# HG changeset patch # User Kevin Walker # Date 1278307968 21600 # Node ID d3b05641e7caadbfa9c7391a1f7c6a6e2b0dc636 # Parent c06a899bd1f07b6acc0712d3b135430b41f5e2e2 making quotation marks consistently "American style" diff -r c06a899bd1f0 -r d3b05641e7ca blob1.tex --- a/blob1.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/blob1.tex Sun Jul 04 23:32:48 2010 -0600 @@ -16,7 +16,7 @@ \maketitle -[revision $\ge$ 414; $\ge$ 3 July 2010] +[revision $\ge$ 417; $\ge$ 4 July 2010] {\color[rgb]{.9,.5,.2} \large \textbf{Draft version, read with caution.}} We're in the midst of revising this, and hope to have a version on the arXiv soon. @@ -25,24 +25,13 @@ \paragraph{To do list} \begin{itemize} -\item[1] (K) tweak intro -\item[2] (S) needs explanation that this will be superseded by the n-cat -definitions in \S 7. -\item[2] (S) incorporate improvements from later -\item[2.3] (S) foreshadow generalising; quotient to resolution -\item[3] (K) look over blob homology section again -\item[4] (S) basic properties, not much to do -\item[5] (K) finish the lemmas in the Hochschild section \item[6] (K) proofs need finishing, then (S) needs to confirm details and try to make more understandable -\item[7] (S) do some work here -- identity morphisms are still imperfect. Say something about the cobordism and stabilization hypotheses \cite{MR1355899} in this setting? Say something about $E_n$ algebras? +\item[7] (S) do some work here -- identity morphisms are still imperfect. Say something about the cobordism and stabilization hypotheses \cite{MR1355899} in this setting? \item[7.6] is new! (S) read \item[8] improve the beginning, finish proof for products, check the argument about maps \item[9] (K) proofs trail off -\item[10] (S) read what's already here -\item[A] may need to weaken statement to get boundaries working (K) finish -\item[B] (S) look at this, decide what to keep \item Work in the references Chris Douglas gave us on the classification of local field theories, \cite{BDH-seminar,DSP-seminar,schommer-pries-thesis,0905.0465}. \nn{KW: Do we need to do this? We don't really classify field theories. @@ -64,9 +53,10 @@ } % end \noop + + \tableofcontents - \input{text/intro} \input{text/tqftreview} diff -r c06a899bd1f0 -r d3b05641e7ca text/a_inf_blob.tex --- a/text/a_inf_blob.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/a_inf_blob.tex Sun Jul 04 23:32:48 2010 -0600 @@ -44,7 +44,7 @@ \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' +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) diff -r c06a899bd1f0 -r d3b05641e7ca text/appendixes/comparing_defs.tex --- a/text/appendixes/comparing_defs.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/appendixes/comparing_defs.tex Sun Jul 04 23:32:48 2010 -0600 @@ -294,4 +294,4 @@ as required (c.f. \cite[p. 6]{MR1854636}). \todo{then the general case.} We won't describe a reverse construction (producing a topological $A_\infty$ category -from a `conventional' $A_\infty$ category), but we presume that this will be easy for the experts. \ No newline at end of file +from a ``conventional" $A_\infty$ category), but we presume that this will be easy for the experts. \ No newline at end of file diff -r c06a899bd1f0 -r d3b05641e7ca text/appendixes/smallblobs.tex --- a/text/appendixes/smallblobs.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/appendixes/smallblobs.tex Sun Jul 04 23:32:48 2010 -0600 @@ -30,9 +30,9 @@ But as noted above, maybe it's best to ignore this.} Nevertheless, we'll begin introducing nomenclature at this point: for configuration $\beta$ of disjoint embedded balls in $M$ we'll associate a one parameter family of homeomorphisms $\phi_\beta : \Delta^1 \to \Homeo(M)$ (here $\Delta^m$ is the standard simplex $\setc{\mathbf{x} \in \Real^{m+1}}{\sum_{i=0}^m x_i = 1}$). For $0$-blobs, where $\beta = \eset$, all these homeomorphisms are just the identity. -When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ `makes $\beta$ small' if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism `makes $\beta$ $\epsilon$-small' if the image of each ball is contained in some open ball of radius $\epsilon$. +When $\beta$ is a collection of disjoint embedded balls in $M$, we say that a homeomorphism of $M$ ``makes $\beta$ small" if the image of each ball in $\beta$ under the homeomorphism is contained in some open set of $\cU$. Further, we'll say a homeomorphism ``makes $\beta$ $\epsilon$-small" if the image of each ball is contained in some open ball of radius $\epsilon$. -On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term `makes $\beta$ small', while the other term `gets the boundary right'. First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small --- in fact, not just small with respect to $\cU$, but $\epsilon/2$-small, where $\epsilon > 0$ is such that every $\epsilon$-ball is contained in some open set of $\cU$. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ $\frac{3\epsilon}{4}$-small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\beta(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\eset(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by +On a $1$-blob $b$, with ball $\beta$, $s$ is defined as the sum of two terms. Essentially, the first term ``makes $\beta$ small", while the other term ``gets the boundary right". First, pick a one-parameter family $\phi_\beta : \Delta^1 \to \Homeo(M)$ of homeomorphisms, so $\phi_\beta(1,0)$ is the identity and $\phi_\beta(0,1)$ makes the ball $\beta$ small --- in fact, not just small with respect to $\cU$, but $\epsilon/2$-small, where $\epsilon > 0$ is such that every $\epsilon$-ball is contained in some open set of $\cU$. Next, pick a two-parameter family $\phi_{\eset \prec \beta} : \Delta^2 \to \Homeo(M)$ so that $\phi_{\eset \prec \beta}(0,x_1,x_2)$ makes the ball $\beta$ $\frac{3\epsilon}{4}$-small for all $x_1+x_2=1$, while $\phi_{\eset \prec \beta}(x_0,0,x_2) = \phi_\beta(x_0,x_2)$ and $\phi_{\eset \prec \beta}(x_0,x_1,0) = \phi_\eset(x_0,x_1)$. (It's perhaps not obvious that this is even possible --- see Lemma \ref{lem:extend-small-homeomorphisms} below.) We now define $s$ by $$s(b) = \restrict{\phi_\beta}{x_0=0}(b) - \restrict{\phi_{\eset \prec \beta}}{x_0=0}(\bdy b).$$ Here, $\restrict{\phi_\beta}{x_0=0} = \phi_\beta(0,1)$ is just a homeomorphism, which we apply to $b$, while $\restrict{\phi_{\eset \prec \beta}}{x_0=0}$ is a one parameter family of homeomorphisms which acts on the $0$-blob $\bdy b$ to give a $1$-blob. To be precise, this action is via the chain map identified in Lemma \ref{lem:CH-small-blobs} with $\cV_0$ the open cover by $\epsilon/2$-balls and $\cV_1$ the open cover by $\frac{3\epsilon}{4}$-balls. From this, it is immediate that $s(b) \in \bc^{\cU}_1(M)$, as desired. @@ -57,7 +57,7 @@ In order to define $s$ on arbitrary blob diagrams, we first fix a sequence of strictly subordinate covers for $\cU$. First choose an $\epsilon > 0$ so every $\epsilon$ ball is contained in some open set of $\cU$. For $k \geq 1$, let $\cV_{k}$ be the open cover of $M$ by $\epsilon (1-2^{-k})$ balls, and $\cV_0 = \cU$. Certainly $\cV_k$ is strictly subordinate to $\cU$. We now chose the chain map $\ev$ provided by Lemma \ref{lem:CH-small-blobs} for the open covers $\cV_k$ strictly subordinate to $\cU$. Note that $\cV_1$ and $\cV_2$ have already implicitly appeared in the description above. -Next, we choose a `shrinking system' for $\left(\cU,\{\cV_k\}_{k \geq 1}\right)$, namely for each increasing sequence of blob configurations +Next, we choose a ``shrinking system" for $\left(\cU,\{\cV_k\}_{k \geq 1}\right)$, namely for each increasing sequence of blob configurations $\beta_1 \prec \cdots \prec \beta_n$, an $n$ parameter family of diffeomorphisms $\phi_{\beta_1 \prec \cdots \prec \beta_n} : \Delta^{n+1} \to \Diff{M}$, such that \begin{itemize} diff -r c06a899bd1f0 -r d3b05641e7ca text/basic_properties.tex --- a/text/basic_properties.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/basic_properties.tex Sun Jul 04 23:32:48 2010 -0600 @@ -89,7 +89,7 @@ Assign to $b$ the acyclic (in positive degrees) subcomplex $T(b) \deq r\bullet\bc_*(S)$. Note that if a diagram $b'$ is part of $\bd b$ then $T(B') \sub T(b)$. Both $f$ and the identity are compatible with $T$ (in the sense of acyclic models), -so $f$ and the identity map are homotopic. \nn{We should actually have a section with a definition of `compatible' and this statement as a lemma} +so $f$ and the identity map are homotopic. \nn{We should actually have a section with a definition of ``compatible" and this statement as a lemma} \end{proof} For the next proposition we will temporarily restore $n$-manifold boundary @@ -111,7 +111,7 @@ } The sum is over all fields $a$ on $Y$ compatible at their ($n{-}2$-dimensional) boundaries with $c$. -`Natural' means natural with respect to the actions of diffeomorphisms. +``Natural" means natural with respect to the actions of diffeomorphisms. } This map is very far from being an isomorphism, even on homology. diff -r c06a899bd1f0 -r d3b05641e7ca text/evmap.tex --- a/text/evmap.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/evmap.tex Sun Jul 04 23:32:48 2010 -0600 @@ -46,7 +46,7 @@ and let $S \sub X$. We say that {\it $f$ is supported on $S$} if $f(p, x) = f(q, x)$ for all $x \notin S$ and $p, q \in P$. Equivalently, $f$ is supported on $S$ if -there is a family of homeomorphisms $f' : P \times S \to S$ and a `background' +there is a family of homeomorphisms $f' : P \times S \to S$ and a ``background" homeomorphism $f_0 : X \to X$ so that \begin{align*} f(p,s) & = f_0(f'(p,s)) \;\;\;\; \mbox{for}\; (p, s) \in P\times S \\ diff -r c06a899bd1f0 -r d3b05641e7ca text/hochschild.tex --- a/text/hochschild.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/hochschild.tex Sun Jul 04 23:32:48 2010 -0600 @@ -107,7 +107,7 @@ quasi-isomorphic to its $0$-th homology (which in turn, by \ref{item:hochschild-coinvariants} above, is just $C$) via the quotient map $\HC_0 \onto \HH_0$. \end{enumerate} -(Together, these just say that Hochschild homology is `the derived functor of coinvariants'.) +(Together, these just say that Hochschild homology is ``the derived functor of coinvariants".) We'll first recall why these properties are characteristic. Take some $C$-$C$ bimodule $M$, and choose a free resolution @@ -130,8 +130,8 @@ \cP_*(F_j) & \xrightarrow{\cP_0(F_j) \onto H_0(\cP_*(F_j))} \coinv(F_j). \end{align*} The cone of each chain map is acyclic. -In the first case, this is because the `rows' indexed by $i$ are acyclic since $\cP_i$ is exact. -In the second case, this is because the `columns' indexed by $j$ are acyclic, since $F_j$ is free. +In the first case, this is because the ``rows" indexed by $i$ are acyclic since $\cP_i$ is exact. +In the second case, this is because the ``columns" indexed by $j$ are acyclic, since $F_j$ is free. Because the cones are acyclic, the chain maps are quasi-isomorphisms. Composing one with the inverse of the other, we obtain the desired quasi-isomorphism $$\cP_*(M) \quismto \coinv(F_*).$$ diff -r c06a899bd1f0 -r d3b05641e7ca text/intro.tex --- a/text/intro.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/intro.tex Sun Jul 04 23:32:48 2010 -0600 @@ -38,7 +38,7 @@ and establishes some of its properties. There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. -At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex +At first we entirely avoid this problem by introducing the notion of a ``system of fields", and define the blob complex associated to an $n$-manifold and an $n$-dimensional system of fields. We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. @@ -50,7 +50,7 @@ We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. -The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. +The basic idea is that each potential definition of an $n$-category makes a choice about the ``shape" of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of @@ -61,10 +61,10 @@ In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). -Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an +Using these definitions, we show how to use the blob complex to ``resolve" any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), -in particular the `gluing formula' of Theorem \ref{thm:gluing} below. +in particular the ``gluing formula" of Theorem \ref{thm:gluing} below. The relationship between all these ideas is sketched in Figure \ref{fig:outline}. @@ -115,7 +115,7 @@ thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalization of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. -The appendixes prove technical results about $\CH{M}$ and the `small blob complex', +The appendixes prove technical results about $\CH{M}$ and the ``small blob complex", and make connections between our definitions of $n$-categories and familiar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. @@ -436,7 +436,7 @@ The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be interesting to investigate if there is a connection with the material here. -Many results in Hochschild homology can be understood `topologically' via the blob complex. +Many results in Hochschild homology can be understood ``topologically" via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details. diff -r c06a899bd1f0 -r d3b05641e7ca text/ncat.tex --- a/text/ncat.tex Sun Jul 04 13:15:03 2010 -0600 +++ b/text/ncat.tex Sun Jul 04 23:32:48 2010 -0600 @@ -271,7 +271,7 @@ 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. @@ -667,7 +667,7 @@ \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 $T$. @@ -704,12 +704,12 @@ \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. -Further, most of these definitions don't even have an agreed-upon notion of `strong duality', which we assume here. +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. \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 @@ -725,7 +725,7 @@ 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. @@ -794,15 +794,15 @@ 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' -\nn{`cofibrant replacement'?} +We think of this as providing a ``free resolution" +\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 $\pi^\infty_{\leq 0}(T) \iso C_* T$. @@ -895,12 +895,12 @@ 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. @@ -909,7 +909,7 @@ 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 \[ W = \bigcup_a X_a , \] @@ -938,7 +938,7 @@ 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. @@ -1740,7 +1740,7 @@ 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