Remark 29.49.13. Here is a generalization of the category of irreducible schemes and dominant rational maps. For a scheme $X$ denote $X^0$ the set of points $x \in X$ with $\dim (\mathcal{O}_{X, x}) = 0$, in other words, $X^0$ is the set of generic points of irreducible components of $X$. Then we can consider the category with

objects are schemes $X$ such that every quasi-compact open has finitely many irreducible components, and

morphisms from $X$ to $Y$ are rational maps $f : U \to Y$ from $X$ to $Y$ such that $f(U^0) = Y^0$.

If $U \subset X$ is a dense open of a scheme, then $U^0 \subset X^0$ need not be an equality, but if $X$ is an object of our category, then this is the case. Thus given two morphisms in our category, the composition is well defined and a morphism in our category.

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