Definition 66.17.1. Let $S$ be a scheme. We say an algebraic space $X$ over $S$ *has property $(\beta )$* if $X$ has the corresponding property of Lemma 66.5.1. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

We say $f$

*has property $(\beta )$*if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times _ Y X$ has property $(\beta )$.We say $f$ is

*decent*if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times _ Y X$ is a decent algebraic space.We say $f$ is

*reasonable*if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times _ Y X$ is a reasonable algebraic space.We say $f$ is

*very reasonable*if for any scheme $T$ and morphism $T \to Y$ the fibre product $T \times _ Y X$ is a very reasonable algebraic space.

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