double fibrations

double fibrations

[Zinovy Diskin]

Dear Categories,
  I'd appreciate references on the following construction. Let f:
E-->B be a double functor between double categories (mapping
2-,1-,0-cels in E to the respective cells in B); and let me call
squares double-arrows and write them as D: S==>T with S a span
(h:B<-- A-->A':u) and T a cospan (h':A'-->B'<--B:v) with h,h' being
horizontal and u,v vertical arrows.

Functor f is called a double fibration if for any double-arrow D:
S==>T in B and a span t over T in E, there is a suitably defined
Cartesian lifting d:s=>t of D. I'm also interested in mixed lifting
being fibrational for horizontal and opfibrational for vertical
arrows.

Did anybody study such things? Or writing it down would be straightforward?

Thanks,
Zinovy


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