blob: comparison pnas/pnas.tex

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-:38532ba5bd0f
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 in a concrete version of the homotopy colimit.)
 We then review some basic properties of the blob complex, and finish by showing how it
 yields a higher categorical and higher dimensional generalization of Deligne's
 conjecture on Hochschild cochains and the little 2-disks operad.
-\nn{maybe this is not necessary?} \nn{let's move this to somewhere later, if we keep it}
+Of course, there are currently many interesting alternative notions of $n$-category and of TQFT.
-In an attempt to forestall any confusion that might arise from different definitions of
+We note that our $n$-categories are both more and less general
-``$n$-category" and ``TQFT", we note that our $n$-categories are both more and less general
 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}.
-More general in that we make no duality assumptions in the top dimension $n+1$.
+They are more general in that we make no duality assumptions in the top dimension $n+1$.
-Less general in that we impose stronger duality requirements in dimensions 0 through $n$.
+They are less general in that we impose stronger duality requirements in dimensions 0 through $n$.
 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while
 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs.
 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. In this paper we attempt to give a clear view of the big picture without getting bogged down in technical details.
 Thus we can have the simplicity of strict associativity in exchange for more morphisms.
 We wish to imitate this strategy in higher categories.
 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with
 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic
 to the standard $k$-ball $B^k$.
-\nn{maybe add that in addition we want functoriality}
 By default our balls are unoriented,
 but it is useful at times to vary this,
 for example by considering oriented or Spin balls.
 We can also consider more exotic structures, such as balls with a map to some target space,
 homeomorphisms which are not the identity on the boundary of the $k$-ball.
 The action of these homeomorphisms gives the ``strong duality" structure.
 As such, we don't subdivide the boundary of a morphism
 into domain and range --- the duality operations can convert between domain and range.
-Later \nn{make sure this actually happens, or reorganise} we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom.
+Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ from arbitrary manifolds to sets. We need the restriction of these functors to $k$-spheres, for $k<n$, for the next axiom.
 \begin{axiom}[Boundaries]\label{nca-boundary}
 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.
 These maps, for various $X$, comprise a natural transformation of functors.
 \end{axiom}
 \begin{property}[Contractibility]
 \label{property:contractibility}%
 The blob complex on an $n$-ball is contractible in the sense
 that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category.
 \begin{equation*}
-\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)}
+\xymatrix{\bc_*(B^n;\cC) \ar[r]^(0.4){\htpy} & H_0(\bc_*(B^n;\cC)) \ar[r]^(0.6)\iso & \cC(B^n)}
 \end{equation*}
 \end{property}
 %\nn{maybe should say something about the $A_\infty$ case}
 \begin{proof}(Sketch)
 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$.
 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send
 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$.
 \end{proof}
-If $\cC$ is an $A-\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$,
+If $\cC$ is an $A_\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$,
 but this is no longer concentrated in degree zero.
 \subsection{Specializations}
 \label{sec:specializations}
 The blob complex has several important special cases.
 \begin{thm}[Skein modules]
 \label{thm:skein-modules}
-\nn{linear n-categories only?}
+Suppose $\cC$ is a linear $n$-category
 The $0$-th blob homology of $X$ is the usual
 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$
 by $\cC$.
 \begin{equation*}
-H_0(\bc_*(X;\cC)) \iso A_{\cC}(X)
+H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X)
 \end{equation*}
 \end{thm}
 This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, or directly from the explicit description of the blob complex.
 \begin{thm}[Hochschild homology when $X=S^1$]

changeset 649	795ec5790b8b
parent 648	38532ba5bd0f
child 650	b17f1f07cba2