nLab
vertical transformation

Contents

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

More explicitly, a vertical transformation α:FG\alpha\colon F\to G consists of

  • For every object cCc\in C, a vertical arrow

    Fc α c Gc \array{ F c \\ \downarrow ^{\alpha_c} \\ G c}

    in DD, which are natural with respect to vertical composition of vertical arrows in CC.

  • For every horizontal arrow p:c 1c 2p\colon c_1 \to c_2 in CC, a square

    Fc 1 Fp Fc 2 α c 1 α p α c 2 Gc 1 Gp Gc 2\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 DD, which are natural with respect to vertical composition of squares in CC.

  • For each cCc\in C, if 1 c:cc1_c\colon c\to c is its horizontal identity, then the square α 1 c\alpha_{1_c} is equal to 1 α c1_{\alpha_c}, the identity square on the arrow α c\alpha_c.

  • For p:c 1c 2p\colon c_1\to c_2 and q:c 2c 3q\colon c_2\to c_3, the horizontal composite of α p\alpha_p and α q\alpha_q is equal to α qp\alpha_{q p}.

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 DblCatDblCat is cartesian closed, and so any two double categories have an exponential D CD^C. The objects of D CD^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 CC and DD are weak in one direction or the other, and/or when FF and GG 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,DC,D,C',D', lax functors F:CDF\colon C\to D and F:CDF'\colon C'\to D', and colax functors G:CCG\colon C\to C' and G:DDG'\colon D\to D', we can define a vertical transformation having the shape

C F D G G C F D\array{C & \overset{F}{\to} & D \\ ^G\downarrow & \Downarrow& \downarrow^{G'}\\ C'& \underset{F'}{\to} & D'}

despite the fact that the composites GFG' \circ F and FGF'\circ G do not exist as double functors of any sort. Such transformations are the squares of a large double category DblDbl 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.

Revised on April 7, 2010 23:09:10 by Mike Shulman (205.208.124.120)