Definition:Property of Morphisms Stable Under Pullback
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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 $g$
- $\begin{xy}\xymatrix@L+2mu@+1em{
X \times_Y Z \ar[r] \ar[d]^{f'} & X \ar[d]^f \\ Z \ar[r]^g & Y }\end{xy}$
we have that $f' \in P$.
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