nLab transfor

Idea

A $k$-transfor is an operation from one $n$-category $C$ to another $D$ (for some value of $n$) that takes objects of $C$ to $k$-morphisms of $D$ (and more generally $j$-morphisms in $C$ to $(j+k)$-morphisms in $D$) in a coherent way. Equivalently, a $k$-transfor is a $k$-cell in an internal-hom $n$-category. Transfors are a common generalisation of:

• $n$-functors, which are 0-transfors
• $n$-natural transformations, which are 1-transfors
• modifications, which are 2-transfors,
• perturbations, which are 3-transfors,
• and so on.

The word “transfor” was coined by Sjoerd Crans in this paper; it is a portmanteau of “functor” and “transformation.” A collection of components which forms a transfor is said to be transforial, as a generalization of “functorial” and “natural.”

Terminology

Once upon a time, there were categories, functors between them, and natural transformations between them. Then when $n$-categories came along, people called the arrows between them ‘$n$-functors’ even though one could just as easily say ‘functors’. In the same vein, people said ‘$n$-transformations’ for natural transformations (that is, 2-transfors) between $n$-categories. At the same time, we saw that we needed modifications between $n$-transformations, and that there would have to be things between higher modifications, and so on. However, due to the prior use of “$n$-transformation” for a 2-transfor between $n$-categories, the natural choice “$k$-transformation” is unavailable to mean a $k$-transfor.

Here are some other possible terms for a $k$-transfor between $n$-categories, which additionally notate the value of $n$ (although this is, strictly speaking, unnecessary).

• $(n,k)$-transformation
• $n$-$k$-transfor
• $n$-dimensional $k$-transfor
• $n$-categorical $k$-transfor
• $n$-natural $k$-transformation

Definitions

We haven't gotten around to saying anything precise yet, but you can see something in the discussion below, or in Crans's paper.

Special cases

See this periodic table of $k$-transfors between $n$-categories for common names for low values of $n$ and $k$. On the $n$-Lab, we tend to omit the prefix $n$- whenever possible (as ironic as that may be).

$k$↓\$n$→	$-1$	$0$	$1$	$2$	$3$	...
$0$	implication	function	functor	$2$-functor	$3$-functor	...
$1$	trivial	equality of functions	natural transformation	$2$-transformation	$3$-transformation	...
$2$	"	trivial	equality of natural transformations	modification	$3$-modification	...
$3$	"	"	trivial	equality of modifications	perturbation	...
$4$	"	"	"	trivial	equality of perturbations	...
$5$	"	"	"	"	trivial	...
⋮	"	"	"	"	"	⋱

Note that the source and target of a $k$-transfor (between $n$-categories) are $(k-1)$-transfors (between the same $n$-categories). Given two fixed source and target $(k-1)$-transfors, the $k$-transfors between them (and the $(k+1)$-transfors between those, and so on) form an $(n-k)$-category.

For n-posets

A similar table periodic table of $k$-transfors between $n$-posets exists for common names for low values of $n$ and $k$.

$k$↓\$n$→	$-1$	$0$	$1$	$2$	$3$	...
$0$	implication	monotonic function	functor	$2$-functor	$3$-functor	...
$1$	trivial	partial order of monotonic functions	natural transformation	$2$-transformation	$3$-transformation	...
$2$	"	trivial	partial order of natural transformations	modification	$3$-modification	...
$3$	"	"	trivial	partial order of modifications	perturbation	...
$4$	"	"	"	trivial	partial order of perturbations	...
$5$	"	"	"	"	trivial	...
⋮	"	"	"	"	"	⋱

Related concepts

• homotopy

• functors

  • functor

  • 2-functor

  • (∞,1)-functor

• transformations

  • natural transformation

  • pseudonatural transformation / lax natural transformation

  • (∞,1)-natural transformation?

  • (∞,n)-natural transformation

• modifications

  • modification

References

• Sjoerd Crans: A Tensor Product for Gray-Categories, Theory and Applications of Categories 5 2 (1999) 12–69 [tac:5-02, pdf]

• Sjoerd Crans: Localizations of Transfors, K-Theory 28 1 (2003) 39-105 [doi:10.1023/A:1024186923002]

• Camell Kachour: Définition algébrique des cellules non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique 1 (2008) 1-68 [numdam:CTGDC_2008__49_1_1_0, pdf]

• Camell Kachour: Steps toward the weak higher category of the weak higher categories in the globular setting, Categories and General Algebraic Structures with Applications 4 1 (2016) 9-42 [cgasa:11180, pdf]