Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



A kk-transfor is an operation from one nn-category CC to another DD (for some value of nn) that takes objects of CC to kk-morphisms of DD (and more generally jj-morphisms in CC to (j+k)(j+k)-morphisms in DD) in a coherent way. Equivalently, a kk-transfor is a kk-cell in an internal-hom nn-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 nn-categories came along, people called the arrows between them ‘nn-functors’ even though one could just as easily say ‘functors’. In the same vein, people said ‘nn-transformations’ for natural transformations (that is, 2-transfors) between nn-categories. At the same time, we saw that we needed modifications between nn-transformations, and that there would have to be things between higher modifications, and so on. However, due to the prior use of “nn-transformation” for a 2-transfor between nn-categories, the natural choice “kk-transformation” is unavailable to mean a kk-transfor.

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

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


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 kk-transfors between nn-categories for common names for low values of nn and kk. On the nn-Lab, we tend to omit the prefix nn- whenever possible (as ironic as that may be).

$1$trivialequality of functionsnatural transformation$2$-transformation$3$-transformation...
$2$\"trivialequality of natural transformationsmodification$3$-modification...
$3$\"\"trivialequality of modificationsperturbation...
$4$\"\"\"trivialequality of perturbations...

Note that the source and target of a kk-transfor (between nn-categories) are (k1)(k-1)-transfors (between the same nn-categories). Given two fixed source and target (k1)(k-1)-transfors, the kk-transfors between them (and the (k+1)(k+1)-transfors between those, and so on) form an (nk)(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 nn-categories into an n+1n+1-category, for 0n20 \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 nn (which Urs gives below).

Actually I was thinking of functions rather than functors, as they are the 1-cells in 0Cat0-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 XX and YY such \infty-categories, the \infty-category of morphisms between them corresponds to the internal hom simplicial set

[X,Y]=Hom SSet(X×Δ ,Y). [X,Y] = Hom_{SSet}(X \times \Delta^\bullet, Y) \,.

This simple formula encodes that pattern that Finn observed. It says that:

  • functors (the 0-cells in [X,Y][X,Y]) are just maps XYX \to Y from cells to cells;

  • natural transformations (the 1-cells in [X,Y][X,Y]) are maps X×Δ 1YX \times \Delta^1 \to Y. Notice that Δ 1\Delta^1 is the interval object in SSetSSet (or at least its Kanification is, but never mind that for the moment). Such maps send nn-cells in XX to (n+1)(n+1)-cells in YY.

  • modifications are maps X×Δ 2YX \times \Delta^2 \to Y, that map nn-cells in XX to (n+2)(n+2)-cells in YY.

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 ω\omega-category of morphisms between strict ω\omega-categories XX and YY is

[X,Y]=Hom ωCat(XG ,Y), [X,Y] = Hom_{\omega Cat}(X \otimes G^\bullet , Y) \,,

where now the tensor product appearing is no longer the cartesian one but the Crans-Gray tensor product and where G nG^n is the standard globular nn-globe. Again G 1G^1 is a model for the interval object and we see that

  • functors are morphisms XYX \to Y;

  • transformations are morphisms XG 1YX \otimes G^1 \to Y

  • modifications are morphisms XG 2YX \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

X[I,Y] X \to [I,Y]

from XX into the “category of cylinders in YY”. This is maybe the most intuitive way: if for instance YY 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 XX to the category of cylinders in YY (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 nn-fold transformations between kk-functors are essentially (in components) (kn)(k-n)-functors with values in nn-cylinders as the “holographic principle” in category theory. That may sound a bit silly, but it is true that in the case the kk-functors in questions are kk-functors on Bord kBord_k respresenting kk-dimensional quantum field theory, then teir transformations, being (k1)(k-1)-functors, represent (k1)(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 nn-transformations (if that’s what they’re called) once I get my head around it properly.

Toby: Unfortunately, ‘nn-transformation’ already (following ‘nn-functor’) means a transformation between functors between nn-categories. See Cheng–Gurski for this, along with ‘nn-modification’ and even ‘nn-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 ‘nn-morphism in nCatn Cat’ (possibly with two different values of nn); you can use ‘nn-cell’ in place of ‘nn-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 ‘nn-transformation’; although it's not unique to them, it may not be too entrenched yet.

(But please let a transformation be a 11-transformation, even though it is a 22-morphism.)

Todd: I think what Urs and Crans both may be suggesting is that, at least in the context of strict nn- and ω\omega-categories, there is a uniform notion of “transformation of depth k between n-functors”, or just (n,k)(n, k)-transformations, where (n,1)(n, 1)-transformations are usual transformations between nn-functors, (n,2)(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 nn is superfluous; the problem is all those other people that are already using it and preventing us from unambiguously saying simply ‘kk-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 kk is at least more interesting than the nn, in that you’re more likely to vary the values of kk than those of nn. However, typing the few extra characters does seem a small price to pay to avoid horrible confusion. I slightly reluctantly vote for (n,k)(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 nn and kk come up, about equally often (e.g., the tensor of a 1-category and an nn-category is an (n+1)(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 kk between strict globular nn-categories XX and YY is an (nk)(n-k)-functor from the truncation XkX{\leq k} of XX to an (nk)(n-k)-category (throwing all higher cells away) to the (nk)(n-k)-category of kk-globes in YY (also truncated)

    η:X k[G k,Y]| k \eta : X_{\leq k} \to [G^k, Y]|_{\leq k}

    satisfying certain naturality conditions (which ensure precisely that η\eta extends uniquely to an nn-functor η:X[G k,Y] \eta : X \to [G^k,Y] ).

JCMcKeown: not meaning to cause annoyance, but how about calling them “+k+k-transformations”, owing to their incrementing dimensions by kk; or if we don’t like the ++ prefix, one might call them kk-vexilors, because they tend to generate flags of period kk.

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 jj-morphisms in CC to (j+k)(j+k)-morphisms in DD”… 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+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 kk, with nn 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 kk). Also (n,k)(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 kk-transfor.

Mike Shulman: Found it

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)(n,k)-transformation’ (or maybe ‘nn-kk-transfor’?) as a possible term, however, since some people might want to specify nn just as some people like to say ‘nn-functor’.

Toby: One could also say ‘nn-natural kk-transformation’, which fits (what Crans claims on page 2 to be standard) ‘22-natural transformation’ for a strict (2,1)(2,1)-transformation. But I still like ‘kk-transfor’ when nn is suppressed (which should be the default).

Mike: What about “nn-categorical kk-transfor” if it is necessary to specify nn?

Toby: That works too. (Well, I don't like ‘categorical’, but that's a separate issue.)

References for the globular approach

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).

Revised on October 22, 2015 21:30:05 by Anonymous Coward (