# HG changeset patch # User Scott Morrison # Date 1275374557 25200 # Node ID ef8fac44a8aa792d01ee5368e3b017e9d6c3dd34 # Parent 5bb1cbe49c40a96190120ed3f76debbf8ae6047d minor diff -r 5bb1cbe49c40 -r ef8fac44a8aa text/blobdef.tex --- a/text/blobdef.tex Mon May 31 17:27:17 2010 -0700 +++ b/text/blobdef.tex Mon May 31 23:42:37 2010 -0700 @@ -39,7 +39,7 @@ \end{itemize} (See Figure \ref{blob1diagram}.) \begin{figure}[t]\begin{equation*} -\mathfig{.9}{definition/single-blob} +\mathfig{.6}{definition/single-blob} \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure} In order to get the linear structure correct, we (officially) define \[ @@ -75,7 +75,7 @@ \end{itemize} (See Figure \ref{blob2ddiagram}.) \begin{figure}[t]\begin{equation*} -\mathfig{.9}{definition/disjoint-blobs} +\mathfig{.6}{definition/disjoint-blobs} \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure} We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$; reversing the order of the blobs changes the sign. @@ -95,7 +95,7 @@ \end{itemize} (See Figure \ref{blob2ndiagram}.) \begin{figure}[t]\begin{equation*} -\mathfig{.9}{definition/nested-blobs} +\mathfig{.6}{definition/nested-blobs} \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure} Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$ (for some $c_1 \in \cC(B_1)$) and @@ -153,7 +153,7 @@ \end{itemize} (See Figure \ref{blobkdiagram}.) \begin{figure}[t]\begin{equation*} -\mathfig{.9}{definition/k-blobs} +\mathfig{.7}{definition/k-blobs} \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} If two blob diagrams $D_1$ and $D_2$ diff -r 5bb1cbe49c40 -r ef8fac44a8aa text/intro.tex --- a/text/intro.tex Mon May 31 17:27:17 2010 -0700 +++ b/text/intro.tex Mon May 31 23:42:37 2010 -0700 @@ -26,15 +26,18 @@ Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) 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. 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 group. +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 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. -In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. 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 Property \ref{property:gluing} below. +In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and 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 $A_\infty$ $n$-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 Property \ref{property:gluing} below. + +The relationship between all these ideas is sketched in Figure \ref{fig:outline}. \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] +\begin{figure}[!ht] {\center \begin{tikzpicture}[align=center,line width = 1.5pt] @@ -69,6 +72,9 @@ \end{tikzpicture} } +\caption{The main gadgets and constructions of the paper.} +\label{fig:outline} +\end{figure} Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, 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 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. @@ -167,7 +173,7 @@ \end{property} As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below. -The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. \todo{exact w.r.t $\cC$?} +The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here. \begin{property}[Disjoint union] \label{property:disjoint-union} @@ -220,19 +226,20 @@ \end{property} In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. -\begin{property}[$C_*(\Homeo(-))$ action] +\begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\ +\vspace{-0.5cm} \label{property:evaluation}% -There is a chain map +\begin{enumerate} +\item There is a chain map \begin{equation*} \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). \end{equation*} -Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. -\nn{should probably say something about associativity here (or not?)} +\item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. -For +\item For any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram -(using the gluing maps described in Property \ref{property:gluing-map}) commutes. +(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). \begin{equation*} \xymatrix@C+2cm{ \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\ @@ -241,15 +248,23 @@ \bc_*(X) \ar[u]_{\gl_Y} } \end{equation*} - -\nn{unique up to homotopy?} +\item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. (That is, any pair of homotopies have a homotopy between them, and so on.) +\item This map is associative, in the sense that the following diagram commutes (up to homotopy). +\begin{equation*} +\xymatrix{ +\CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ +\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) +} +\end{equation*} +\end{enumerate} \end{property} -Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map +Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ +for any homeomorphic pair $X$ and $Y$, satisfying corresponding conditions. -In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. +In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] \label{property:blobs-ainfty} diff -r 5bb1cbe49c40 -r ef8fac44a8aa text/ncat.tex --- a/text/ncat.tex Mon May 31 17:27:17 2010 -0700 +++ b/text/ncat.tex Mon May 31 23:42:37 2010 -0700 @@ -95,7 +95,7 @@ Morphisms are modeled on balls, so their boundaries are modeled on spheres. In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for $1\le k \le n$. -At first might seem that we need another axiom for this, but in fact once we have +At first it might seem that we need another axiom for this, but in fact once we have all the axioms in the subsection for $0$ through $k-1$ we can use a coend construction, as described in Subsection \ref{ss:ncat-coend} below, to extend $\cC_{k-1}$ to spheres (and any other manifolds): @@ -107,6 +107,7 @@ homeomorphisms to the category of sets and bijections. \end{prop} +We postpone the proof of this result until after we've actually given all the axioms. Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, along with the data described in other Axioms at lower levels. %In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point. @@ -515,6 +516,8 @@ (Note that homotopy invariance implies isotopy invariance.) For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. + +Recall we described a system of fields and local relations based on maps to $T$ in Example \ref{ex:maps-to-a-space(fields)} above. Constructing a system of fields from $\pi_{\leq n}(T)$ recovers that example. \end{example} \begin{example}[Maps to a space, with a fiber] @@ -556,6 +559,9 @@ Define $\cC(X; c)$, for $X$ an $n$-ball, 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' $C$ in Example \ref{ex:traditional-n-categories(fields)} above. Constructing a system of fields from $\cC$ recovers that example. \end{example} Finally, we describe a version of the bordism $n$-category suitable to our definitions. diff -r 5bb1cbe49c40 -r ef8fac44a8aa text/tqftreview.tex --- a/text/tqftreview.tex Mon May 31 17:27:17 2010 -0700 +++ b/text/tqftreview.tex Mon May 31 23:42:37 2010 -0700 @@ -30,14 +30,20 @@ Before finishing the definition of fields, we give two motivating examples (actually, families of examples) of systems of fields. -The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps +\begin{example} +\label{ex:maps-to-a-space(fields)} +Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps from X to $B$. +\end{example} -The second examples: Fix an $n$-category $C$, and let $\cC(X)$ be +\begin{example} +\label{ex:traditional-n-categories(fields)} +Fix an $n$-category $C$, and let $\cC(X)$ be the set of sub-cell-complexes of $X$ with codimension-$j$ cells labeled by $j$-morphisms of $C$. One can think of such sub-cell-complexes as dual to pasting diagrams for $C$. This is described in more detail below. +\end{example} Now for the rest of the definition of system of fields. \begin{enumerate} @@ -262,8 +268,23 @@ \subsection{Local relations} \label{sec:local-relations} +Local relations are certain subspaces of the fields on balls, which form an ideal under gluing. Again, we give the examples first. +\addtocounter{prop}{-2} +\begin{example}[contd.] +For maps into spaces, $U(B; c)$ is generated by fields of the form $a-b \in \lf(B; c)$, +where $a$ and $b$ are maps (fields) which are homotopic rel boundary. +\end{example} +\begin{example}[contd.] +For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map +$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into +domain and range. +\end{example} + +These motivate the following definition. + +\begin{defn} A {\it local relation} is a collection subspaces $U(B; c) \sub \lf(B; c)$, for all $n$-manifolds $B$ which are homeomorphic to the standard $n$-ball and all $c \in \cC(\bd B)$, @@ -277,17 +298,9 @@ \item ideal with respect to gluing: if $B = B' \cup B''$, $x\in U(B')$, and $c\in \cC(B'')$, then $x\bullet r \in U(B)$ \end{enumerate} -See \cite{kw:tqft} for details. - - -For maps into spaces, $U(B; c)$ is generated by things of the form $a-b \in \lf(B; c)$, -where $a$ and $b$ are maps (fields) which are homotopic rel boundary. +\end{defn} +See \cite{kw:tqft} for further details. -For $n$-category pictures, $U(B; c)$ is equal to the kernel of the evaluation map -$\lf(B; c) \to \mor(c', c'')$, where $(c', c'')$ is some (any) division of $c$ into -domain and range. - -\nn{maybe examples of local relations before general def?} \subsection{Constructing a TQFT} \label{sec:constructing-a-tqft}