blob: comparison text/ncat.tex

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 which is natural with respect to the actions of homeomorphisms.
 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product
 becomes a normal product.)
 \end{lem}
-\begin{figure}[!ht] \centering
+\begin{figure}[t] \centering
 \begin{tikzpicture}[%every label/.style={green}
 ]
 \node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {};
 \node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {};
 \draw (S) arc  (-90:90:1);
 or if $k=n$ and we are in the $A_\infty$ case,
 we require that $\gl_Y$ is injective.
 (For $k=n$ in the ordinary (non-$A_\infty$) case, see below.)
 \end{axiom}
-\begin{figure}[!ht] \centering
+\begin{figure}[t] \centering
 \begin{tikzpicture}[%every label/.style={green},
 				x=1.5cm,y=1.5cm]
 \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {};
 \node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {};
 \draw (S) arc  (-90:90:1);
 Given any splitting of a ball $B$ into smaller balls
 $$\bigsqcup B_i \to B,$$
 any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result.
 \end{axiom}
-\begin{figure}[!ht]
+\begin{figure}[t]
 $$\mathfig{.65}{ncat/strict-associativity}$$
 \caption{An example of strict associativity.}\label{blah6}\end{figure}
 We'll use the notation  $a\bullet b$ for the glued together field $\gl_Y(a, b)$.
 In the other direction, we will call the projection from $\cC(B)\trans E$ to $\cC(B_i)\trans E$
 map from an appropriate subset (like a fibered product)
 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$,
 and these various $m$-fold composition maps satisfy an
 operad-type strict associativity condition (Figure \ref{fig:operad-composition}).}
-\begin{figure}[!ht]
+\begin{figure}[t]
 $$\mathfig{.8}{ncat/operad-composition}$$
 \caption{Operad composition and associativity}\label{fig:operad-composition}\end{figure}
 The next axiom is related to identity morphisms, though that might not be immediately obvious.
 \begin{eqnarray*}
 	\psi_{Y,J}: \cC(X) &\to& \cC(X) \\
 	a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) .
 \end{eqnarray*}
 (See Figure \ref{glue-collar}.)
-\begin{figure}[!ht]
+\begin{figure}[t]
 \begin{equation*}
 \begin{tikzpicture}
 \def\rad{1}
 \def\srad{0.75}
 \def\gap{4.5}
 to be the dual Hilbert space $A(X\times F; c)$.
 (See \S\ref{sec:constructing-a-tqft}.)
 \end{example}
-\begin{example}[The bordism $n$-category of $d$-manifolds, ordinary version]
+\begin{example}[te bordism $n$-category of $d$-manifolds, ordinary version]
 \label{ex:bord-cat}
 \rm
 \label{ex:bordism-category}
 For a $k$-ball $X$, $k<n$, define $\Bord^{n,d}(X)$ to be the set of all $(d{-}n{+}k)$-dimensional PL
 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$.
 is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
 It's easy to see that with $n=0$, the corresponding system of fields is just
 linear combinations of connected components of $T$, and the local relations are trivial.
 There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$.
-\begin{example}[The bordism $n$-category of $d$-manifolds, $A_\infty$ version]
+\begin{example}[te bordism $n$-category of $d$-manifolds, $A_\infty$ version]
 \rm
 \label{ex:bordism-category-ainf}
 As in Example \ref{ex:bord-cat}, for $X$ a $k$-ball, $k<n$, we define $\Bord^{n,d}_\infty(X)$
 to be the set of all $(d{-}n{+}k)$-dimensional
 submanifolds $W$ of $X\times \Real^\infty$ such that $\bd W = W \cap \bd X \times \Real^\infty$.
 The poset $\cell(W)$ has objects the permissible decompositions of $W$,
 and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$.
 See Figure \ref{partofJfig} for an example.
 \end{defn}
-\begin{figure}[!ht]
+\begin{figure}[t]
 \begin{equation*}
 \mathfig{.63}{ncat/zz2}
 \end{equation*}
 \caption{A small part of $\cell(W)$}
 \label{partofJfig}
 (see Example \ref{ex:maps-with-fiber}).
 (The union is along $N\times \bd W$.)
 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be
 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.)
-\begin{figure}[!ht]
+\begin{figure}[t]
 $$\mathfig{.55}{ncat/boundary-collar}$$
 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure}
 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$.
 Call such a thing a {marked $k{-}1$-hemisphere}.
 We require two sorts of composition (gluing) for modules, corresponding to two ways
 of splitting a marked $k$-ball into two (marked or plain) $k$-balls.
 (See Figure \ref{zzz3}.)
-\begin{figure}[!ht]
+\begin{figure}[t]
 \begin{equation*}
 \mathfig{.4}{ncat/zz3}
 \end{equation*}
 \caption{Module composition (top); $n$-category action (bottom).}
 \label{zzz3}
 Note that the above associativity axiom applies to mixtures of module composition,
 action maps and $n$-category composition.
 See Figure \ref{zzz1b}.
-\begin{figure}[!ht]
+\begin{figure}[t]
 \begin{equation*}
 \mathfig{0.49}{ncat/zz0} \mathfig{0.49}{ncat/zz1}
 \end{equation*}
 \caption{Two examples of mixed associativity}
 \label{zzz1b}

changeset 774	b88c4c4af945
parent 773	775b5ca42bed
child 775	9ea10b1adfaa