Definition:Local Membership Relation

Definition

Let $\mathbf C$ be a metacategory.

Let $C$ be an object of $\mathbf C$.

The local membership relation $\in_C$ between variable elements $z: Z \to C$ and subobjects $m: M \to C$ of $C$ is defined by:

$z \in_C m$ if and only if there exists an $f: Z \to M$: $z = m \circ f$.

If $z \in_C m$, one says that $z$ is a local member of $m$.

Also known as

Abusing notation, some authors write the more suggestive $z \in_C M$ in place of $z \in_C m$.

Also see

• Local Membership Relation Extends Inclusion Relation on Subobjects

• Definition:Membership Relation

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.3$: Definition $1.1$