nLab
simplicial Stein site

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

The category of second countable complex manifolds and holomorphic maps is simplicially enriched: Map Δ(X,Y)Map_\Delta(X,Y) consists of all singular simplices Δ opMap(X,Y)\Delta^{op}\to Map(X,Y) where Map(X,Y)Map(X,Y) is the space of holomorphic maps from XX to YY 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 SS is a family of holomorphic maps {X αS} α\{X_\alpha\to S\}_\alpha such that we can deform each of them (by a homotopy within Map(X α,S)Map(X_\alpha,S)) to a biholomorphic map onto a Stein open subset in SS, such that these Stein open subsets cover SS.

This simplicial site is called the simplicial Stein site Stein ΔStein_\Delta.

References

Revised on December 8, 2010 23:58:27 by David Roberts (203.24.207.11)