blob: comparison text/ncat.tex

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 Most of the examples of $n$-categories we are interested in are enriched in the following sense.
 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and
 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category
 (e.g.\ vector spaces, or modules over some ring, or chain complexes),
+\nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?}
 and all the structure maps of the $n$-category should be compatible with the auxiliary
 category structure.
 Note that this auxiliary structure is only in dimension $n$;
 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$.
 (We require that the interiors of the little balls be disjoint, but their
 boundaries are allowed to meet.
 Note in particular that the space for $k=1$ contains a copy of $\Diff(B^n)$, namely
 the embeddings of a ``little" ball with image all of the big ball $B^n$.
 \nn{should we warn that the inclusion of this copy of $\Diff(B^n)$ is not a homotopy equivalence?})
-The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad.
+The operad $\cE\cB_n$ is homotopy equivalent to the standard framed little $n$-ball operad:
-By shrinking the little balls (precomposing them with dilations),
+by shrinking the little balls (precomposing them with dilations),
 we see that both operads are homotopic to the space of $k$ framed points
 in $B^n$.
 It is easy to see that $n$-fold loop spaces $\Omega^n(T)$  have
 an action of $\cE\cB_n$.
 \nn{add citation for this operad if we can find one}
 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 $1 \leq k \leq n$.
 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
+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 $n$-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}
+Define a {\it permissible decomposition} of $W$ to be 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.
+\nn{need to define this more carefully}
 Given permissible decompositions $x$ and $y$, we say that $x$ is a refinement
 of $y$, or write $x \le y$, if each $k$-ball of $y$ is a union of $k$-balls of $x$.
-The category $\cell(W)$ has objects the permissible decompositions of $W$,
+\begin{defn}
+The category (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]
 \end{equation*}
 \caption{A small part of $\cell(W)$}
 \label{partofJfig}
 \end{figure}
 An $n$-category $\cC$ determines
 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.
 \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}
 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).
 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}
-When the $n$-category $\cC$ is enriched in some symmetric monoidal category $(A,\boxtimes)$, and $W$ is a
+If $k=n$ in the above definition and we are enriching in some auxiliary category,
-closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and
+we need to say a bit more.
-we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$.
+We can rewrite Equation \ref{eq:psi-C} as
-(Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.)
+\begin{equation} \label{eq:psi-CC}
-Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we
+	\psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) ,
-fix a field on $\bd W$
+\end{equation}
-(i.e. fix an element of the colimit associated to $\bd W$).
+where $\beta$ runs through labelings of the $k{-}1$-skeleton of the decomposition
+(which are compatible when restricted to the $k{-}2$-skeleton), and $\cC(X_a; \beta)$
+means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$.
+If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in
+$\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate
+operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect).
 Finally, we construct $\cC(W)$ as the appropriate colimit of $\psi_{\cC;W}$.
 \begin{defn}[System of fields functor]
 \label{def:colim-fields}

changeset 419	a571e37cc68d
parent 418	a96f3d2ef852
child 421	a896ec294254