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:
The word “transfor” was coined by Sjoerd Crans in this paper; it is a portmanteu of “functor” and “transformation.” A collection of components which forms a transfor is said to be transforial, as a generalization of “functorial” and “natural.”
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).
We haven't gotten around to saying anything precise yet, but you can see something in the discussion below, or in Crans's paper.
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.
This discussion was originally at modification. It discusses both terminology and definitions.
Finn: There is a pattern here: functors are indexed collections of objects, natural transformations are i.c.s of 1-cells, modifications i.c.s of 2-cells; and these are what make the collection of all $n$-categories into an $n+1$-category, for $0 \leq n \leq 2$ anyway. Any references for the pattern in higher dimensions?
Toby: Do you mean for the terminology or for the appropriate coherence laws? (the details that you've been leaving out). Not that I have either …
Incidentally, I corrected ‘function’ to ‘functor’ in you question above; I hope that's OK.
Finn: I meant terminology and/or an explanation for arbitrary $n$ (which Urs gives below).
Actually I was thinking of functions rather than functors, as they are the 1-cells in $0-Cat$. But of course functions are just functors between discrete categories, and thinking of them as the latter probably makes more sense when moving to higher dimensions.
Toby: Now, I would either have said ‘functors are indexed collections of objects’ or ‘functions are indexed collections of elements’; your mixture confused me! (^_^)
Finn: Ah! Point taken. In any case, I should have said ‘0-cell’ instead of ‘object’. But I think ‘functor’ is better anyway, as I said.
Urs: the pattern that Finn is looking for is that embodied in the nature of the internal hom of the closed monoidal structure on presheaves.
In its most general form, consider an infinity-category modeled as a simplicial set with certain properties. Being a simplicial set, this is a presheaf on the simplex category. Hence for $X$ and $Y$ such $\infty$-categories, the $\infty$-category of morphisms between them corresponds to the internal hom simplicial set
This simple formula encodes that pattern that Finn observed. It says that:
functors (the 0-cells in $[X,Y]$) are just maps $X \to Y$ from cells to cells;
natural transformations (the 1-cells in $[X,Y]$) are maps $X \times \Delta^1 \to Y$. Notice that $\Delta^1$ is the interval object in $SSet$ (or at least its Kanification is, but never mind that for the moment). Such maps send $n$-cells in $X$ to $(n+1)$-cells in $Y$.
modifications are maps $X \times \Delta^2 \to Y$, that map $n$-cells in $X$ to $(n+2)$-cells in $Y$.
It may be helpful to realize the same pattern in the globular context of, for instance, strict omega-category. These are certain presheaves not on the simplex category but on the globe category, but the pattern is the same: the internal hom strict $\omega$-category of morphisms between strict $\omega$-categories $X$ and $Y$ is
where now the tensor product appearing is no longer the cartesian one but the Crans-Gray tensor product and where $G^n$ is the standard globular $n$-globe. Again $G^1$ is a model for the interval object and we see that
functors are morphisms $X \to Y$;
transformations are morphisms $X \otimes G^1 \to Y$
modifications are morphisms $X \otimes G^2 \to Y$
etc. Same logic as before.
When thinking about this, it may be useful to explicitly apply the hom-adjunction everywhere and think for instance of a natural transformation as a morphism
from $X$ into the “category of cylinders in $Y$”. This is maybe the most intuitive way: if for instance $Y$ happens to be just a 2-category, then this says that a transformation between functors between 2-categories is a 1-functor from the 1-category underlying $X$ to the category of cylinders in $Y$ (satisfying some property). Which is exactly what it is, in components.
When in a certain mood, I like to think of this basic fact, that $n$-fold transformations between $k$-functors are essentially (in components) $(k-n)$-functors with values in $n$-cylinders as the “holographic principle” in category theory. That may sound a bit silly, but it is true that in the case the $k$-functors in questions are $k$-functors on $Bord_k$ respresenting $k$-dimensional quantum field theory, then teir transformations, being $(k-1)$-functors, represent $(k-1)$-dim QFT, and this relation between higher and lower dim QFT is called “holography” in phyiscs.
Finn: Cool! Thanks, Urs. I might move this section to an article on $n$-transformations (if that’s what they’re called) once I get my head around it properly.
Toby: Unfortunately, ‘$n$-transformation’ already (following ‘$n$-functor’) means a transformation between functors between $n$-categories. See Cheng–Gurski for this, along with ‘$n$-modification’ and even ‘$n$-perturbation’ (gee, that doesn't conflict with anything else, does it?), along with the claim that there is ‘no existing terminology’ thereafter.
I would probably say ‘$n$-morphism in $n Cat$’ (possibly with two different values of $n$); you can use ‘$n$-cell’ in place of ‘$n$-morphism’ if you like. But it would be nice to find something more specific that's not already taken. Or we could just throw out the Cheng–Gurski meaning of ‘$n$-transformation’; although it's not unique to them, it may not be too entrenched yet.
(But please let a transformation be a $1$-transformation, even though it is a $2$-morphism.)
Todd: I think what Urs and Crans both may be suggesting is that, at least in the context of strict $n$- and $\omega$-categories, there is a uniform notion of “transformation of depth k between n-functors”, or just $(n, k)$-transformations, where $(n, 1)$-transformations are usual transformations between $n$-functors, $(n, 2)$-transformations are modifications, and so on. Surely this usage won’t conflict with Cheng-Gurski.
Toby: Yeah, that would work, so we could write (n,k)-transformation. My only disgruntlement is that the $n$ is superfluous; the problem is all those other people that are already using it and preventing us from unambiguously saying simply ‘$k$-transformation’!
Finn: Probably tiros like me shouldn’t have a say in this sort of thing, but I would tend to agree with Toby here, that the $k$ is at least more interesting than the $n$, in that you’re more likely to vary the values of $k$ than those of $n$. However, typing the few extra characters does seem a small price to pay to avoid horrible confusion. I slightly reluctantly vote for $(n,k)$.
Todd: I’m not crazy about it either, but I agree it’s a small price. I’ll note (in case it helps) that in the general theory of Crans-Gray tensor products, both variations in $n$ and $k$ come up, about equally often (e.g., the tensor of a 1-category and an $n$-category is an $(n+1)$-category).
Urs: yes, so to summarize what I think the main points are
there is a systematic notion of “transformation of depth k between n-functors” for geometric definition of higher category in terms of simplicial sets;
the corresponding notion in the (strict) globular context is formalized by Crans’ construction;
unwrapping what this says, it yields in particular that a transformation of depth $k$ between strict globular $n$-categories $X$ and $Y$ is an $(n-k)$-functor from the truncation $X{\leq k}$ of $X$ to an $(n-k)$-category (throwing all higher cells away) to the $(n-k)$-category of $k$-globes in $Y$ (also truncated)
satisfying certain naturality conditions (which ensure precisely that $\eta$ extends uniquely to an $n$-functor $\eta : X \to [G^k,Y]$).
JCMcKeown: not meaning to cause annoyance, but how about calling them “$+k$-transformations”, owing to their incrementing dimensions by $k$; or if we don’t like the $+$ prefix, one might call them $k$-vexilors, because they tend to generate flags of period $k$.
Toby: Interesting; can you explain more about how they generate flags? (Maybe that's something to put in a new section here, or you could just give a reference.)
JCMcKeown: Just from reading above “… and more generally $j$-morphisms in $C$ to $(j+k)$-morphisms in $D$”… ahah! Now I see what you’re getting at. I’ve got my head fixed on endo-functors; where if you wanted to (I don’t mean it’s a good idea. Who knows?) you can consider iterations of the underlying function that is the $+k$-transformation.
Mike Shulman: FWIW, Sjoerd Crans has called these things k-transfors, and speaks of something being transforial as a general term including both “functorial” and “natural.”
Toby: I'm inclined to say that we should go with that!
Mike Shulman: I’m not sure how serious you are… but I’ve always thought it was a proposal that deserved to be taken more seriously than it seems to have been. The reference is “Localizations of Transfors,” K-Theory 2004 (I can’t find a free version online).
Toby: I'm perfectly serious. The term should be indexed primarily by $k$, with $n$ only if one really insists. I didn't want to make up my own word, but if Crans has published one, then why not use it? I should be able to check that reference the next time that I visit the UCR library (about once a week).
Mike Shulman: No argument here (about indexing by $k$). Also $(n,k)$-transformation sounds to me like something to do with (n,r)-categories, but there of course the comma denotes something completely different.
Todd Trimble: I like $k$-transfor.
Toby: Excellent! Since Finn and JCMcKeown have not been active lately, I'll move it over with that paper as a guide (or you can).
I would like to also mention ‘$(n,k)$-transformation’ (or maybe ‘$n$-$k$-transfor’?) as a possible term, however, since some people might want to specify $n$ just as some people like to say ‘$n$-functor’.
Toby: One could also say ‘$n$-natural $k$-transformation’, which fits (what Crans claims on page 2 to be standard) ‘$2$-natural transformation’ for a strict $(2,1)$-transformation. But I still like ‘$k$-transfor’ when $n$ is suppressed (which should be the default).
Mike: What about “$n$-categorical $k$-transfor” if it is necessary to specify $n$?
Toby: That works too. (Well, I don't like ‘categorical’, but that's a separate issue.)
functors
transformations
(∞,1)-natural transformation?
modifications
Camell Kachour: Kamel Kachour, Définition algébrique des cellules non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique (2008), volume 1, pages 1–68.
Camell Kachour: Steps toward the Weak ω-category of the Weak ω-categories in the globular setting, Published Categories and General Algebraic Structures with Applications (2015).