blob: comparison text/ncat.tex

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 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always
 omitted.
 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by
 gluing subintervals together and/or omitting some of the rightmost subintervals.
 (See Figure \ref{fig:lmar}.)
-\begin{figure}[t]$$
+\begin{figure}[t] \centering
 \definecolor{arcolor}{rgb}{.75,.4,.1}
 \begin{tikzpicture}[line width=1pt]
 \fill (0,0) circle (.1);
 \draw (0,0) -- (2,0);
 \draw (1,0.1) -- (1,-0.1);
 \foreach \x in {1.0, 1.5} {
 	\draw (\x,1.1) -- (\x,0.9);
 }
 \end{tikzpicture}
-$$
 \caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure}
 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$.
 The underlying vector space is
 \[
 \]
 for each left-marked interval $K$.
 These are required to commute with gluing;
 for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes:
 \[ \xymatrix{
-	\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id}
+	\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_1}\ot \id}
 							\ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q)
 								\ar[d]^{\gl} \\
 	\cX(K) \ar[r]^{h_{K}} & \cY(K)
 } \]
 Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows.
 The product is pinched over the boundary of $J$.
 The set $\cD$ breaks into ``blocks" according to the restrictions to the pinched points of $X\times J$
 (see Figure \ref{feb21b}).
 These restrictions are 0-morphisms $(a, b)$ of $\cA$ and $\cB$.
-\begin{figure}[t]
+\begin{figure}[t] \centering
-$$
 \begin{tikzpicture}[blue,line width=2pt]
 \draw (0,1) -- (0,-1) node[below] {$X$};
 \draw (2,0) -- (4,0) node[below] {$J$};
 \fill[red] (3,0) circle (0.1);
 \fill (b) circle (0.1) node[right] {\color{green!50!brown} $b$};
 \path (bottom) node[below]{$X \times J$};
 \end{tikzpicture}
-$$
 \caption{The pinched product $X\times J$}
 \label{feb21b}
 \end{figure}
 More generally, consider an interval with interior marked points, and with the complements
 (See Figure \ref{feb21c}.)
 To this data we can apply the coend construction as in \S\ref{moddecss} above
 to obtain an $\cA_0$-$\cA_l$ $0$-sphere module and, forgetfully, an $n{-}1$-category.
 This amounts to a definition of taking tensor products of $0$-sphere modules over $n$-categories.
-\begin{figure}[t]
+\begin{figure}[t] \centering
-$$
 \begin{tikzpicture}[baseline,line width = 2pt]
 \draw[blue] (0,0) -- (6,0);
 \foreach \x/\n in {0.5/0,1.5/1,3/2,4.5/3,5.5/4} {
 	\path (\x,0)  node[below] {\color{green!50!brown}$\cA_{\n}$};
 }
 \foreach \q/\n in {60/0,120/1,-120/2} {
 	\fill[red] (\q:2) circle (0.1);
 	\path (\q:2.4) node {\color{green!50!brown}$\cM_{\n}$};
 }
 \end{tikzpicture}
-$$
 \caption{Marked and labeled 1-manifolds}
 \label{feb21c}
 \end{figure}
 We could also similarly mark and label a circle, obtaining an $n{-}1$-category
 A 1-marked $k$-ball can be decomposed in various ways into smaller balls, which are either
 (a) smaller 1-marked $k$-balls, (b) 0-marked $k$-balls, or (c) plain $k$-balls.
 (See Figure \nn{need figure}.)
 We now proceed as in the above module definitions.
-\begin{figure}[!ht]
+\begin{figure}[t] \centering
-$$
 \begin{tikzpicture}[baseline,line width = 2pt]
 \draw[blue][fill=blue!15!white] (0,0) circle (2);
 \fill[red] (0,0) circle (0.1);
 \foreach \qm/\qa/\n in {70/-30/0, 120/95/1, -120/180/2} {
 	\draw[red] (0,0) -- (\qm:2);
 	\path (\qa:1) node {\color{green!50!brown} $\cA_\n$};
 	\path (\qm+20:2.5) node(M\n) {\color{green!50!brown} $\cM_\n$};
 	\draw[line width=1pt, green!50!brown, ->] (M\n.\qm+135) to[out=\qm+135,in=\qm+90] (\qm+5:1.3);
 }
 \end{tikzpicture}
-$$
 \caption{Cone on a marked circle}
 \label{feb21d}
 \end{figure}
 A $n$-category 1-sphere module is, among other things, an $n{-}2$-category $\cD$ with

changeset 530	b236746e8e4d
parent 529	8e055b7c0768
child 531	da9bf150bf3d