Adjoining companions

Adjoining companions

[Alexander Campbell]

Dear categorists,

I have encountered the following pair of thorny questions in the course of
an investigation into double categories, to which I would like to know the
answers. I would be grateful for any help you may have to offer.

Given a horizontal morphism f in a double category A, we may construct a
new double category A[f_*] by freely adjoining a vertical companion for f.
This new double category comes equipped with a double functor A --> A[f_*],
which is the pushout of the inclusion H(2) --> Sq(2) along the double
functor H(2) --> A that picks out the horizontal morphism f. (Here H(2) and
Sq(2) denote the "free-living horizontal morphism" and the "free-living
companion pair" respectively.)

Question 1: Is the double functor A --> A[f_*] fully faithful on squares?

A related question concerns the "double category of squares" functor Sq :
2-Cat --> DblCat and its left adjoint St : DblCat --> 2-Cat.

Question 2: Is the unit double functor A --> Sq(St(A)) fully faithful on
squares?

Best regards,
Alexander


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