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I finally went (only now) to look what the statement is in Kock 11 regarding expressing the concept of compactly supported distributions in terms of monads on cartesian closed categories. Now I realize it takes a bit of work to unwind the hints. While I am looking into this, maybe somebody here has already absorbed this and can give me a quick summary?!
I gather the claim about the relation happens in section 11 of Kock 11. I suppose this is secretly discussing the Cahiers topos $\mathbf{H}$ where it says
There exist cartesian closed categories $\mathcal{E}$ which contain the category of smooth manifolds, and also contain the category of convenient vector spaces (with smooth, not necessarily linear, maps)
Then I suppose the statement is that for $X \in \mathbf{H}$ a smooth manifold, the space $\mathcal{E}'(X)$ of compactly supported distributions (distributional densities, really, I suppose), regarded as a convenient vector space and as such as an object of $\mathbf{H}$, is equivalently
$\mathcal{E}'(X) \simeq [[X,\mathbb{R}], \mathbb{R}]_{\mathbb{R}} \,,$where $[-,-]$ denotes the internal hom and $[-,-]_{\mathbb{R}}$ the subobject of $\mathbb{R}$-linear maps (with $\mathbb{R} \in \mathbf{H}$ the image under Yoneda of the smooth real line).
Is that right?
That statement (minus the general monadic infrastructure) would seem to more or less coincide with prop. 3.2.11 in Frédéric Paugam’s book (pdf), except that there the statement is just on the level of the underlying sets of the above equivalence. Is that right?
I would like to have this sorted out cleanly. And if $[X,\mathbb{R}]_c, [X,\mathbb{R}]_s \hookrightarrow [X,\mathbb{R}]$ are the sub-objects of compactly supported smooth functions, resp. smooth functions with rapidly decaying derivatives, what’s the relation between $[[X,\mathbb{R}]_c, \mathbb{R}]_{\mathbb{R}}$ and $\mathcal{D}'(X)$ (plain distributions) as well as $[[X,\mathbb{R}]_s,\mathbb{R}]_{\mathbb{R}}$ and $\mathcal{S}'(X)$ (tempered distributions)? I suppose there must be some subtlety hidden in these cases?
Kock explicitly references this book by Frölicher and Kriegl, and the category $Lip^\infty$ he mentions is in section 1.4. It seems to me to be a Lipschitz version of Fröhlicher spaces (cf Definition 1.1.1). Note that the category of smooth spaces in 1.4 is Frölicher spaces.
Right, I was looking at theorem 5.1.1 in that book, the one that Kock 11 points to. But maybe the theorem number referenced is wrong, or else I need to think harder about what this theorem says about distributions.
By the way, I now discover that Anders Kock has earlier articles that make the setup more explicit: Kock-Reyes 03, Kock-Reyes 04. (and of course there is also Moerdijk-Reyes 91, prop. II 3.6)
I just went rooting about various discussions we’ve had over the years about how functional analysis might be treated category theoretically, but can’t see anything specifically on items a the end of #1. Of course, Lawvere talks much about this.
By the way, I came across a page – algebraic theories in functional analysis – in need of some loving care.
Lawvere talks much about this.
He talks about the general idea of spaces of distributions being covariant functors in their domain (“extensive”), but does he state any internal topos-theoretic (re-)constructions of the usual spaces of distributions?
Not that I could see. Closest I saw is a context for which “the theory of bornological spaces would have to be done internally” in this interview.
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