Definition:Property of Morphisms Stable Under Composition

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:

$g \circ f \mathop \in P$