What is the Grothendieck topology used for the site of open subsets of Euclidean spaces, in the definition of a diffeology?

69 Views Asked by At

In this nLab page on diffeological spaces, Definition 2.1 refers to a site $\mathcal{O}_\mathcal{P}$ whose objects are open subsets of Euclidean spaces, and whose morphisms are smooth maps between them.

However, the nLab page doesn't elaborate on the site structure (via e.g. a coverage or Grothendieck topology) used on the category described above.

I was wondering, could anyone help explain what is the appropriate site structure to use in the context of Definition 2.1 in the first link, or suggest a reference that discusses this?

1

There are 1 best solutions below

0
On BEST ANSWER

The Grothendieck topology on $\mathcal{Op}$ is generated by the coverage of open covers, i.e., a family of maps $\{U_i\to X\}_{i\in I}$ is a covering family if every map $U_i\to X$ is an open embedding and the union of the images of $U_i$ in $X$ equals $X$.