition. So the question is, from a category theory perspective, what's the right way to pin down the additional structure that this function preserves?
The measure theory example is a generalisation of this, since a measure is a functor from a sigma algebra to the reals (considered as an ordered set), but, in a similar way, it preserves much more structure than just order. I'm hoping for a category theory flavoured way to pin down exactly the structure that measures preserve. I imagine this would be a statement of the form "measures are exactly the functors from sigma algebras to ___, such that [specification in terms of their relationship with other objects rather than their internal structure]". But it may be that the right way to do it is something else.
Instead of viewing $\sigma$-algebras as categories, I will rather say partial order, as it's mapped to another partial order in $\Bbb R_{\ge 0}$ by a measure.
Consequently, in these terms, what you are looking for is an order preserving homomorphism. (Well, you can call these functors as well, as a functor between partial ordered sets is just an order preserving map.)
My proposal for the algebra part is to consider the emptyset as a constant operation, and the following infinitary partial operation ${\bf U}$ on a $\sigma$-algebra $\mathcal A$:
On the other side, we'd rather consider the extentended set of nonnegative reals: $\overline{\Bbb R}_{\ge 0}:=\,\Bbb R_{\ge 0}\cup\{\infty\}\,=\,[0,\infty]$, with its usual order and $\bf U$ can be the (total operation of) sum.
Then, measures from $\mathcal A$ are exactly the structure preserving maps, i.e. maps that preserve the zero constant, the partial order and the partial operation ${\bf U}$ (in that, whenever ${\bf U}(A_0,A_1,\dots)$ is defined, its result is mapped to the sum of the images of $A_n$'s).
[By the way, note that both the empty set and the inclusion are definable using only ${\bf U}$ in a $\sigma$-algebra:
- For $A,B\in\mathcal A$, we have $A\subseteq B$ iff there is a $C\in\mathcal A$ for which ${\bf U}(A,C,\emptyset,\emptyset,\dots)=B$,
- The empty set is the only element $o\in\mathcal A$ for which ${\bf U}(o,o,o,\dots)$ is defined and equals to $o$, or, we can say that $\emptyset$ is the neutral element of $\bf U$, in that ${\bf U}(\emptyset,A_0,A_1,\dots)={\bf U}(A_0,A_1,\dots)$. ]