--- a/pnas/pnas.tex Mon Mar 28 18:07:36 2011 -0700
+++ b/pnas/pnas.tex Fri May 06 15:32:55 2011 -0700
@@ -65,6 +65,19 @@
\usepackage{amssymb,amsfonts,amsmath,amsthm}
+% fiddle with fonts
+
+\usepackage{microtype}
+
+\usepackage{ifxetex}
+\ifxetex
+\usepackage{xunicode,fontspec,xltxtra}
+\setmainfont[Ligatures={}]{Linux Libertine O}
+\usepackage{unicode-math}
+\setmathfont{Asana Math}
+\fi
+
+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% OPTIONAL MACRO FILES
%% Insert self-defined macros here.
@@ -159,6 +172,8 @@
\def\spl{_\pitchfork}
+
+
% equations
\newcommand{\eq}[1]{\begin{displaymath}#1\end{displaymath}}
\newcommand{\eqar}[1]{\begin{eqnarray*}#1\end{eqnarray*}}
@@ -337,6 +352,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{article}
+
\begin{abstract}
We summarize our axioms for higher categories, and describe the ``blob complex".
Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$.
@@ -494,7 +510,7 @@
While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory
of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories.
In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a
-{\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$.
+{\it strictly associative} composition $\Omega_r \times \Omega_s \to \Omega_{r+s}$.
Thus we can have the simplicity of strict associativity in exchange for more morphisms.
We wish to imitate this strategy in higher categories.
Because we are mainly interested in the case of pivotal $n$-categories, we replace the intervals $[0,r]$ not with
@@ -743,7 +759,7 @@
Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$
-with $\du_b Y_b = M_i$ for some $i$.
+with $\du_b Y_b = M_i$ for some $i$, and each $M_j$ with $j<i$ is also a disjoint union of balls.
\begin{defn}
The poset $\cell(W)$ has objects the permissible decompositions of $W$,
@@ -906,6 +922,7 @@
\end{equation*}
\end{property}
+
If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary,
write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$.
\begin{property}[Gluing map]
@@ -916,9 +933,9 @@
%\end{equation*}
Given a gluing $X \to X \bigcup_{Y}\selfarrow$, there is
a map
-\[
- \bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow),
-\]
+$
+ \bc_*(X) \to \bc_*\left(X \bigcup_{Y} \selfarrow\right),
+$
natural with respect to homeomorphisms, and associative with respect to iterated gluings.
\end{property}
Binary file talks/2011-Teichner/Section7-Scott_Morrison.pdf has changed
--- a/text/a_inf_blob.tex Mon Mar 28 18:07:36 2011 -0700
+++ b/text/a_inf_blob.tex Fri May 06 15:32:55 2011 -0700
@@ -134,7 +134,7 @@
We want to find 2-simplices which fill in this cycle.
Choose a decomposition $M$ which has common refinements with each of
$K$, $KL$, $L$, $K'L$, $K'$, $K'L'$, $L'$ and $KL'$.
-(We also also require that $KLM$ antirefines to $KM$, etc.)
+(We also require that $KLM$ antirefines to $KM$, etc.)
Then we have 2-simplices, as shown in Figure \ref{zzz5}, which do the trick.
(Each small triangle in Figure \ref{zzz5} can be filled with a 2-simplex.)
@@ -192,7 +192,7 @@
We have $\psi(r) = 0$ since $\psi$ is zero on $(\ge 1)$-simplices.
Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models.
-To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above.
+To each generator $(b, \ol{K})$ of $\cl{\cC_F}(Y)$ we associate the acyclic subcomplex $D(b)$ defined above.
Both the identity map and $\phi\circ\psi$ are compatible with this
collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps
are homotopic.
@@ -248,12 +248,12 @@
A fiber bundle $F\to E\to Y$ gives an example of a $k$-category over $Y$:
assign to $p:D\to Y$ the blob complex $\bc_*(p^*(E))$, if $\dim(D) = k$,
or the fields $\cE(p^*(E))$, if $\dim(D) < k$.
-($p^*(E)$ denotes the pull-back bundle over $D$.)
+(Here $p^*(E)$ denotes the pull-back bundle over $D$.)
Let $\cF_E$ denote this $k$-category over $Y$.
We can adapt the homotopy colimit construction (based decompositions of $Y$ into balls) to
get a chain complex $\cl{\cF_E}(Y)$.
The proof of Theorem \ref{thm:product} goes through essentially unchanged
-to show that
+to show the following result.
\begin{thm}
Let $F \to E \to Y$ be a fiber bundle and let $\cF_E$ be the $k$-category over $Y$ defined above.
Then
@@ -270,7 +270,7 @@
$D\widetilde{\times} M$ is a manifold of dimension $n-k+j$ with a collar structure along the boundary of $D$.
(If $D\to Y$ is an embedding then $D\widetilde{\times} M$ is just the part of $M$
lying above $D$.)
-We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which a re good with respect to $M$.
+We can define a $k$-category $\cF_M$ based on maps of balls into $Y$ which are good with respect to $M$.
We can again adapt the homotopy colimit construction to
get a chain complex $\cl{\cF_M}(Y)$.
The proof of Theorem \ref{thm:product} again goes through essentially unchanged
--- a/text/appendixes/comparing_defs.tex Mon Mar 28 18:07:36 2011 -0700
+++ b/text/appendixes/comparing_defs.tex Fri May 06 15:32:55 2011 -0700
@@ -48,12 +48,12 @@
The base case is for oriented manifolds, where we obtain no extra algebraic data.
For 1-categories based on unoriented manifolds,
-there is a map $*:c(\cX)^1\to c(\cX)^1$
+there is a map $\dagger:c(\cX)^1\to c(\cX)^1$
coming from $\cX$ applied to an orientation-reversing homeomorphism (unique up to isotopy)
from $B^1$ to itself.
Topological properties of this homeomorphism imply that
-$a^{**} = a$ (* is order 2), * reverses domain and range, and $(ab)^* = b^*a^*$
-(* is an anti-automorphism).
+$a^{\dagger\dagger} = a$ ($\dagger$ is order 2), $\dagger$ reverses domain and range, and $(ab)^\dagger = b^\dagger a^\dagger$
+($\dagger$ is an anti-automorphism).
For 1-categories based on Spin manifolds,
the the nontrivial spin homeomorphism from $B^1$ to itself which covers the identity
--- a/text/deligne.tex Mon Mar 28 18:07:36 2011 -0700
+++ b/text/deligne.tex Fri May 06 15:32:55 2011 -0700
@@ -29,7 +29,7 @@
of Figure \ref{delfig1} and ending at the topmost interval.
\begin{figure}[t]
$$\mathfig{.9}{deligne/intervals}$$
-\caption{Little bigons, though of as encoding surgeries}\label{delfig1}\end{figure}
+\caption{Little bigons, thought of as encoding surgeries}\label{delfig1}\end{figure}
The surgeries correspond to the $k$ bigon-shaped ``holes".
We remove the bottom interval of each little bigon and replace it with the top interval.
To convert this topological operation to an algebraic one, we need, for each hole, an element of
@@ -89,7 +89,8 @@
cylinders.
More specifically, we impose the following two equivalence relations:
\begin{itemize}
-\item If $g: R_i\to R'_i$ is a homeomorphism, we can replace
+\item If $g: R_i\to R'_i$ is a homeomorphism which restricts to the identity on
+$\bd R_i = \bd R'_i = E_0\cup \bd M_i$, we can replace
\begin{eqnarray*}
(\ldots, R_{i-1}, R_i, R_{i+1}, \ldots) &\to& (\ldots, R_{i-1}, R'_i, R_{i+1}, \ldots) \\
(\ldots, f_{i-1}, f_i, \ldots) &\to& (\ldots, g\circ f_{i-1}, f_i\circ g^{-1}, \ldots),
--- a/text/intro.tex Mon Mar 28 18:07:36 2011 -0700
+++ b/text/intro.tex Fri May 06 15:32:55 2011 -0700
@@ -552,8 +552,9 @@
Thomas Tradler,
Kevin Costello,
Chris Douglas,
+Alexander Kirillov,
and
-Alexander Kirillov
+Michael Shulman
for many interesting and useful conversations.
Peter Teichner ran a reading course based on an earlier draft of this paper, and the detailed feedback
we got from the student lecturers lead to very many improvements in later drafts.
--- a/text/ncat.tex Mon Mar 28 18:07:36 2011 -0700
+++ b/text/ncat.tex Fri May 06 15:32:55 2011 -0700
@@ -37,7 +37,7 @@
Strictly speaking, before we can state the axioms for $k$-morphisms we need all the axioms
for $k{-}1$-morphisms.
-So readers who prefer things to be presented in a strictly logical order should read this subsection $n$ times, first imagining that $k=0$, then that $k=1$, and so on until they reach $k=n$.
+Readers who prefer things to be presented in a strictly logical order should read this subsection $n+1$ times, first setting $k=0$, then $k=1$, and so on until they reach $k=n$.
\medskip
@@ -476,6 +476,17 @@
}
\end{scope}
\end{tikzpicture}
+\qquad
+\begin{tikzpicture}[baseline=0]
+\begin{scope}
+\path[clip] (0,0) arc (135:45:4) arc (-45:-135:4);
+\draw[blue,line width=2pt] (0,0) arc (135:45:4) arc (-45:-135:4);
+\draw[blue] (2.82,-5) -- (2.83,5);
+\foreach \x in {0, 0.5, ..., 6} {
+ \draw[green!50!brown] (\x,-2) -- (\x,2);
+}
+\end{scope}
+\end{tikzpicture}
$$
\caption{Five examples of unions of pinched products}\label{pinched_prod_unions}
\end{figure}
@@ -512,7 +523,13 @@
Product morphisms are compatible with gluing (composition).
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$
be pinched products with $E = E_1\cup E_2$.
+(See Figure \ref{pinched_prod_unions}.)
+Note that $X_1$ and $X_2$ can be identified with subsets of $X$,
+but $X_1 \cap X_2$ might not be codimension 1, and indeed we might have $X_1 = X_2 = X$.
+We assume that there is a decomposition of $X$ into balls which is compatible with
+$X_1$ and $X_2$.
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\sub X$.
+(We assume that $a$ is splittable with respect to the above decomposition of $X$ into balls.)
Then
\[
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) .
@@ -674,6 +691,12 @@
\medskip
+We define a $j$ times monoidal $n$-category to be an $(n{+}j)$-category $\cC$ where
+$\cC(X)$ is a trivial 1-element set if $X$ is a $k$-ball with $k<j$.
+See Example \ref{ex:bord-cat}.
+
+\medskip
+
The alert reader will have already noticed that our definition of a (ordinary) $n$-category
is extremely similar to our definition of a system of fields.
There are two differences.
@@ -806,20 +829,6 @@
(See \S\ref{sec:constructing-a-tqft}.)
\end{example}
-\noop{
-\nn{shouldn't this go elsewhere? we haven't yet discussed constructing a system of fields from
-an n-cat}
-Recall we described a system of fields and local relations based on a ``traditional $n$-category"
-$C$ in Example \ref{ex:traditional-n-categories(fields)} above.
-\nn{KW: We already refer to \S \ref{sec:fields} above}
-Constructing a system of fields from $\cC$ recovers that example.
-\todo{Except that it doesn't: pasting diagrams v.s. string diagrams.}
-\nn{KW: but the above example is all about string diagrams. the only difference is at the top level,
-where the quotient is built in.
-but (string diagrams)/(relations) is isomorphic to
-(pasting diagrams composed of smaller string diagrams)/(relations)}
-}
-
\begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version]
\label{ex:bord-cat}
@@ -834,6 +843,9 @@
The case $n=d$ captures the $n$-categorical nature of bordisms.
The case $n > 2d$ captures the full symmetric monoidal $n$-category structure.
\end{example}
+\begin{remark}
+Working with the smooth bordism category would require careful attention to either collars, corners or halos.
+\end{remark}
%\nn{the next example might be an unnecessary distraction. consider deleting it.}
@@ -987,8 +999,11 @@
and we will define $\cl{\cC}(W)$ as a suitable colimit
(or homotopy colimit in the $A_\infty$ case) of this functor.
We'll later give a more explicit description of this colimit.
-In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain complexes to $n$-balls with boundary data),
-then the resulting colimit is also enriched, that is, the set associated to $W$ splits into subsets according to boundary data, and each of these subsets has the appropriate structure (e.g. a vector space or chain complex).
+In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain
+complexes to $n$-balls with boundary data),
+then the resulting colimit is also enriched, that is, the set associated to $W$ splits into
+subsets according to boundary data, and each of these subsets has the appropriate structure
+(e.g. a vector space or chain complex).
Recall (Definition \ref{defn:gluing-decomposition}) that a {\it ball decomposition} of $W$ is a
sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls
@@ -1005,7 +1020,8 @@
Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement
of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$
-with $\du_b Y_b = M_i$ for some $i$.
+with $\du_b Y_b = M_i$ for some $i$,
+and with $M_0,\ldots, M_i$ each being a disjoint union of balls.
\begin{defn}
The poset $\cell(W)$ has objects the permissible decompositions of $W$,
@@ -1036,7 +1052,7 @@
\psi_{\cC;W}(x) \sub \prod_a \cC(X_a)\spl
\end{equation}
where the restrictions to the various pieces of shared boundaries amongst the cells
-$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells).
+$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n{-}1$-cells).
If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
\end{defn}
@@ -1270,8 +1286,7 @@
Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$.
If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces),
-then $\cM(M; c)$ should be an object in that category for each marked $n$-ball $M$
-and $c\in \cC(\bd M)$.
+then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category.
\begin{lem}[Boundary from domain and range]
{Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$),
@@ -1300,7 +1315,7 @@
(for both modules and $n$-categories)
we have for each marked $k$-ball $(B, N)$ and each $k{-}1$-ball $Y\sub \bd B \setmin N$
a natural map from a subset of $\cM(B, N)$ to $\cC(Y)$.
-The subset is the subset of morphisms which are appropriately splittable (transverse to the
+This subset $\cM(B,N)\trans{\bdy Y}$ is the subset of morphisms which are appropriately splittable (transverse to the
cutting submanifolds).
This fact will be used below.
@@ -1326,11 +1341,11 @@
and $Y = M_1\cap M_2$ is a marked $k{-}1$-ball.
Let $E = \bd Y$, which is a marked $k{-}2$-hemisphere.
Note that each of $M$, $M_1$ and $M_2$ has its boundary split into two marked $k{-}1$-balls by $E$.
-We have restriction (domain or range) maps $\cM(M_i)_E \to \cM(Y)$.
-Let $\cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E$ denote the fibered product of these two maps.
+We have restriction (domain or range) maps $\cM(M_i)\trans E \to \cM(Y)$.
+Let $\cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E$ denote the fibered product of these two maps.
Then (axiom) we have a map
\[
- \gl_Y : \cM(M_1)_E \times_{\cM(Y)} \cM(M_2)_E \to \cM(M)_E
+ \gl_Y : \cM(M_1) \trans E \times_{\cM(Y)} \cM(M_2) \trans E \to \cM(M) \trans E
\]
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
to the intersection of the boundaries of $M$ and $M_i$.
@@ -1350,11 +1365,11 @@
$X$ is a plain $k$-ball,
and $Y = X\cap M'$ is a $k{-}1$-ball.
Let $E = \bd Y$, which is a $k{-}2$-sphere.
-We have restriction maps $\cM(M')_E \to \cC(Y)$ and $\cC(X)_E\to \cC(Y)$.
-Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E$ denote the fibered product of these two maps.
+We have restriction maps $\cM(M') \trans E \to \cC(Y)$ and $\cC(X) \trans E\to \cC(Y)$.
+Let $\cC(X)\trans E \times_{\cC(Y)} \cM(M') \trans E$ denote the fibered product of these two maps.
Then (axiom) we have a map
\[
- \gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')_E \to \cM(M)_E
+ \gl_Y :\cC(X)\trans E \times_{\cC(Y)} \cM(M')\trans E \to \cM(M) \trans E
\]
which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions
to the intersection of the boundaries of $X$ and $M'$.