nLab
simplicial Stein site

Definition

The category of second countable complex manifolds and holomorphic maps is simplicially enriched: ${\mathrm{Map}}_{\Delta }(X,Y)$ consists of all singular simplices ${\Delta }^{\mathrm{op}}\to \mathrm{Map}(X,Y)$ where $\mathrm{Map}(X,Y)$ 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 $\mathrm{Map}(X_{\alpha },S)$) 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 ${\mathrm{Stein}}_{\Delta }$.

References

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