nLab category of monic maps

Context

Higher category theory

higher category theory

Definition

Given a dagger 2-poset $A$, the category of monic maps $MonoMap(A)$ is the sub-2-poset whose objects are the objects of $A$ and whose morphisms are the injective maps of $A$.

In every dagger 2-poset, given two injective maps $f:hom_A(a,b)$ and $g:hom_A(a,b)$, if $f \leq g$, then $f = g$. This means that the sub-2-poset $Map(A)$ is a category and trivially a 2-poset.

Examples

• For the dagger 2-poset Rel of sets and relations, the category of monic maps $MonoMap(Rel)$ is equivalent to the category of sets and injections.