nLab category of maps

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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 maps of $A$. In every dagger 2-poset, given two maps $f:Map_A(a,b)$ and $g:Map_A(a,b)$, 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 maps $Map(Rel)$ is equivalent to the category Set of sets and functions.

See also

• map in a dagger 2-poset
• bicategory of maps
• 2-poset of partial maps
• category of monic maps