double functor




A double functor is a functor between double categories. Since if CC and DD are double categories, the only possible sort of functor CDC\to D is a double functor, it is unambiguous to leave off the adjective and simply speak about “functors” between double categories.

However, just like 2-functors, double functors do come in different flavors: strict, pseudo/strong, lax, and oplax. Moreover, these various flavors can be chosen more or less independently in the two directions of a double category (vertical and horizontal). Thus we can have functors which are strict in both directions, strict in one direction and pseudo in the other, pseudo in both directions, strict in one direction and lax in the other, and so on. (It’s not clear whether lax+lax or lax+oplax are sensible, though.)


If CC and DD are strict double categories, i.e. internal categories in CatCat, then a strict double functor CDC\to D is simply an internal functor in CatCat. Thus, it takes objects, arrows of both sorts, and squares in CC to the same structures in DD, preserving sources and targets and also preserving all identities and composites.


Double pseudofunctors are discussed in

  • Michael Shulman, section 6 of Comparing composites of left and right derived functors, New York Journal of Mathematics 17 (2011), 75-125 (arXiv:0706.2868)

Last revised on July 12, 2018 at 05:45:32. See the history of this page for a list of all contributions to it.