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\section*{transfor}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references_for_the_globular_approach}{References for the globular approach}\dotfill \pageref*{references_for_the_globular_approach} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{$k$-transfor} is an operation from one $n$-[[n-category|category]] $C$ to another $D$ (for some value of $n$) that takes [[object|objects]] of $C$ to $k$-[[k-morphism|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:

\begin{itemize}%
\item $n$-[[n-functor|functor]]s, which are 0-transfors
\item $n$-[[natural transformation]]s, which are 1-transfors
\item [[modifications]], which are 2-transfors,
\item [[perturbation]]s, which are 3-transfors,
\item and so on.

\end{itemize}
The word ``transfor'' was coined by Sjoerd Crans in \href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.506&rep=rep1&type=pdf}{this paper}; it is a \emph{portmanteau} of ``functor'' and ``transformation.'' A collection of components which forms a transfor is said to be \emph{transforial}, as a generalization of ``functorial'' and ``natural.''

\hypertarget{terminology}{}\subsection*{{Terminology}}\label{terminology}

Once upon a time, there were [[categories]], [[functor|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).

\begin{itemize}%
\item $(n,k)$-transformation
\item $n$-$k$-transfor
\item $n$-dimensional $k$-transfor
\item $n$-categorical $k$-transfor
\item $n$-natural $k$-transformation

\end{itemize}
\hypertarget{definitions}{}\subsection*{{Definitions}}\label{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.

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{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).

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.

\hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion}

This discussion was originally at [[modification]]. It discusses both terminology and definitions.

[[Finn Lawler|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?

\emph{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 \ldots{}

Incidentally, I corrected `function' to `functor' in you question above; I hope that's OK.

[[Finn Lawler|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.

\emph{Toby}: Now, I would either have said `functors are indexed collections of objects' or `functions are indexed collections of elements'; your mixture confused me! ({\tt \symbol{94}}\_{\tt \symbol{94}})

\emph{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 Schreiber|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

\begin{displaymath}
[X,Y] = Hom_{SSet}(X \times \Delta^\bullet, Y)
  \,.
\end{displaymath}
This simple formula encodes that pattern that Finn observed. It says that:

\begin{itemize}%
\item [[functor]]s (the 0-cells in $[X,Y]$) are just maps $X \to Y$ from cells to cells;


\item [[natural transformation]]s (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 $Y$.


\item modifications are maps $X \times \Delta^2 \to Y$, that map $n$-cells in $X$ to $(n+2)$-cells in $Y$.



\end{itemize}
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

\begin{displaymath}
[X,Y] = Hom_{\omega Cat}(X \otimes G^\bullet , Y)
  \,,
\end{displaymath}
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

\begin{itemize}%
\item functors are morphisms $X \to Y$;


\item transformations are morphisms $X \otimes G^1 \to Y$


\item modifications are morphisms $X \otimes G^2 \to Y$



\end{itemize}
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

\begin{displaymath}
X \to [I,Y]
\end{displaymath}
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 Lawler|Finn]]: Cool! Thanks, Urs. I might move this section to an article on $n$-[[n-transformation|transformations]] (if that's what they're called) once I get my head around it properly.

\emph{Toby}: Unfortunately, `$n$-transformation' already (following `$n$-functor') means a transformation between functors between $n$-categories. See \href{http://cheng.staff.shef.ac.uk/degeneracy/cheng-gurski-degeneracy.pdf}{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 Trimble|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.

\emph{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'!

\emph{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 $(n,k)$.

\emph{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 Schreiber|Urs]]: yes, so to summarize what I think the main points are

\begin{itemize}%
\item there is a systematic notion of ``transformation of depth k between n-functors'' for [[geometric definition of higher category]] in terms of simplicial sets;


\item the corresponding notion in the (strict) globular context is formalized by Crans' construction;


\item 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)

\begin{displaymath}
\eta : X_{\leq k} \to [G^k, Y]|_{\leq k}
\end{displaymath}
satisfying certain naturality conditions (which ensure precisely that $\eta$ extends uniquely to an $n$-functor $\eta : X \to [G^k,Y]$).



\end{itemize}
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 $k$.

\emph{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 ``\ldots{} and more generally $j$-morphisms in $C$ to $(j+k)$-morphisms in $D$''\ldots{} ahah! Now I see what you're getting at. I've got my head fixed on \emph{endo}-functors; where if you wanted to (I don't mean it's a \emph{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 \textbf{k-transfors}, and speaks of something being \emph{transforial} as a general term including both ``functorial'' and ``natural.''

\emph{Toby}: I'm inclined to say that we should go with that!

[[Mike Shulman]]: I'm not sure how serious you are\ldots{} 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,'' \emph{K-Theory} 2004 (I can't find a free version online).

\emph{Toby}: I'm perfectly serious. The term \emph{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.

[[Mike Shulman]]: \href{http://home.tiscali.nl/secrans/papers/lotr.html}{Found it}

\emph{Toby}: Excellent! Since [[Finn Lawler|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 \emph{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'.

\emph{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).

\emph{Mike}: What about ``$n$-categorical $k$-transfor'' if it is necessary to specify $n$?

\emph{Toby}: That works too. (Well, I don't like `categorical', but that's a separate issue.)

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[homotopy]]


\item functors

\begin{itemize}%
\item [[functor]]


\item [[2-functor]]


\item [[(∞,1)-functor]]



\end{itemize}

\item transformations

\begin{itemize}%
\item [[natural transformation]]


\item [[pseudonatural transformation]] / [[lax natural transformation]]


\item [[(∞,1)-natural transformation]]


\item [[(∞,n)-natural transformation]]



\end{itemize}

\item modifications

\begin{itemize}%
\item [[modification]]

\end{itemize}


\end{itemize}
\hypertarget{references_for_the_globular_approach}{}\subsection*{{References for the globular approach}}\label{references_for_the_globular_approach}

[[Camell Kachour]]: Kamel Kachour, D\'e{}finition alg\'e{}brique des cellules non-strictes, Cahiers de Topologie et de G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}gorique (2008), volume 1, pages 1--68.

[[Camell Kachour]]: Steps toward the Weak $\omega$-category of the Weak $\omega$-categories in the globular setting, Published Categories and General Algebraic Structures with Applications (2015).

[[!redirects (n,j)-transformation]] [[!redirects j-transformation]] [[!redirects k-transformation]] [[!redirects n-transformation]] [[!redirects (n,j)-transformations]] [[!redirects (n,k)-transformation]] [[!redirects (n,k)-transformations]] [[!redirects j-transformations]] [[!redirects k-transformations]] [[!redirects n-transformations]] [[!redirects k-transfor]] [[!redirects k-transfors]] [[!redirects transfors]]

[[!redirects n-natural transformation]] [[!redirects n-natural transformations]]



\end{document}