nLab
category of open subsets

Category of open subsets

Definition

Given a topological space XX, the category of open subsets Op(X)Op(X) of XX is the category whose

  • objects are the open subsets UXU \hookrightarrow X of XX;

  • morphisms are the inclusions V U X \array{ V &&\hookrightarrow && U \\ & \searrow && \swarrow \\ && X } of open subsets into each other.

Properties

  • The category Op(X)Op(X) is a poset, in fact a frame (dually a locale): it is the frame of opens of XX.

  • The category Op(X)Op(X) is naturally equipped with the structure of a site, where a collection {U iU} i\{U_i \to U\}_i of morphisms is a cover precisely if their union in XX equals UU:

    iU i=U. \bigcup_i U_i = U .

    The category of sheaves on Op(X)Op(X) equipped with this site structure is usually written

    Sh(X):=Sh(Op(X)). Sh(X) := Sh(Op(X)) \,.

Revised on August 31, 2012 14:27:37 by Urs Schreiber (89.204.139.6)