# HG changeset patch # User Kevin Walker # Date 1279421866 21600 # Node ID 901a7c79976b7bedc2c1c6ddd3ffec0fb3d6a76b # Parent c3fb6e8a71366f3d093714771f0510e7743cd7fe# Parent 45807ce1561501766e105abfef97c95570a1f2af Automated merge with https://tqft.net/hg/blob/ diff -r c3fb6e8a7136 -r 901a7c79976b text/a_inf_blob.tex --- a/text/a_inf_blob.tex Fri Jul 16 17:24:20 2010 -0600 +++ b/text/a_inf_blob.tex Sat Jul 17 20:57:46 2010 -0600 @@ -7,12 +7,16 @@ We will show below in Corollary \ref{cor:new-old} -that when $\cC$ is obtained from a topological $n$-category $\cD$ as the blob complex of a point, this agrees (up to homotopy) with our original definition of the blob complex -for $\cD$. +that when $\cC$ is obtained from a system of fields $\cD$ +as the blob complex of an $n$-ball (see Example \ref{ex:blob-complexes-of-balls}), +$\cl{\cC}(M)$ is homotopy equivalent to +our original definition of the blob complex $\bc_*^\cD(M)$. -An important technical tool in the proofs of this section is provided by the idea of `small blobs'. +\medskip + +An important technical tool in the proofs of this section is provided by the idea of ``small blobs". Fix $\cU$, an open cover of $M$. -Define the `small blob complex' $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set. +Define the ``small blob complex" $\bc^{\cU}_*(M)$ to be the subcomplex of $\bc_*(M)$ of all blob diagrams in which every blob is contained in some open set of $\cU$, and moreover each field labeling a region cut out by the blobs is splittable into fields on smaller regions, each of which is contained in some open set of $\cU$. \begin{thm}[Small blobs] \label{thm:small-blobs} The inclusion $i: \bc^{\cU}_*(M) \into \bc_*(M)$ is a homotopy equivalence. @@ -22,42 +26,27 @@ \subsection{A product formula} \label{ss:product-formula} -\noop{ -Let $Y$ be a $k$-manifold, $F$ be an $n{-}k$-manifold, and -\[ - E = Y\times F . -\] -Let $\cC$ be an $n$-category. -Let $\cF$ be the $k$-category of Example \ref{ex:blob-complexes-of-balls}, -\[ - \cF(X) = \cC(X\times F) -\] -for $X$ an $m$-ball with $m\le k$. -} -\nn{need to settle on notation; proof and statement are inconsistent} +Given a system of fields $\cE$ and a $n{-}k$-manifold $F$, recall from +Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\cC_F$ +defined by $\cC_F(X) = \cE(X\times F)$ if $\dim(X) < k$ and +$\cC_F(X) = \bc_*^\cE(X\times F)$ if $\dim(X) = k$. + \begin{thm} \label{thm:product} -Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from -Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by -\begin{equation*} -\bc_*(F; C)(B) = \cB_*(F \times B; C). -\end{equation*} -Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the ``old-fashioned'' -blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled" -(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: -\begin{align*} -\cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y) -\end{align*} -\end{thm} - +Let $Y$ be a $k$-manifold. +Then there is a homotopy equivalence between ``old-fashioned" (blob diagrams) +and ``new-fangled" (hocolimit) blob complexes +\[ + \cB_*(Y \times F) \htpy \cl{\cC_F}(Y) . +\]\end{thm} \begin{proof} -We will use the concrete description of the colimit from \S\ref{ss:ncat_fields}. +We will use the concrete description of the homotopy colimit from \S\ref{ss:ncat_fields}. First we define a map \[ - \psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) . + \psi: \cl{\cC_F}(Y) \to \bc_*(Y\times F;C) . \] In filtration degree 0 we just glue together the various blob diagrams on $X_i\times F$ (where $X_i$ is a component of a permissible decomposition of $Y$) to get a blob diagram on @@ -68,7 +57,7 @@ In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ and a map \[ - \phi: G_* \to \cl{\bc_*(F; C)}(Y) . + \phi: G_* \to \cl{\cC_F}(Y) . \] Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding @@ -81,9 +70,9 @@ projections to $Y$ are contained in some disjoint union of balls.) Note that the image of $\psi$ is equal to $G_*$. -We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models. +We will define $\phi: G_* \to \cl{\cC_F}(Y)$ using the method of acyclic models. Let $a$ be a generator of $G_*$. -Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ generated by all $(b, \ol{K})$ +Let $D(a)$ denote the subcomplex of $\cl{\cC_F}(Y)$ generated by all $(b, \ol{K})$ such that $a$ splits along $K_0\times F$ and $b$ is a generator appearing in an iterated boundary of $a$ (this includes $a$ itself). (Recall that $\ol{K} = (K_0,\ldots,K_l)$ denotes a chain of decompositions; @@ -189,7 +178,7 @@ \end{proof} We are now in a position to apply the method of acyclic models to get a map -$\phi:G_* \to \cl{\bc_*(F; C)}(Y)$. +$\phi:G_* \to \cl{\cC_F}(Y)$. We may assume that $\phi(a)$ has the form $(a, K) + r$, where $(a, K)$ is in filtration degree zero and $r$ has filtration degree greater than zero. diff -r c3fb6e8a7136 -r 901a7c79976b text/ncat.tex --- a/text/ncat.tex Fri Jul 16 17:24:20 2010 -0600 +++ b/text/ncat.tex Sat Jul 17 20:57:46 2010 -0600 @@ -819,7 +819,7 @@ where $\bc^\cE_*$ denotes the blob complex based on $\cE$. \end{example} -This example will be essential for Theorem \ref{thm:product} below, which allows us to compute the blob complex of a product. +This example will be used in 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"