In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not.
I thought to myself, "well, could we at least approximate the space with a category?" I couldn't see any obvious reason why not. It then occurred to me to ask, "what properties of $\mathscr{L}^1(\mathbb{R}^k)$ might I want to preserve if I try to approximate $\mathscr{L}^1(\mathbb{R}^k)$ with a category?"
So I had a look around and stumbled on vector-lattices (which are defined here). They appear to have many of the properties I was interested in in the question I provided a link to above. So:
Do vector-lattices form a category $\mathcal{C}$? Is $\mathscr{L^1(\mathbb{R}^k)}$ a vector-lattice?
I believe so (in both cases, considering them as objects with structure-preserving linear transformations as morphisms) but I don't think I'm up to proving it. Please help. If the answer to the first question is "yes," but the second, "no," then does $\mathcal{C}$ contain an object that is approximately $\mathscr{L^1(\mathbb{R}^k)}$?
What do I mean by "approximately $\mathscr{L^1(\mathbb{R}^k)}$"?
Okay, I'm not sure what I mean, but I guess "such that enough of the salient features of $\mathscr{L^1(\mathbb{R}^k)}$ are reflected" is about right. How close can we get?
This bit can be ignored.
Furthermore, - and to provide some extra motivation for these questions - there seems to be a heavy use of characteristic functions in the theory of Lebesgue Integration, which, to me, calls to mind topoi. I'm not sure why. So, yeah, wouldn't it be cool if we could find a sub-category of the category of vector-lattices (let's say over $\mathbb{R}^k$) that's also a topos, that has an object a bit like $\mathscr{L}^1(\mathbb{R}^k)$?
Does such a thing exist?
Nope: See the first comment! $\ddot\smile$
I find these questions interesting in their own right and I hope you agree :)
