# Definition:Property of Morphisms Stable Under Composition

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It has been suggested that this page be renamed.In particular: the term "closed" is used consistently in $\mathsf{Pr} \infty \mathsf{fWiki}$, as it is an obviously better term 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 composition** if and only if:

- for any three objects $X$, $Y$ and $Z$
- any two morphisms $f, g \in P$ such that $f: X \to Y$ and $g: Y \to Z$

we have:

- $g \circ f \mathop \in P$