blob: comparison text/intro.tex

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 \draw[->] (FU) -- node[right=10pt] {$\cF(M)/\cU$} (A);
 \draw[->] (C) -- node[left=10pt] {
 	Example \ref{ex:traditional-n-categories(fields)} \\ and \S \ref{ss:ncat_fields}
-	%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker \ev: \cC(c) \to \cC(B)$
+	%$\displaystyle \cF(M) = \DirectSum_{c \in\cell(M)} \cC(c)$ \\ $\displaystyle \cU(B) = \DirectSum_{c \in \cell(B)} \ker e: \cC(c) \to \cC(B)$
 } (FU);
 \draw[->] (BC) -- node[left] {$H_0$} node[right] {c.f. Theorem \ref{thm:skein-modules}} (A);
 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs);
 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs);
 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.
 Appendix \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 $M$.
-Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes)
+%%%% this is said later in the intro
-\nn{...}
+%Throughout the paper we typically prefer concrete categories (vector spaces, chain complexes)
+%even when we could work in greater generality (symmetric monoidal categories, model categories, etc.).
-%\item related: we are being unsophisticated from a homotopy theory point of
-%view and using chain complexes in many places where we could get by with spaces
 %\item ? one of the points we make (far) below is that there is not really much
 %difference between (a) systems of fields and local relations and (b) $n$-cats;
 %thus we tend to switch between talking in terms of one or the other
 \subsection{Motivations}
 \label{sec:motivations}
 We will briefly sketch our original motivation for defining the blob complex.
-\nn{this is adapted from an old draft of the intro; it needs further modification
-in order to better integrate it into the current intro.}
 As a starting point, consider TQFTs constructed via fields and local relations.
 (See \S\ref{sec:tqftsviafields} or \cite{kw:tqft}.)
 This gives a satisfactory treatment for semisimple TQFTs
 (i.e.\ TQFTs for which the cylinder 1-category associated to an
 Our main motivating example (though we will not develop it in this paper)
 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology.
 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together
 with a link $L \subset \bd W$.
 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$.
-\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S}
+%\todo{I'm tempted to replace $A_{Kh}$ with $\cl{Kh}$ throughout this page -S}
 How would we go about computing $A_{Kh}(W^4, L)$?
 For the Khovanov homology of a link in $S^3$ the main tool is the exact triangle (long exact sequence)
 relating resolutions of a crossing.
 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt
 According to the gluing theorem for TQFTs, gluing along $B^3 \subset \bd B^4$
 corresponds to taking a coend (self tensor product) over the cylinder category
 associated to $B^3$ (with appropriate boundary conditions).
 The coend is not an exact functor, so the exactness of the triangle breaks.
+The obvious solution to this problem is to replace the coend with its derived counterpart,
-The obvious solution to this problem is to replace the coend with its derived counterpart.
+Hochschild homology.
 This presumably works fine for $S^1\times B^3$ (the answer being the Hochschild homology
 of an appropriate bimodule), but for more complicated 4-manifolds this leaves much to be desired.
 If we build our manifold up via a handle decomposition, the computation
 would be a sequence of derived coends.
 A different handle decomposition of the same manifold would yield a different
 sequence of derived coends.
 To show that our definition in terms of derived coends is well-defined, we
-would need to show that the above two sequences of derived coends yield the same answer.
+would need to show that the above two sequences of derived coends yield
+isomorphic answers, and that the isomorphism does not depend on any
+choices we made along the way.
 This is probably not easy to do.
 Instead, we would prefer a definition for a derived version of $A_{Kh}(W^4, L)$
 which is manifestly invariant.
 (That is, a definition that does not
 The solution is to replace $A_{Kh}(W^4, L)$, which is a quotient
 \[
 \text{linear combinations of fields} \;\big/\; \text{local relations} ,
 \]
-with an appropriately free resolution (the ``blob complex")
+with an appropriately free resolution (the blob complex)
 \[
 	\cdots\to \bc_2(W, L) \to \bc_1(W, L) \to \bc_0(W, L) .
 \]
 Here $\bc_0$ is linear combinations of fields on $W$,
 $\bc_1$ is linear combinations of local relations on $W$,
 $\bc_2$ is linear combinations of relations amongst relations on $W$,
 and so on.
-None of the above ideas depend on the details of the Khovanov homology example,
-so we develop the general theory in this paper and postpone specific applications
-to later papers.
 \subsection{Formal properties}
 \label{sec:properties}
 The blob complex enjoys the following list of formal properties.
 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cF)$;
 this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:evaluation} below.
 The blob complex is also functorial (indeed, exact) with respect to $\cF$,
 although we will not address this in detail here.
+\nn{KW: what exactly does ``exact in $\cF$" mean?
+Do we mean a similar statement for module labels?}
 \begin{property}[Disjoint union]
 \label{property:disjoint-union}
-The blob complex of a disjoint union is naturally the tensor product of the blob complexes.
+The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes.
 \begin{equation*}
 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2)
 \end{equation*}
 \end{property}
 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings).
 \end{property}
 \begin{property}[Contractibility]
 \label{property:contractibility}%
-With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology.
+With field coefficients, the blob complex on an $n$-ball is contractible in the sense
-Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cF$ to balls.
+that it is homotopic to its $0$-th homology.
-\begin{equation*}
+Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces
-\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & \cF(B^n)}
+associated by the system of fields $\cF$ to balls.
+\begin{equation*}
+\xymatrix{\bc_*(B^n;\cF) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cF)) \ar[r]^(0.6)\iso & A_\cF(B^n)}
 \end{equation*}
 \end{property}
 Properties \ref{property:functoriality} will be immediate from the definition given in
 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there.
-Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
+Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and
+\ref{property:contractibility} are established in \S \ref{sec:basic-properties}.
 \subsection{Specializations}
 \label{sec:specializations}
 The blob complex is a simultaneous generalization of the TQFT skein module construction and of Hochschild homology.
 \begin{thm:hochschild}[Hochschild homology when $X=S^1$]
 The blob complex for a $1$-category $\cC$ on the circle is
 quasi-isomorphic to the Hochschild complex.
 \begin{equation*}
-\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).}
+\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).}
 \end{equation*}
 \end{thm:hochschild}
 Theorem \ref{thm:skein-modules} is immediate from the definition, and
 Theorem \ref{thm:hochschild} is established in \S \ref{sec:hochschild}.
-We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of certain commutative algebras thought of as an $n$-category.
+We also note \S \ref{sec:comm_alg} which describes the blob complex when $\cC$ is a one of
+certain commutative algebras thought of as $n$-categories.
 \subsection{Structure of the blob complex}
 \label{sec:structure}
 \begin{thm:CH}[$C_*(\Homeo(-))$ action]
 \label{thm:evaluation}%
 There is a chain map
 \begin{equation*}
-\ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
+e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X).
 \end{equation*}
 such that
 \begin{enumerate}
 \item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}.
 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 (up to homotopy).
 \begin{equation*}
 \xymatrix@C+2cm{
 \CH{X} \otimes \bc_*(X)
-\ar[r]_{\ev_{X}}  \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &
+\ar[r]_{e_{X}}  \ar[d]^{\gl^{\Homeo}_Y \otimes \gl_Y}  &
 \bc_*(X) \ar[d]_{\gl_Y} \\
-\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)
+\CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}}    & \bc_*(X \bigcup_Y \selfarrow)
 }
 \end{equation*}
 \end{enumerate}
 Moreover any such chain map is unique, up to an iterated homotopy.
 (That is, any pair of homotopies have a homotopy between them, and so on.)
+\nn{revisit this after proof below has stabilized}
 \end{thm:CH}
 \newtheorem*{thm:CH-associativity}{Theorem \ref{thm:CH-associativity}}
 Further,
 \begin{thm:CH-associativity}
 The chain map of Theorem \ref{thm:CH} 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 \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor e_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{e_X} \\
-\CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X)
+\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X)
 }
 \end{equation*}
 \end{thm:CH-associativity}
 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps

changeset 482	6ba3a46a0b50
parent 481	7caafccef7e8
child 483	2cb4fa7c5d0a