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A geometric dagger 2-poset is a dagger 2-poset whose category of maps is a geometric category.
A geometric dagger 2-poset is a dagger 2-poset such that
There is an object such that for each object , there is a monic map such that for each object with a monic map , there is a monic map such that .
For each object , type , and family of objects , with monic maps
there is an object
with monic maps
such that
and for every object with monic maps , such that and , there is a monic map
such that
For each object , , with monic maps , , there is an object with monic maps , , , such that and , and for every object with monic maps , such that and , there is a monic map such that .
For each object , , with monic maps and type , and family of objects , with monic maps
For each object , the identity function is a monic map, and for each object with a monic map , .
The unitary isomorphisms classes of monic maps into every object is a frame. Since every monic map is a map, the category of maps is a geometric category?.
The dagger 2-poset of sets and relations is a coherent dagger 2-poset.