Definition 67.17.1. Let $S$ be a scheme. We say an algebraic space $X$ over $S$ has property $(\beta )$ if $X$ has the corresponding property of Lemma 67.5.1. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

1. 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 $(\beta )$.

2. 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.

3. 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.

4. 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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