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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]