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A double functor is a functor between double categories. Since if and are double categories, the only possible sort of functor is a double functor, it is unambiguous to leave off the adjective and simply speak about βfunctorsβ 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. (Itβs not clear whether lax+lax or lax+oplax are sensible, though.)
If and are strict double categories, i.e. internal categories in , then a strict double functor is by definition an internal functor in . Thus, it takes objects, arrows of both sorts, and squares in to the same structures in , preserving sources and targets and also preserving all identities and composites.
In explicit terms:
Let and be double categories. Denote by (resp. ) and (resp. ) the horizontal and vertical categories, respectively, underlying (resp. ). Moreover, let us denote by the horizontal composition of squares and by the vertical one.
A double functor consists of the following data:
in , a square
in ,
obeying the following axioms:
In practice, , and are all denoted with the same symbol and the correct assignment is deduced from the type of object it is applied to.
Definitions of double categories, double functors and more can be found in
Double pseudofunctors are discussed in
Last revised on February 6, 2024 at 20:30:42. See the history of this page for a list of all contributions to it.