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 Cβ†’DC\to D is a double functor, it is unambiguous to leave off the adjective and simply speak about β€œfunctors” 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 π’ž\mathcal{C} and π’Ÿ\mathcal{D} are strict double categories, i.e. internal categories in Cat\mathbf{Cat}, then a strict double functor π’žβ†’π’Ÿ\mathcal{C} \to \mathcal{D} is by definition an internal functor in Cat\mathbf{Cat}. Thus, it takes objects, arrows of both sorts, and squares in π’ž\mathcal{C} to the same structures in π’Ÿ\mathcal{D}, preserving sources and targets and also preserving all identities and composites.

In explicit terms:


Let π’ž\mathcal{C} and π’Ÿ\mathcal{D} be double categories. Denote by vπ’žv\mathcal{C} (resp. vπ’Ÿv\mathcal{D}) and hπ’žh\mathcal{C} (resp. hπ’Ÿh\mathcal{D}) the horizontal and vertical categories, respectively, underlying π’ž\mathcal{C} (resp. π’Ÿ\mathcal{D}). Moreover, let us denote by |\vert the horizontal composition of squares and by xxxx\frac{\phantom{xx}}{\phantom{xx}} the horizontal one.

A double functor F:π’žβ†’π’ŸF: \mathcal{C} \to \mathcal{D} consists of the following data:

  • a function F:Obπ’žβ†’Obπ’ŸF : Ob \mathcal{C} \to Ob \mathcal{D}, called the object part,
  • a functor vF:vπ’žβ†’vπ’ŸvF : v\mathcal{C} \to v\mathcal{D}, called the vertical part, coinciding with FF on objects,
  • a functor hF:hπ’žβ†’hπ’ŸhF : h\mathcal{C} \to h\mathcal{D}, called the horizontal part, coinciding with FF on objects,
  • for every square
    X β†’f Z g ↓ Ξ± ↓ gβ€² Y β†’fβ€² W \array{& X & \overset{f}\rightarrow & Z & \\ g & \downarrow & \alpha &\downarrow & g'\\ &Y & \underset{f'}\rightarrow& W & \\ }

    in π’ž\mathcal{C}, a square

    FX β†’hF(f) FZ hF(g) ↓ β–‘FΞ± ↓ vF(gβ€²) FY β†’hF(fβ€²) FW \array{& FX & \overset{hF(f)}\rightarrow & FZ & \\ hF(g) & \downarrow & \square F\alpha &\downarrow & vF(g')\\ &FY & \underset{hF(f')}\rightarrow& FW & \\ }

    in π’Ÿ\mathcal{D},

obeying the following axioms:

  • β–‘F(Ξ±|Ξ²)=β–‘F(Ξ±)|β–‘F(Ξ²)\square F(\alpha \vert \beta) = \square F(\alpha) \vert \square F(\beta),
  • β–‘F(Ξ±Ξ²)=β–‘F(Ξ±)β–‘F(Ξ²)\square F\left(\frac{\alpha}{\beta}\right) = \frac{\square F(\alpha)}{\square F(\beta)},
  • For each horizontal map f:Xβ†’Zf : X \to Z, β–‘F(1 f h)=1 hF(f) h\square F(1^h_f) = 1^h_{hF(f)} (β–‘F\square F preserves horizontal identity squares),
  • For each vertical map g:Xβ†’Yg : X \to Y, β–‘F(1 g v)=1 vF(g) v\square F(1^v_g) = 1^v_{vF(g)} (β–‘F\square F preserves vertical identity squares)

In practice, vFvF, hFhF and β–‘F\square F are all denoted with the same symbol FF 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

  • 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 September 16, 2021 at 18:42:36. See the history of this page for a list of all contributions to it.