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.
If $C$ and $D$ are strict double categories regarded as internal categories in $Cat$ and $F,G\colon C\to D$ are double functors regarded as internal functors in $Cat$, then a transformation between them is simply an internal natural transformation in $Cat$. Whether this is a vertical or horizontal transformation depends on how we identify double categories with internal categories in $Cat$ (there being two ways).
More explicitly, a vertical transformation $\alpha\colon F\to G$ consists of
For every object $c\in C$, a vertical arrow
in $D$, which are natural with respect to vertical composition of vertical arrows in $C$.
For every horizontal arrow $p\colon c_1 \to c_2$ in $C$, a square
in $D$, which are natural with respect to vertical composition of squares in $C$.
For each $c\in C$, if $1_c\colon c\to c$ 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 $p\colon c_1\to c_2$ and $q\colon c_2\to c_3$, the horizontal composite of $\alpha_p$ and $\alpha_q$ is equal to $\alpha_{q p}$.
The notion of horizontal transformation is dual.
Another characterization of transformations between double categories comes from observing that the 1-category $DblCat$ 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.
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 $C,D,C',D'$, lax functors $F\colon C\to D$ and $F'\colon C'\to D'$, and colax functors $G\colon C\to C'$ and $G'\colon D\to D'$, we can define a vertical transformation having the shape
despite the fact that the composites $G' \circ F$and $F'\circ G$ do not exist as double functors of any sort. Such transformations are the squares of a large double category $Dbl$ 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.
Last revised on April 7, 2010 at 23:09:10. See the history of this page for a list of all contributions to it.