nLab jointly monic morphisms

Definition

Given a category $C$ and objects $a, b \in Ob(C)$, a pair of morphisms $f, g \in C(a, b)$ is jointly monic if for every object $c: Ob(C)$ and pair of morphisms $h, k \in C(c, a)$, $f \circ h = f \circ k$ and $g \circ h = g \circ k$ imply that $h = k$.

In a well-pointed category $C$, given objects $a, b \in Ob(C)$, a pair of morphisms $f, g \in C(a, b)$ is jointly injective if for every global element $h, k \in C(1, a)$, $f \circ h = f \circ k$ and $g \circ h = g \circ k$ imply that $h = k$.

More generally, we can consider a jointly monic family of morphisms, where we indexed over a set $I$. When $I$ is a singleton set, this reduces to a monomorphism, and when $I$ is a two-element set, this reduces to a jointly monic pair. (And similarly for joint injectivity.)

In tablular allegories

In every tabular allegory, a relation $R$ could be factored into jointly monic maps $f$ and $g$ such that $f^\dagger \circ g = R$.

Examples

• Taking $C = Set$, and ${ F_i : X \to Y }_{i \in I}$ to be a family of functors, joint monicity of $(F_i)_{x, x'} : X(x, x') \to Y(F_i x, F_i x')$ is the notion of joint faithfulness (which specialises to the notion of faithful functor when $I$ is a singleton).

See also

• monomorphism

• faithful functor

• internal relation

• regular category

• tabular allegory

• bicategory of relations

• Rel

• SEAR

• ETCR

• digraph

• loop digraph object

• Theory of units and tabulations in allegories

References

• Peter Freyd, Andre Scedrov, Categories, Allegories, Mathematical Library Vol 39, North-Holland (1990). ISBN 978-0-444-70368-2