nLab simplicial Stein site

Definition

The category of second countable complex manifolds and holomorphic maps is simplicially enriched: $Map_\Delta(X,Y)$ consists of all singular simplices $\Delta^{op}\to Map(X,Y)$ where $Map(X,Y)$ is the space of holomorphic maps from $X$ to $Y$ with compact-open topology.

The full simplicially enriched subcategory spanned by Stein manifolds has a Grothendieck topology on the category of homotopy components: namely a cover of a Stein manifold $S$ is a family of holomorphic maps $\{X_\alpha\to S\}_\alpha$ such that we can deform each of them (by a homotopy within $Map(X_\alpha,S)$) to a biholomorphic map onto a Stein open subset in $S$, such that these Stein open subsets cover $S$.

This simplicial site is called the simplicial Stein site $Stein_\Delta$.

References

• Finnur Larusson, Model structures and Oka principle, math.CV/0303355