# HG changeset patch # User Kevin Walker # Date 1290441737 25200 # Node ID 9fbd8e63ab2ebd1463665ef938256d0fe83a7d6b # Parent 28592849a474c34dbf76835a5e96e0621bbe8b69 fixing single quotes and long lines diff -r 28592849a474 -r 9fbd8e63ab2e pnas/pnas.tex --- a/pnas/pnas.tex Sun Nov 21 15:24:53 2010 -0800 +++ b/pnas/pnas.tex Mon Nov 22 09:02:17 2010 -0700 @@ -137,7 +137,7 @@ \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. @@ -271,10 +271,10 @@ %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). @@ -339,7 +339,7 @@ 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} @@ -385,7 +385,8 @@ $$\bigsqcup B_i \to B,$$ 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} @@ -500,7 +501,7 @@ \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 $C$. @@ -562,12 +563,12 @@ \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 $\psi_{\cC;W}(x)$. @@ -607,9 +608,10 @@ 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 @@ -638,7 +640,7 @@ \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. @@ -808,7 +810,7 @@ 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.