Kenji Fukaya's Lecture series at Simons center

Question

In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry.

Kenji Fukaya is teaching a course on this subject at Simons Center and the course is being recorded. Here is the link to the first lecture. http://scgp.stonybrook.edu/archives/10004

In order to clarify some comments and claims, for myself and may be others, I will gradually post some questions on mathoverflow. Hopefully, I will gather the results of these discussions in a Lecture note.

1) At some point (min 33-34), John Morgan comments that for an arbitrary compact subset of Euclidean space, $K\subset \mathbb{R}^m$, and for small enough open set $U\subset \mathbb{R}^m$, $U\cap K$ can be realized as the zero set of some smooth map $f\colon U \to \mathbb{R}^n$. How is the proof? For example if this compact set if the contour set in $\mathbb{R}$.

Comment: zero locus of smooth maps + extra structure, are the bulding blocks of Kuranishi spaces

2) A Delign-Mumford stack is somehow a category itself. Then, is there a category of Deligne-Mumford stacks that includes fiber products, ...?

Comment: A Dream of Kuranishi theory is to build a category out of Kuranihsi spaces.

Answer 1

#1 is a classical statement: it suffices to have $K$ closed and $U=\mathbb{R}^m$. Here is a proof I can think of right now. For each point $x\in V:=\mathbb{R}^n\setminus K$ there is a smooth function $f_x\colon\mathbb{R}^m\to[0,1]$ such that $f_x(x)=1$ and $f_x|_K=0$. It can be chosen with compact support, so that all derivatives are bounded. The sets $f_x\ne0$ form an open covering of $V$; pick a countable subcovering and consider the series $ \sum a_if_i$ with $a_i\to0$ sufficiently fast. It really is classical, but I don't remember the reference; it is usually given as a homework in topology courses :)

Answer 2

ad (1) : this is a corollary to the classical Theorem on the existence of a $C^\infty$ partition of unity (subordinate to a given open cover of a smooth manifold) see F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Cie 1971,Thm. 1.11, p. 10 and Corollary on p.11. Here $K$ does not need be compact, but only a closed subset of a smooth manifold $M$.