Is there a notion of a complex/analytic diffeological space?

Question

I have a bit of a general question. This seems like something you can do, but I can't seem to find much reference for this.. Perhaps something like this already exists in a different guise. But, is there an concept of a complex diffeology analogous to a smooth one?

That is: plots should be defined in terms of maps from open subsets $U \subset \mathbb{C}^{n}$. Axioms concerning compatibility of plots should be defined in terms of analytic maps between $U$ and $V$..etc.

Perhaps there should be extra conditions, or a modification of the conditions, but essentially I'm asking about the existence of a sort of space described by a sheaf on the site of open subsets of $\mathbb{C}^{n}$, or maybe they should be Stein.. If anybody could point me in the direction of anything that may seem relevant, that would be much appreciated.

Thanks!

EDIT:

I have realized that you can (I think!) endow a smooth diffeological space with a "complex/holomorphic structure". It should go something like this:

If $\mathfrak{X}$ is a diffeological space, defined via a collection of plots $ U \rightarrow \mathfrak{X}$, then we can define a complex structure on $\mathfrak{X}$ if there exists a subcollection of plots (perhaps called, holomorphic plots) $ U \rightarrow \mathfrak{X}$ such that:

• $ U \hookrightarrow \mathbb{C}^{n}$ for some $n \geq 0 $ ( and hence $U$ has an induced complex structure )
• If $U$ and $V$ are complex domains, then the composition $ V \rightarrow U \rightarrow \mathfrak{X}$ is a holomorphic plot of $\mathfrak{X}$ if $V \rightarrow U$ is a holomorphic map.

Note that if $V \rightarrow U$ is a holomorphic map, then the composition $V \rightarrow U \rightarrow \mathfrak{X}$ is a (smooth) plot. But it is not guaranteed to be contained in our subcollection of holomorphic plots.

Then, you can also define morphisms of complex diffeological spaces $\mathfrak{X}$, $\mathfrak{Y}$ (i.e. diffeological spaces with complex structures) to be morphisms of diffeological spaces $\mathfrak{X} \rightarrow \mathfrak{Y}$ that is compatible with the choices of holomorphic plots. That is, if $U \rightarrow \mathfrak{X}$ is a holomorphic plot, then $ U \rightarrow \mathfrak{X} \rightarrow \mathfrak{Y}$ should also be a holomorphic plot.

I believe that this is sound, and you can embed complex manifolds into diffeologies with complex structure this way. However, if this is indeed sound, I wonder what the relationship would be if you started off by defining a concrete sheaf on the concrete site of open domains in $\mathbb{C}^{n}$ (or Stein manifolds..etc), as in Enxin Wu's thesis linked in ARA's comment below.

Answer 1

This is an answer the question posed in the last paragraph.

There is a canonical forgetful functor from the site of open subsets of ${\bf C}^n$ and holomorphic maps to the site of open subsets of ${\bf R}^n$ and smooth maps.

This forgetful functor induces a canonical cocontinuous forgetful functor $F$ from holomorphic spaces to diffeological spaces.

Now a complex structure on a diffeological space $\def\fX{{\frak X}}\fX$ can be identified with the data of a holomorphic space $\def\fH{{\frak H}}\fH$ together with an isomorphism $F(\fH)→\fX$. The plots of $\fH$ get mapped to the holomorphic plots of $\fX$ under this isomorphism.

Answer 2

Perhaps the general solution to your question can be found in "A theory of plots" by Atsushi Yamaguchi here? Slides from his talk at the last conference on diffeology and differential homotopy are here.