diff -r a02a6158f3bd -r a8b8ebcf07ac text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sat Jun 26 16:31:28 2010 -0700 +++ b/text/a_inf_blob.tex Sat Jun 26 17:22:53 2010 -0700 @@ -2,18 +2,13 @@ \section{The blob complex for $A_\infty$ $n$-categories} \label{sec:ainfblob} +Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we make the anticlimactically tautological definition of the blob +complex $\bc_*(M;\cC)$ to be the homotopy colimit $\cl{\cC}(M)$ of Section \ref{ss:ncat_fields}. -Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob -complex $\bc_*(M)$ to the be the homotopy colimit $\cC(M)$ of Section \ref{sec:ncats}. -\nn{say something about this being anticlimatically tautological?} We will show below in Corollary \ref{cor:new-old} -that this agrees (up to homotopy) with our original definition of the blob complex -in the case of plain $n$-categories. -When we need to distinguish between the new and old definitions, we will refer to the -new-fangled and old-fashioned blob complex. - -\medskip +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 $\cD$. An important technical tool in the proofs of this section is provided by the idea of `small blobs'. Fix $\cU$, an open cover of $M$. @@ -44,15 +39,15 @@ \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 $C^{\times F}$ defined by +Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by \begin{equation*} -C^{\times F}(B) = \cB_*(B \times F, C). +\bc_*(F; C) = \cB_*(B \times F, 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 $C^{\times F}$: +(i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: \begin{align*} -\cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) +\cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y) \end{align*} \end{thm} @@ -62,7 +57,7 @@ First we define a map \[ - \psi: \bc_*^\cF(Y) \to \bc_*^C(Y\times F) . + \psi: \cl{\bc_*(F; C)}(Y) \to \bc_*(Y\times F;C) . \] 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 @@ -70,25 +65,25 @@ In filtration degrees 1 and higher we define the map to be zero. It is easy to check that this is a chain map. -In the other direction, we will define a subcomplex $G_*\sub \bc_*^C(Y\times F)$ +In the other direction, we will define a subcomplex $G_*\sub \bc_*(Y\times F;C)$ and a map \[ - \phi: G_* \to \bc_*^\cF(Y) . + \phi: G_* \to \cl{\bc_*(F; C)}(Y) . \] Given a decomposition $K$ of $Y$ into $k$-balls $X_i$, let $K\times F$ denote the corresponding decomposition of $Y\times F$ into the pieces $X_i\times F$. -Let $G_*\sub \bc_*^C(Y\times F)$ be the subcomplex generated by blob diagrams $a$ such that there +Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. -It follows from Proposition \ref{thm:small-blobs} that $\bc_*^C(Y\times F)$ is homotopic to a subcomplex of $G_*$. +It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. (If the blobs of $a$ are small with respect to a sufficiently fine cover then their projections to $Y$ are contained in some disjoint union of balls.) Note that the image of $\psi$ is equal to $G_*$. -We will define $\phi: G_* \to \bc_*^\cF(Y)$ using the method of acyclic models. +We will define $\phi: G_* \to \cl{\bc_*(F; C)}(Y)$ using the method of acyclic models. Let $a$ be a generator of $G_*$. -Let $D(a)$ denote the subcomplex of $\bc_*^\cF(Y)$ generated by all $(b, \ol{K})$ +Let $D(a)$ denote the subcomplex of $\cl{\bc_*(F; C)}(Y)$ 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; @@ -194,13 +189,13 @@ \end{proof} We are now in a position to apply the method of acyclic models to get a map -$\phi:G_* \to \bc_*^\cF(Y)$. +$\phi:G_* \to \cl{\bc_*(F; C)}(Y)$. 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. We now show that $\phi\circ\psi$ and $\psi\circ\phi$ are homotopic to the identity. -$\psi\circ\phi$ is the identity on the nose: +First, $\psi\circ\phi$ is the identity on the nose: \[ \psi(\phi(a)) = \psi((a,K)) + \psi(r) = a + 0. \] @@ -208,10 +203,10 @@ $\psi$ glues those pieces back together, yielding $a$. We have $\psi(r) = 0$ since $\psi$ is zero in positive filtration degrees. -$\phi\circ\psi$ is the identity up to homotopy by another MoAM argument. +Second, $\phi\circ\psi$ is the identity up to homotopy by another argument based on the method of acyclic models. To each generator $(b, \ol{K})$ of $G_*$ we associate the acyclic subcomplex $D(b)$ defined above. Both the identity map and $\phi\circ\psi$ are compatible with this -collection of acyclic subcomplexes, so by the usual MoAM argument these two maps +collection of acyclic subcomplexes, so by the usual method of acyclic models argument these two maps are homotopic. This concludes the proof of Theorem \ref{thm:product}. @@ -221,10 +216,10 @@ \medskip -\todo{rephrase this} \begin{cor} \label{cor:new-old} -The new-fangled and old-fashioned blob complexes are homotopic. +The blob complex of a manifold $M$ with coefficients in a topological $n$-category $\cC$ is homotopic to the homotopy colimit invariant of $M$ defined using the $A_\infty$ $n$-category obtained by applying the blob complex to a point: +$$\bc_*(M; \cC) \htpy \cl{\bc_*(pt; \cC)}(M).$$ \end{cor} \begin{proof} Apply Theorem \ref{thm:product} with the fiber $F$ equal to a point.