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 \begin{article}
 \begin{abstract}
 We explain the need for new axioms for topological quantum field theories that include ideas from derived
-categories and homotopy theory. We summarize our axioms for higher categories, and describe the `blob complex'.
+categories and homotopy theory. We summarize our axioms for higher categories, and describe the ``blob complex".
 Fixing an $n$-category $\cC$, the blob complex associates a chain complex $\bc_*(W;\cC)$ to any $n$-manifold $W$.
 The $0$-th homology of this chain complex recovers the usual TQFT invariants of $W$.
 The higher homology groups should be viewed as generalizations of Hochschild homology.
 The blob complex has a very natural definition in terms of homotopy colimits along decompositions of the manifold $W$.
 We outline the important properties of the blob complex, and sketch the proof of a generalization of
 %of the main theorems; and (2) specify a minimal set of generators and/or axioms.
 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second.
 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.}
 We will define two variations simultaneously,  as all but one of the axioms are identical in the two cases.
-These variations are `isotopy $n$-categories', where homeomorphisms fixing the boundary
+These variations are ``isotopy $n$-categories", where homeomorphisms fixing the boundary
 act trivially on the sets associated to $n$-balls
 (and these sets are usually vector spaces or more generally modules over a commutative ring)
-and `$A_\infty$ $n$-categories',  where there is a homotopy action of
+and ``$A_\infty$ $n$-categories",  where there is a homotopy action of
 $k$-parameter families of homeomorphisms on these sets
 (which are usually chain complexes or topological spaces).
 There are five basic ingredients
 \cite{life-of-brian} of an $n$-category definition:
 This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure
 of an object of $\cS$, and all of the structure maps of the category (above and below) are
 compatible with the $\cS$ structure on $\cC_n(X; c)$.
-Given two hemispheres (a `domain' and `range') that agree on the equator, we need to be able to
+Given two hemispheres (a ``domain" and ``range") that agree on the equator, we need to be able to
 assemble them into a boundary value of the entire sphere.
 \begin{lem}
 \label{lem:domain-and-range}
 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$,
 The gluing maps above are strictly associative.
 Given any decomposition of a ball $B$ into smaller balls
 $$\bigsqcup B_i \to B,$$
 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result.
 \end{axiom}
-This axiom is only reasonable because the definition assigns a set to every ball; any identifications would limit the extent to which we can demand associativity.
+This axiom is only reasonable because the definition assigns a set to every ball;
+any identifications would limit the extent to which we can demand associativity.
 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices.
 \begin{axiom}[Product (identity) morphisms]
 \label{axiom:product}
 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$),
 there is a map $\pi^*:\cC(X)\to \cC(E)$.
 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$
 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes).
 \subsection{Example (string diagrams)}
-Fix a `traditional' $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category).
+Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category).
 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$;
 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$.
 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$.
 Boundary restrictions and gluing are again straightforward to define.
 Define product morphisms via product cell decompositions.
 %\label{eq:psi-C}
 	\psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl
 \end{equation*}
 where the restrictions to the various pieces of shared boundaries amongst the balls
 $X_a$ all agree (this is a fibered product of all the labels of $k$-balls over the labels of $k-1$-manifolds).
-When $k=n$, the `subset' and `product' in the above formula should be
+When $k=n$, the ``subset" and ``product" in the above formula should be
 interpreted in the appropriate enriching category.
 If $x$ is a refinement of $y$, the map $\psi_{\cC;W}(x) \to \psi_{\cC;W}(y)$ is given by the composition maps of $\cC$.
 \end{defn}
-We will use the term `field on $W$' to refer to a point of this functor,
+We will use the term ``field on $W$" to refer to a point of this functor,
 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$.
 \subsubsection{Colimits}
 Recall that our definition of an $n$-category is essentially a collection of functors
 homotopy colimit via the cone-product polyhedra in $\cell(W)$.
 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the
 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs,
 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root,
 and taking product identifies the roots of several trees.
-The `local homotopy colimit' is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$.
+The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$.
 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required.
-A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product.
+A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization.
+In the local homotopy colimit we can further require that any composition of morphisms in a directed tree is not expressible as a product.
 %When $\cC$ is a topological $n$-category,
 %the flexibility available in the construction of a homotopy colimit allows
 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
 %\todo{either need to explain why this is the same, or significantly rewrite this section}
 \begin{itemize}
 \item a permissible collection of $k$ embedded balls, and
 \item for each resulting piece of $W$, a field,
 \end{itemize}
 such that for any innermost blob $B$, the field on $B$ goes to zero under the gluing map from $\cC$.
-We call such a field a `null field on $B$'.
+We call such a field a ``null field on $B$".
 The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs.
 We now spell this out for some small values of $k$. For $k=0$, the $0$-blob group is simply fields on $W$.
 For $k=1$, a generator consists of a field on $W$ and a ball, such that the restriction of the field to that ball is a null field.
 More specifically, a generator of $\cB\cT_k(X)$ is an $i$-parameter family of $j$-blob diagrams, with $i+j=k$.
 An essential step in the proof of this equivalence is a result to the effect that a $k$-parameter
 family of homeomorphisms can be localized to at most $k$ small sets.
 With this alternate version in hand, the theorem is straightforward.
-By functoriality (Property \cite{property:functoriality}) $Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this
+By functoriality (Property \ref{property:functoriality}) $\Homeo(X)$ acts on the set $BD_j(X)$ of $j$-blob diagrams, and this
 induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$
 and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$.
 It is easy to check that $e_X$ thus defined has the desired properties.
 \end{proof}

changeset 657	9fbd8e63ab2e
parent 656	28592849a474
child 658	c56a3fe75d1e