Definition:Property of Morphisms Stable Under Pullback

It has been suggested that this page be renamed. In particular: use "closed" rather than "stable" To discuss this page in more detail, feel free to use the talk page.

Definition

Let $\mathbf C$ be a category.

Let $P$ be a property of morphisms of $\mathbf C$.

Then $P$ is stable under pullback if and only if:

for all morphisms $f : X \to Y$ with $f \mathop \in P$

for all morphisms $g : Z \to Y$ for which the pullback of $f$ and $g$

$\begin{xy}\[email protected]+2mu@+1em{ X \times_Y Z \ar[r] \ar[d]^{f'} & X \ar[d]^f \\ Z \ar[r]^g & Y }\end{xy}$

exists

we have that $f' \in P$.