A dagger 2-poset with tabulations, such as a tabular allegory.

Definition

A tabular dagger 2-poset is a dagger 2-poset$C$ such that for every object $A:Ob(C)$ and $B:Ob(C)$ and morphism $R:Hom(A,B)$ , there exist is an object$|R|:\mathrm{Ob}(C)$ \vert R \vert \vert:Ob(C) called and atabulation of $R$ and maps$f:Hom(\vert R \vert, A)$ , and$g:\mathrm{Hom}(|R|,\mathrm{BA})$ g:Hom(\vert R \vert, B) A) , such that$R=g\circ {f}^{\u2020}\circ g$ R = g \circ f^\dagger \circ g , and for every object$E:Ob(C)$ and maps $h:\mathrm{Hom}(E,A|R|)$ h:Hom(E,A) h:Hom(E,\vert R \vert) and $k:\mathrm{Hom}(E,B|R|)$ k:Hom(E,B) k:Hom(E,\vert R \vert), $\mathrm{kf}\circ {h}^{\u2020}h\le =\mathrm{Rf}\circ k$ k f \circ h^\dagger h \leq = R f \circ k if and only if there exists a unique map$\mathrm{jg}:\circ \mathrm{Eh}\to =|gR\circ |k$ j:E g \to \circ \vert h R = \vert g \circ k such imply that$h=\mathrm{fk}\circ j$ h = f k \circ j and $k = g \circ j$.

Properties

The category of maps of a tabular dagger 2-poset has all pullback?s.

Examples

The dagger 2-poset of sets and relations is a tabular dagger 2-poset.