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A coherent dagger 2-poset is a dagger 2-poset whose category of maps is a coherent category.
A coherent dagger 2-poset is a dagger 2-poset such that
There is an object such that for each object , there is a functional monic map dagger monomorphism such that for each object such with that a for monic each map object , with there is a functional monic dagger map monomorphism , there such is that a functional dagger monomorphism such that .
For each object , , with functional monic dagger maps monomorphisms, , there is an object with functional monic dagger maps monomorphisms, , , such that and ,and for every object with functional monic dagger maps monomorphisms , such that and , there is a functional monic dagger map monomorphism such that
For each object , , with functional monic dagger maps monomorphisms, , there is an object with functional monic dagger maps monomorphisms, , , such that and , and for every object with functional monic dagger maps monomorphisms , such that and , there is a functional monic dagger map monomorphism such that .
For each object , , , with functional monic dagger maps monomorphisms, , , there is a unitary isomorphism
For each object , the identity function is a functional monic dagger map, monomorphism, and for each object with a functional monic dagger map monomorphism, .
The unitary isomorphisms classes of functional monic dagger maps monomorphisms into every object is a distributive lattice . Since every functional monic dagger map monomorphism is a map, thecategory of maps is a coherent category?.
The dagger 2-poset of sets and relations is a coherent dagger 2-poset.