Ordered blueprinted space

Definition

An ordered blueprinted space, or OBlpr-space, is a topological space $X$ together with a sheaf $\mathcal O_X$ of ordered blueprints.

A morphism

(f,f^\#):(X,\mathcal O_X)\longrightarrow(Y,\mathcal O_Y)

consists of a continuous map $f:X\to Y$ and a morphism of sheaves $f^\#:\mathcal O_Y\to f_*\mathcal O_X$ such that, for every $x\in X$ , the induced stalk map

f_x^\#:\mathcal O_{Y,f(x)}\longrightarrow\mathcal O_{X,x}

sends nonunits to nonunits. These objects and morphisms form the category $\mathbf{OBlprSp}$ .

Unlike the definition of a locally blueprinted space, the definition does not require every stalk to have a unique maximal ideal. An ordered blue scheme is an OBlpr-space that is locally isomorphic to the spectrum of an ordered blueprint.

References

Matthew Baker and Oliver Lorscheid, The moduli space of matroids, §4.1.3.