\documentclass[draft]{article}
\usepackage{amssymb}
\usepackage{fancybox}
\usepackage[all]{xy}
\input 15851
\input theorem
\input homeworknodia
\input prooftree

\def\rid{\mathrm{id}}
\def\easyrule#1#2{
\begin{prooftree}
#1 \justifies #2
\end{prooftree}}

\title{Speculations on Weak $\omega$-structures}
\author{Jason Reed}
\def\lol{\multimap}
\def\rtype{\mathop{\rm type}}

\begin{document}
\maketitle
\begin{abstract}
There are some interesting definitions of algebraic structures that
can be made with relatively little machinery beyond the usual
definition of strict $n$-categories, which yield {\em something} like
weak $\omega$-categories, not that I'm claiming any particular
equivalence.  Of particular interest is how higher associativity laws
and coherence properties possibly `fall out' of composability of cells
one dimension up.

The main question is, in what sense is might this definition be `too
strict', say, at dimensions 3 and above where not every weak category
can be strictified?
\end{abstract}

{\bf General Disclaimer}: I am not an expert in weak $\omega$-category
theory. It is entirely possible what is discussed here is not
original, not correct, or both.
\section{Preliminaries}
\def\ncat{n\mathbf{Cat}}
The tower of $n$-categories is defined as usual by saying that a $0$-category is
a set, and an $n+1$-category is a category enriched over the evident category
$\ncat$ of $n$-categories for every $n\ge 0$.

A strict $\omega$ category is a globular set $(S_n)_{n\in\N}$ of $n$-cells
such that every prefix of it forms a strict $n$-category in a coherent way.

We write the maps that return the codomain and domain of cells as $c$
and $d$. We write $\partial$ as a variable varying over the set
$\{c,d\}$. By the definition of globular sets, $cc = cd$ and $dd =
dc$. For this reason, we can unambiguously write $c^n$ and $d^n$ for a
string of $n$ (co)domain operations, where the outermost is,
respectively, codomain or domain. For the composition that is `$n-1$
times more horizontal' than `purely vertical' composition, we write
$\circ_n$. We write $\circ_1$ (`purely vertical' composition)
sometimes as $\circ$ and sometimes as mere juxtaposition. It makes
sense when the two cells being composed share a boundary which is $n$
dimensions smaller than them: $f \circ_n g$ is well-defined if $c^n g
= d^n f$. We have of course that and $d(f\circ g) = dg$ and $c(f\circ
g) = cf$ and $\partial(f \circ_{n+1} g) = \partial f \circ_{n}
\partial g$ for $n \ge 1$.

The notation $\rid^n A$ is defined in the obvious way by $\rid^0 A = A$ and $\rid^{n+1} A = \rid_{\rid^n A}$.

\section{Equivalence}
Inside a strict $\omega$-category, some arrows are very strongly
invertible: those for which there is an infinite proof tree using the
following rule.

$$
\easyrule
{d : A \to B \qquad d^* : B \to A \qquad m_A : dd^* \equiv \rid_B \qquad m_B : d^*d \equiv \rid_A}
{d : A \equiv B}
$$

That is, $d$ is an equivalence between $A$ and $B$ if it has a
pseudoinverse $d^*$ such that both composites of $d$ and $d^*$ admit
an equivalence to the appropriate identity arrow. That $\equiv$
is an equivalence relation can be easily shown by a coinductive
argument.

\section{Weak $\omega$-structures}

\subsection{Definition} 
A {\em weak $\omega$-structure} is a strict $\omega$-category $(S_n)_{n\in\N}$
together with specified subsets $(W_n)_{n\in\N}$ such that
\begin{enumerate}
\item[(i)] If $w\in W_{n+1}$, then $\partial w \in W_n$, for all $n \in \N$.
\item[(ii)] %If $s\in S_0$, then there exist $w\in W_0$ and $d : s\equiv w$.\\
If $s\in S_{n}$
% and $\partial s \in W_{n}$,
 then there 
exist $w\in W_{n}$ and $d\in S_{n+1}$ such that $d : s \equiv w$.
\end{enumerate}

The intuition is that the cells $W_n$ are the `real' cells that belong
to the structure, but that in order to talk about composition, we want
to talk about pasting diagrams constructed out of those cells. The
compositions of the strict $n$-category are the operations with which
we construct pasting diagrams; they satisfy associative, interchange
and unit laws on the nose. The axiom (i) is simple closure of the
`real' cells under domain and comdain, and (ii) is a sort of
reflection principle that finds an image of the strict category in the
weak structure, implying that every pasting diagram has a (not
necessarily unique, but appropriately canonical) composite.

\subsection{Composition} 

Na\"ively, to take a composite in the weak structure, we just take the
composite in the strict category and reflect: say\footnote{Really what
is meant here despite the equals sign is a ternary relation on
$f,g,h$. Weak composition is absolutely {\em not} a unique function
that outputs $h$ from $f,g$, but a relation which characterizes which
$h$ count as composites of $f,g$} $f *_n g = h$ if $ f \circ_n g\equiv
h$. However, this doesn't quite seem appropriate for compositions that
are at all horizontal, i.e. for $n \ge 2$.  This is because the
(co)domain $\partial(f *_{n+1} g)$ of $f *_{n+1} g$ should be
$\partial f *_n \partial g$, not the result of {\em strict}
composition $\partial f \circ_n \partial g$, which is what we get from
the above recipe.

Instead we want to allow some `slackening' equivalence cells to mediate between
the strict composition and the weak composition of the domain and codomain
of the cells being horizontally composed. For example, for the $*_2$ (`once horizontal') composite of
two cells, it should be any cell that is equivalent to the pasting diagram
on the left side of Figure~\ref{wc}.
\begin{figure}
$$
\fbox{
\xymatrix{
 % FIRST HALF
\bullet
\ar@/^3pc/[rr]^{f' * f}="gf"
\ar@/_3pc/[rr]_{g' * g}="ggff"
\ar@/^/[r]^f="f" 
\ar@/_/[r]_g="g" 
& \bullet
\ar@{}"gf";[]|(0.3){\,}="s3"
\ar@{}"gf";[]|(0.8){\,}="e3"
\ar@2{<->}"s3";"e3"|-\equiv
\ar@{}"ggff";[]|(0.3){\,}="s4"
\ar@{}"ggff";[]|(0.8){\,}="e4"
\ar@2{<->}"s4";"e4"|-\equiv
\ar@/^/[r]^{f'}="ff" 
\ar@/_/[r]_{g'}="gg" 
& \bullet
\ar@{=>}"f";"g"_\alpha
\ar@{=>}"ff";"gg"_\beta 
&
\ar@{}[l];[r]|(0.2){\,}="s"
\ar@{}[l];[r]|(0.8){\,}="e"
\ar@3{<->}"s";"e"^\equiv
& % SECOND HALF
 \bullet 
\ar@/^3pc/[rr]^{f' * f}="gf"
\ar@/_3pc/[rr]_{g' * g}="ggff"
&
\ar@{}"gf";"ggff"|(0.2){\,}="s2"
\ar@{}"gf";"ggff"|(0.8){\,}="e2"
\ar@2{->}"s2";"e2"|-{\beta *_2 \alpha}
& \bullet
}}
$$
\caption{Weak Composition}\label{wc}
\end{figure}
The blame for this problem could be laid on $\equiv$ for only relating
cells that have the same domain and codomain; we therefore generalize
$\equiv$ to $\equiv_n$ (with $\equiv_1$ being $\equiv$) which is a relation
between cells $x,y$ that satisfy $\partial^n x = \partial^n y$. For $n \ge 1$,
this is defined by
$$\easyrule
{\alpha : d^n y \equiv d^n x \qquad \beta : c^n x \equiv c^n y \qquad d : \rid^{n-1}_\beta \circ_n x \circ_n \rid^{n-1}_\alpha \equiv_n y}
{d : x \equiv_{n+1} y}
$$

We can now officially define weak composition as a three-place relation by
$$\easyrule{ f \circ_n g\equiv_n h}{*_n(f, g, h)}$$

We also should revise condition (ii) above to
\begin{enumerate}
\item[(ii')] 
If $s\in S_{n+m}$ and $\partial^m s \in W_{n}$,
 then there 
exist $w\in W_{n+m}$ and $d \in S_{n+m+1}$ such that $d : s \equiv_m w$.
\end{enumerate}

(This will correctly guarantee that all composites at least {\em exist})

Finally we can define the relation of what it means to be a weak identity:
$$\easyrule{f\equiv \rid_A}{ \rid (A, f)}$$

\begin{question}
How reasonable is it to make the following definition?

A {\em weak $\omega$-category} is a globular set of cells
$(W_n)_{n\in\N}$ together with relations $*_n(f, g, h)$ and $\rid(A, f)$
that arises from the construction above, forgetting which $S_n$ was used.
\end{question}

In any such structure, we do get some laws which are satsfied just by
our ability to compose higher-dimensional cells. For instance, $*$ is
weakly associative. For if we have some $n$-cells $f,g,h$ sitting around, and
we are able to compose them and find out that $*(f, g, m_{fg})$ and
$*(g, h, m_{gh})$ and $*(f, m_{gh}, m_{f(gh)})$ and $*(m_{fg}, h,
m_{(fg)h})$, then we can fill in the diagram in Figure~\ref{assoc}.

%% there are $(n+1)$-cells going between $m_{(fg)h}$
%% and $m_{f(gh)}$, whose composites are equivalent to the identity,
%% which means there are cells going back and forth between those
%% composites and the identity, which when composed are equivalent to
%% higher identites, etc.

\begin{figure}
$$\fbox{\xymatrix{
 % FIRST HALF
\bullet
\ar[r]|-h
\ar@/^4pc/[rrr]|-{m_{f(gh)}}="f gh"
\ar@/^2pc/[rr]|-{m_{gh}}="gh"
\ar@/_4pc/[rrr]|-{m_{(fg)h}}="fg h"
& \bullet
\ar@2{<->}"gh";[]|-\equiv
\ar[r]|-g
\ar@/_2pc/[rr]|-{m_{fg}}="fg"
& \bullet
\ar@2{<->}"fg";[]|-\equiv
\ar[r]|-f
& \bullet
\ar@2{<->}"gh";"f gh"|-\equiv
\ar@2{<->}"fg";"fg h"|-\equiv
}}$$
\caption{Weak Associativity}\label{assoc}
\end{figure}

Note that the strict composite $f\circ g \circ h$ may not exist as a
cell in $W_n$ at all, but we can nonetheless build up (in $S_{n+1}$) the
cell depicted by the whole diagram which stretches from $m_{f(gh)}$
to $m_{(fg)h}$, (which by assumption exist in $W_n$)
and so that cell must have some reflection in $W_{n+1}$.

\section{Conclusion}

So this notion of weak $\omega$-category is not completely obviously
broken yet --- we don't require associativity on the nose, for
instance. But it's hard to picture what goes on at higher dimensions, and
specifically I don't know how to formulate the following formally:
\begin{question}
If we chop everything off at dimension $1,2,3$, is this at
least equivalent to the commonly accepted definition of category,
bicategory, tricategory? 
\end{question}
An immediate obstacle is that composition is relational rather than functional.
But I seem to recall that this was not unheard-of in the weak $n$-category literature.
\end{document}