Double Categories – Profunctors

A profunctor $\ct F:\ct A\ag\ct B$ is functorial iff every object of $\ct A$ has a reflection in $\ct B$ within $\ct F$ . Fixing (arbitrarily) one reflection arrow for each object $\ct A$ yields a functor $F:\ct A\to\ct B$ such that $F_*\cong \ct F$ .
Dually, $\ct F$ is co-functorial iff every object of $\ct B$ has a coreflection in $\ct A$ , yielding a functor $G:\ct B\to\ct A$ such that $G^*\cong\ct F$ .
If both are satisfied, then $F$ above is left adjoint to $G$ .

This analogy continues in dimension 2 (for bicategories or double categories), yielding the colax functors as double profunctors with the reflection property and the lax functors as double profunctors with the coreflection property. If a double profunctor has both property, it determines a colax/lax adjunction.

Continue reading “Co/lax functors without apriori comparison cells”