nLab
vertical transformation

Idea

A vertical transformation is an analogue of a natural transformation which goes between double functors of double categories, and whose components are vertical arrows and squares. There is a dual notion of horizontal transformation.

Definition

If $C$ and $D$ are strict double categories regarded as internal categories in $\mathrm{Cat}$ and $F,G:C\to D$ are double functors regarded as internal functors in $\mathrm{Cat}$, then a transformation between them is simply an internal natural transformation in $\mathrm{Cat}$. Whether this is a vertical or horizontal transformation depends on how we identify double categories with internal categories in $\mathrm{Cat}$ (there being two ways).

More explicitly, a vertical transformation $\alpha :F\to G$ consists of

• For every object $c\in C$, a vertical arrow

  $$\begin{array}{c}Fc\\ {\downarrow }^{{\alpha }_c}\\ Gc\end{array}$$\array{ F c \\ \downarrow ^{\alpha_c} \\ G c}

  in $D$, which are natural with respect to vertical composition of vertical arrows in $C$.

• For every horizontal arrow $p:c_1\to c_2$ in $C$, a square

  $$\begin{array}{ccc}F{c}_1& \stackrel{Fp}{\to }& F{c}_2\\ {}^{{\alpha }_{c_1}}\downarrow & {\Downarrow }^{{\alpha }_p}& {\downarrow }^{{\alpha }_{c_2}}\\ G{c}_1& \underset{Gp}{\to }& G{c}_2\end{array}$$\array{F c_1 & \overset{F p}{\to} & F c_2\\ ^{\alpha_{c_1}}\downarrow & \Downarrow^{\alpha_p}& \downarrow^{\alpha_{c_2}}\\ G c_1& \underset{G p}{\to} & G c_2}

  in $D$, which are natural with respect to vertical composition of squares in $C$.

• For each $c\in C$, if $1_c:c\to c$ is its horizontal identity, then the square ${\alpha }_{1_c}$ is equal to $1_{{\alpha }_c}$, the identity square on the arrow ${\alpha }_c$.

• For $p:c_1\to c_2$ and $q:c_2\to c_3$, the horizontal composite of ${\alpha }_p$ and ${\alpha }_q$ is equal to ${\alpha }_{qp}$.

The notion of horizontal transformation is dual.

The double category of double functors

Another characterization of transformations between double categories comes from observing that the 1-category $\mathrm{DblCat}$ is cartesian closed, and so any two double categories have an exponential $D^C$. The objects of $D^C$ are double functors, its vertical arrows are vertical transformations, and its horizontal arrows are horizontal transformations. Its squares are a sort of “square modification” relating a pair of vertical and a pair of horizontal transformations.

Generalizations

It is easy to modify the explicit definition to handle the cases when $C$ and $D$ are weak in one direction or the other, and/or when $F$ and $G$ are pseudo functors in one direction or the other, by composing with appropriate coherence constraints. In this way, we obtain many 2-categories of double categories.

It is also easy to define vertical transformations between double functors which are horizontally lax or colax, and dually. In fact, given double categories $C,D,C\prime ,D\prime$, lax functors $F:C\to D$ and $F\prime :C\prime \to D\prime$, and colax functors $G:C\to C\prime$ and $G\prime :D\to D\prime$, we can define a vertical transformation having the shape

$$\begin{array}{ccc}C& \stackrel{F}{\to }& D\\ {}^G\downarrow & \Downarrow & {\downarrow }^{G\prime }\\ C\prime & \underset{F\prime }{\to }& D\prime \end{array}$$\array{C & \overset{F}{\to} & D \\ ^G\downarrow & \Downarrow& \downarrow^{G'}\\ C'& \underset{F'}{\to} & D'}

despite the fact that the composites $G\prime \circ F$and $F\prime \circ G$ do not exist as double functors of any sort. Such transformations are the squares of a large double category $\mathrm{Dbl}$ whose objects are double categories, whose horizontal arrows are lax functors, and whose vertical arrows are colax functors. This, in turn, is a special case of a construction which works for algebras over any 2-monad.

Finally, we can also define vertical transformations between functors of (horizontally) virtual double categories.