## Definition ## Given a [[dagger 2-poset]] $A$, the **category of maps** $Map(A)$ is the sub-2-poset whose objects are the objects of $A$ and whose morphisms are the [[map in a dagger 2-poset|maps]] of $A$. In every dagger 2-poset, given two 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 of sets and relations $Rel$, the category of maps $Map(Rel)$ is equivalent to the category of sets and functions $Set$. ## See also ## * [[map in a dagger 2-poset]] * [[2-poset of partial maps]] category: category theory