blob: comparison text/ncat.tex

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 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$.
 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls.
 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from
 the smaller balls to $X$.
-We  say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'.
+We  say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$".
 In situations where the subdivision is notationally anonymous, we will write
 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to)
 the unnamed subdivision.
 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$;
 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous
 We now describe several classes of examples of $n$-categories satisfying our axioms.
 \begin{example}[Maps to a space]
 \rm
 \label{ex:maps-to-a-space}%
-Fix a `target space' $T$, any topological space.
+Fix a ``target space" $T$, any topological space.
 We define $\pi_{\leq n}(T)$, the fundamental $n$-category of $T$, as follows.
 For $X$ a $k$-ball with $k < n$, define $\pi_{\leq n}(T)(X)$ to be the set of
 all continuous maps from $X$ to $T$.
 For $X$ an $n$-ball define $\pi_{\leq n}(T)(X)$ to be continuous maps from $X$ to $T$ modulo
 homotopies fixed on $\bd X$.
 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy
 $h: X\times F\times I \to T$, then $a = \alpha(h)b$.
 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully}
 \end{example}
-The next example is only intended to be illustrative, as we don't specify which definition of a `traditional $n$-category' we intend.
+The next example is only intended to be illustrative, as we don't specify which definition of a ``traditional $n$-category" we intend.
-Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here.
+Further, most of these definitions don't even have an agreed-upon notion of ``strong duality", which we assume here.
 \begin{example}[Traditional $n$-categories]
 \rm
 \label{ex:traditional-n-categories}
-Given a `traditional $n$-category with strong duality' $C$
+Given a ``traditional $n$-category with strong duality" $C$
 define $\cC(X)$, for $X$ a $k$-ball with $k < n$,
 to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}).
 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear
 combinations of $C$-labeled embedded cell complexes of $X$
 modulo the kernel of the evaluation map.
 to be the set of all $C$-labeled embedded cell complexes of $X\times F$.
 Define $\cC(X; c)$, for $X$ an $n$-ball,
 to be the dual Hilbert space $A(X\times F; c)$.
 \nn{refer elsewhere for details?}
-Recall we described a system of fields and local relations based on a `traditional $n$-category'
+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,
 \end{example}
 This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product.
 Notice that with $F$ a point, the above example is a construction turning a topological
 $n$-category $\cC$ into an $A_\infty$ $n$-category which we'll denote by $\bc_*(\cC)$.
-We think of this as providing a `free resolution'
+We think of this as providing a ``free resolution"
-\nn{`cofibrant replacement'?}
+\nn{``cofibrant replacement"?}
 of the topological $n$-category.
 \todo{Say more here!}
 In fact, there is also a trivial, but mostly uninteresting, way to do this:
 we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$,
 and take $\CD{B}$ to act trivially.
-Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$.
+Be careful that the ``free resolution" of the topological $n$-category $\pi_{\leq n}(T)$ 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, $A_\infty$ version]
 so that we can state the boundary axiom for $\cC$ on $k+1$-balls.
 In the case of plain $n$-categories, this construction factors into a construction of a
 system of fields and local relations, followed by the usual TQFT definition of a
 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}.
 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead.
-Recall that we can take a plain $n$-category $\cC$ and pass to the `free resolution',
+Recall that we can take a plain $n$-category $\cC$ and pass to the ``free resolution",
 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls (recall Example \ref{ex:blob-complexes-of-balls} above).
 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant
 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the same as the original blob complex  for $M$ with coefficients in $\cC$.
-We will first define the `cell-decomposition' poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
+We will first define the ``cell-decomposition" poset $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$.
 An $n$-category $\cC$ provides a functor from this poset to the category of sets,
 and we  will define $\cC(W)$ 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 $n$-manifolds 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).
 \begin{defn}
-Say that a `permissible decomposition' of $W$ is a cell decomposition
+Say that a ``permissible decomposition" of $W$ is a cell decomposition
 \[
 	W = \bigcup_a X_a ,
 \]
 where each closed top-dimensional cell $X_a$ is an embedded $k$-ball.
 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
 a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets
 (possibly with additional structure if $k=n$).
 Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls,
 and, as described above, we have a subset $\cC(X)\spl \sub \cC(X)$ of morphisms whose boundaries
 are splittable along this decomposition.
-%For a $k$-cell $X$ in a cell composition of $W$, we can consider the `splittable fields' $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
+%For a $k$-cell $X$ in a cell composition of $W$, we can consider the ``splittable fields" $\cC(X)_{\bdy X}$, the subset of $\cC(X)$ consisting of fields which are splittable with respect to each boundary $k-1$-cell.
 \begin{defn}
 Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows.
 For a decomposition $x = \bigcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset
 \begin{equation}
 If $k=n$ these sets should be enriched to the extent $\cA$ and $\cB$ are.
 Given a decomposition of a $0$-marked $k$-ball $X$ into smaller balls $X_i$, we have
 morphism sets $\cA_k(X_i)$ (if $X_i$ lies on the $\cA$-labeled side)
 or $\cB_k(X_i)$ (if $X_i$ lies on the $\cB$-labeled side)
 or $\cM_k(X_i)$ (if $X_i$ intersects the marking and is therefore a smaller 0-marked ball).
-Corresponding to this decomposition we have a composition (or `gluing') map
+Corresponding to this decomposition we have a composition (or ``gluing") map
 from the product (fibered over the boundary data) of these various sets into $\cM_k(X)$.
 \medskip
 Part of the structure of an $n$-category 0-sphere module $\cM$  is captured by saying it is

changeset 417	d3b05641e7ca
parent 416	c06a899bd1f0
child 418	a96f3d2ef852