Let $k$ be a field and let $L$ be a field extension of $k$.

1. i)

   The Zariski-Riemann space $\boldsymbol{P}_{L/k}$ is the set of valuation rings in $L$ which contain $k$ and whose quotient field is $L$ endowed with the coarsest topology such that all sets of the form $\boldsymbol{P}_{L/k}\{f\}:=\left\{R\in\boldsymbol{P}_{L/k}\;\Big|\;f\in R\right\}$ with $f\in L$ are open.

2. ii)

   The category $(\textrm{bir}_{k})$ is the following: The objects are triples $(X,L,\phi)$ where $X$ is a connected quasi-compact and quasi-separated topological space, $L$ is a field extension of $k$ and $\phi:X\rightarrow\boldsymbol{P}_{L/k}$ is a local homeomorphism. A morphism $(X,L,\phi)\rightarrow(Y,M,\psi)$ is a pair $(h,i)$ where $h:X\rightarrow Y$ is a continuous map and $i:M\rightarrow L$ is a morphism of field extensions of $k$ such that $\psi\circ h=i^{\#}\circ\phi$ where $i^{\#}:\boldsymbol{P}_{L/k}\rightarrow\boldsymbol{P}_{M/k}$ is the morphism induced by $i$.

3. iii)

   A morphism $(h,i):(X,L,\phi)\rightarrow(Y,M,\psi)$ is called proper if the map $X\rightarrow Y\times_{\boldsymbol{P}_{M/k}}\boldsymbol{P}_{L/k}$ is bijective.