We construct a model structure on the category
of
double categories and double functors, whose trivial fibrations are the double
functors that are surjective on objects, full on horizontal and vertical morphisms, and
fully faithful on squares; and whose fibrant objects are the weakly horizontally
invariant double categories.
We show that the functor
,
a more homotopical version of the usual horizontal embedding
, is right
Quillen and homotopically fully faithful when considering Lack’s model structure on
. In particular,
exhibits a levelwise fibrant
replacement of
. Moreover,
Lack’s model structure on
is right-induced along
from the model structure for weakly horizontally invariant double categories.
We also show that this model structure is monoidal with respect to Böhm’s Gray
tensor product. Finally, we prove a Whitehead theorem characterizing the weak
equivalences with fibrant source as the double functors which admit a pseudoinverse
up to horizontal pseudonatural equivalence.
Keywords
model structure, double categories, 2–categories, monoidal
model structure, Whitehead theorem