# HG changeset patch # User Kevin Walker # Date 1278514070 21600 # Node ID a571e37cc68d4650435068487eb2391262fb95e5 # Parent a96f3d2ef852821fa1dee7c212e5c01a0af393ec a few more ncat revisions diff -r a96f3d2ef852 -r a571e37cc68d text/blobdef.tex --- a/text/blobdef.tex Mon Jul 05 07:47:23 2010 -0600 +++ b/text/blobdef.tex Wed Jul 07 08:47:50 2010 -0600 @@ -137,7 +137,8 @@ behavior} \nn{need to allow the case where $B\to X$ is not an embedding on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ -and blobs are allowed to meet $\bd X$.} +and blobs are allowed to meet $\bd X$. +Also, the complement of the blobs (and regions between nested blobs) might not be manifolds.} Now for the general case. A $k$-blob diagram consists of diff -r a96f3d2ef852 -r a571e37cc68d text/ncat.tex --- a/text/ncat.tex Mon Jul 05 07:47:23 2010 -0600 +++ b/text/ncat.tex Wed Jul 07 08:47:50 2010 -0600 @@ -127,6 +127,7 @@ 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 $n$; @@ -844,8 +845,8 @@ 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. -By shrinking the little balls (precomposing them with dilations), +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), 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 @@ -913,22 +914,23 @@ 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} -Say that a ``permissible decomposition" of $W$ is a cell decomposition +Define a {\it permissible decomposition} of $W$ to be 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} @@ -941,15 +943,12 @@ \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. @@ -963,13 +962,18 @@ 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 -closed $n$-manifold, the functor $\psi_{\cC;W}$ has target $A$ and -we replace the cartesian product of sets appearing in Equation \eqref{eq:psi-C} with the monoidal product $\boxtimes$. -(Moreover, $\psi_{\cC;W}(x)$ might be a subobject, rather than a subset, of the product.) -Similar things are true if $W$ is an $n$-manifold with non-empty boundary and we -fix a field on $\bd W$ -(i.e. fix an element of the colimit associated to $\bd W$). +If $k=n$ in the above definition and we are enriching in some auxiliary category, +we need to say a bit more. +We can rewrite Equation \ref{eq:psi-C} as +\begin{equation} \label{eq:psi-CC} + \psi_{\cC;W}(x) \deq \coprod_\beta \prod_a \cC(X_a; \beta) , +\end{equation} +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 $\psi_{\cC;W}$.