author | Kevin Walker <kevin@canyon23.net> |
Wed, 17 Nov 2010 11:16:39 -0800 | |
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of the National Academy of Sciences of the United States of America} |
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%\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
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\title{$n$-categories, colimits and the blob complex} |
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\author{Scott Morrison\affil{1}{Miller Institute for Basic Research, UC Berkeley, CA 94704, USA} \and Kevin Walker\affil{2}{Microsoft Station Q, 2243 CNSI Building, UC Santa Barbara, CA 93106, USA}} |
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\contributor{Submitted to Proceedings of the National Academy of Sciences |
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of the United States of America} |
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\keywords{n-categories | topological quantum field theory | hochschild homology} |
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% \abbreviations{TQFT, topological quantum field theory} |
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%\dropcap{I}n this article we study the evolution of ''almost-sharp'' fronts |
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\dropcap{T}he aim of this paper is to describe a derived category version of TQFTs. |
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For our purposes, an $n{+}1$-dimensional TQFT is a locally defined system of |
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invariants of manifolds of dimensions 0 through $n+1$. |
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The TQFT invariant $A(Y)$ of a closed $k$-manifold $Y$ is a linear $(n{-}k)$-category. |
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If $Y$ has boundary then $A(Y)$ is a collection of $(n{-}k)$-categories which afford |
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a representation of the $(n{-}k{+}1)$-category $A(\bd Y)$. |
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(See \cite{1009.5025} and \cite{kw:tqft}; |
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for a more homotopy-theoretic point of view see \cite{0905.0465}.) |
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We now comment on some particular values of $k$ above. |
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By convention, a linear 0-category is a vector space, and a representation |
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of a vector space is an element of the dual space. |
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So a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
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and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
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In fact we will be mainly be interested in so-called $(n{+}\epsilon)$-dimensional |
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TQFTs which have nothing to say about $(n{+}1)$-manifolds. |
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For the remainder of this paper we assume this case. |
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When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
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and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
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The gluing rule for the TQFT in dimension $n$ states that |
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$A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
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where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$. |
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When $k=0$ we have an $n$-category $A(pt)$. |
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This can be thought of as the local part of the TQFT, and the full TQFT can be constructed of $A(pt)$ |
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via colimits (see below). |
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We call a TQFT semisimple if $A(S)$ is a semisimple 1-category for all $(n{-}1)$-manifolds $S$ |
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and $A(Y)$ is a finite-dimensional vector space for all $n$-manifolds $Y$. |
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Examples of semisimple TQFTS include Witten-Reshetikhin-Turaev theories, |
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Turaev-Viro theories, and Dijkgraaf-Witten theories. |
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These can all be given satisfactory accounts in the framework outlined above. |
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(The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories in order to be |
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extended all the way down to 0-manifolds.) |
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For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
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\nn{temp} |
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For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory |
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involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
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Long exact sequences are important computational tools in these theories, |
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and also in Khovanov homology, but the colimit construction breaks exactness. |
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For these reasons and others, it is desirable to |
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extend to above framework to incorporate ideas from derived categories. |
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One approach to such a generalization might be to simply define a |
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TQFT via its gluing formulas, replacing tensor products with |
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derived tensor products. |
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\nn{maybe cite Kh's paper on links in $S^1\times S^2$} |
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However, it is probably difficult to prove |
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the invariance of such a definition, as the object associated to a manifold |
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will a priori depend on the explicit presentation used to apply the gluing formulas. |
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We instead give a manifestly invariant construction, and |
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deduce from it the gluing formulas based on $A_\infty$ tensor products. |
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This paper is organized as follows. |
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We first give an account of our version of $n$-categories. |
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According to our definition, $n$-categories are, among other things, |
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functorial invariants of $k$-balls, $0\le k \le n$, which behave well with respect to gluing. |
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We then describe how to use [homotopy] colimits to extend $n$-categories |
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from balls to arbitrary $k$-manifolds. |
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This extension is the desired derived version of a TQFT, which we call the blob complex. |
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(The name comes from the ``blobs" which feature prominently |
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in a concrete version of the homotopy colimit.) |
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\nn{In many places we omit details; they can be found in MW. |
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(Blanket statement in order to avoid too many citations to MW.)} |
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\nn{perhaps say something explicit about the relationship of this paper to big blob paper. |
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like: in this paper we try to give a clear view of the big picture without getting bogged down in details} |
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\nn{diff w/ lurie} |
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\section{Definitions} |
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\subsection{$n$-categories} \mbox{} |
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\nn{rough draft of n-cat stuff...} |
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\nn{maybe say something about goals: well-suited to TQFTs; avoid proliferation of coherency axioms; |
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non-recursive (n-cats not defined n terms of (n-1)-cats; easy to show that the motivating |
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examples satisfy the axioms; strong duality; both plain and infty case; |
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(?) easy to see that axioms are correct, in the sense of nothing missing (need |
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to say this better if we keep it)} |
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\nn{maybe: the typical n-cat definition tries to do two things at once: (1) give a list of basic properties |
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which are weak enough to include the basic examples and strong enough to support the proofs |
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of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
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We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
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More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
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\nn{say something about defining plain and infty cases simultaneously} |
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There are five basic ingredients |
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\cite{life-of-brian} of an $n$-category definition: |
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$k$-morphisms (for $0\le k \le n$), domain and range, composition, |
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identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
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in some auxiliary category, or strict associativity instead of weak associativity). |
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We will treat each of these in turn. |
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To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
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\cite[\S 2.2]{MR505692}. |
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In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
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so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
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of higher associativity relations. |
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While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
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of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
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In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
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{\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
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Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
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We wish to imitate this strategy in higher categories. |
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Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
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a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
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to the standard $k$-ball $B^k$. |
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\nn{maybe add that in addition we want functoriality} |
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We haven't said precisely what sort of balls we are considering, |
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because we prefer to let this detail be a parameter in the definition. |
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It is useful to consider unoriented, oriented, Spin and $\mbox{Pin}_\pm$ balls. |
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Also useful are more exotic structures, such as balls equipped with a map to some target space, |
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or equipped with $m$ independent vector fields. |
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(The latter structure would model $n$-categories with less duality than we usually assume.) |
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%In fact, the axioms here may easily be varied by considering balls with structure (e.g. $m$ independent vector fields, a map to some target space, etc.). Such variations are useful for axiomatizing categories with less duality, and also as technical tools in proofs. |
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\begin{axiom}[Morphisms] |
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\label{axiom:morphisms} |
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For each $0 \le k \le n$, we have a functor $\cC_k$ from |
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the category of $k$-balls and |
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homeomorphisms to the category of sets and bijections. |
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\end{axiom} |
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Note that the functoriality in the above axiom allows us to operate via |
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homeomorphisms which are not the identity on the boundary of the $k$-ball. |
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The action of these homeomorphisms gives the ``strong duality" structure. |
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As such, we don't subdivide the boundary of a morphism |
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into domain and range --- the duality operations can convert between domain and range. |
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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. |
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\begin{axiom}[Boundaries]\label{nca-boundary} |
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For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
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These maps, for various $X$, comprise a natural transformation of functors. |
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\end{axiom} |
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For $c\in \cl{\cC}_{k-1}(\bd X)$ we define $\cC_k(X; c) = \bd^{-1}(c)$. |
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|
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Many of the examples we are interested in are enriched in some auxiliary category $\cS$ |
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(e.g. vector spaces or rings, or, in the $A_\infty$ case, chain complexes or topological spaces). |
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This means that in the top dimension $k=n$ the sets $\cC_n(X; c)$ have the structure |
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of an object of $\cS$, and all of the structure maps of the category (above and below) are |
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compatible with the $\cS$ structure on $\cC_n(X; c)$. |
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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. |
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\begin{lem} |
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\label{lem:domain-and-range} |
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Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
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$B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
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Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
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two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
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Then we have an injective map |
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\[ |
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\gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
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\] |
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which is natural with respect to the actions of homeomorphisms. |
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%(When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
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%becomes a normal product.) |
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\end{lem} |
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If $\bdy B = S$, we denote $\bdy^{-1}(\im(\gl_E))$ by $\cC(B)_E$. |
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|
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\begin{axiom}[Gluing] |
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\label{axiom:composition} |
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Let $B = B_1 \cup_Y B_2$, where $B$, $B_1$ and $B_2$ are $k$-balls ($0\le k\le n$) |
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and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
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Let $E = \bd Y$, which is a $k{-}2$-sphere. |
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%Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
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We have restriction maps $\cC(B_i)_E \to \cC(Y)$. |
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Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
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We have a map |
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\[ |
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\gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
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\] |
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which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
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to the intersection of the boundaries of $B$ and $B_i$. |
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If $k < n$, |
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or if $k=n$ and we are in the $A_\infty$ case, |
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we require that $\gl_Y$ is injective. |
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(For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
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\end{axiom} |
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\begin{axiom}[Strict associativity] \label{nca-assoc} |
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The gluing maps above are strictly associative. |
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Given any decomposition of a ball $B$ into smaller balls |
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$$\bigsqcup B_i \to B,$$ |
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any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
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\end{axiom} |
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For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
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\begin{axiom}[Product (identity) morphisms] |
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\label{axiom:product} |
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For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
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there is a map $\pi^*:\cC(X)\to \cC(E)$. |
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These maps must be |
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\begin{enumerate} |
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\item natural with respect to maps of pinched products, |
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370 |
\item functorial with respect to composition of pinched products, |
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371 |
\item compatible with gluing and restriction of pinched products. |
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372 |
\end{enumerate} |
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373 |
|
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374 |
%%% begin noop %%% |
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375 |
% this was the original list of conditions, which I've replaced with the much terser list above -S |
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376 |
\noop{ |
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377 |
These maps must satisfy the following conditions. |
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378 |
\begin{enumerate} |
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379 |
\item |
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380 |
If $\pi:E\to X$ and $\pi':E'\to X'$ are pinched products, and |
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381 |
if $f:X\to X'$ and $\tilde{f}:E \to E'$ are maps such that the diagram |
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382 |
\[ \xymatrix{ |
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383 |
E \ar[r]^{\tilde{f}} \ar[d]_{\pi} & E' \ar[d]^{\pi'} \\ |
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384 |
X \ar[r]^{f} & X' |
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385 |
} \] |
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386 |
commutes, then we have |
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387 |
\[ |
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388 |
\pi'^*\circ f = \tilde{f}\circ \pi^*. |
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389 |
\] |
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390 |
\item |
585
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391 |
Product morphisms are compatible with gluing. |
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392 |
Let $\pi:E\to X$, $\pi_1:E_1\to X_1$, and $\pi_2:E_2\to X_2$ |
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393 |
be pinched products with $E = E_1\cup E_2$. |
611 | 394 |
Let $a\in \cC(X)$, and let $a_i$ denote the restriction of $a$ to $X_i\subset X$. |
575
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395 |
Then |
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396 |
\[ |
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397 |
\pi^*(a) = \pi_1^*(a_1)\bullet \pi_2^*(a_2) . |
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398 |
\] |
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399 |
\item |
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400 |
Product morphisms are associative. |
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401 |
If $\pi:E\to X$ and $\rho:D\to E$ are pinched products then |
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402 |
\[ |
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403 |
\rho^*\circ\pi^* = (\pi\circ\rho)^* . |
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404 |
\] |
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405 |
\item |
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406 |
Product morphisms are compatible with restriction. |
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407 |
If we have a commutative diagram |
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408 |
\[ \xymatrix{ |
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409 |
D \ar@{^(->}[r] \ar[d]_{\rho} & E \ar[d]^{\pi} \\ |
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410 |
Y \ar@{^(->}[r] & X |
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411 |
} \] |
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412 |
such that $\rho$ and $\pi$ are pinched products, then |
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413 |
\[ |
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414 |
\res_D\circ\pi^* = \rho^*\circ\res_Y . |
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415 |
\] |
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416 |
\end{enumerate} |
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} %%% end \noop %%% |
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418 |
\end{axiom} |
604
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419 |
|
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420 |
To state the next axiom we need the notion of {\it collar maps} on $k$-morphisms. |
611 | 421 |
Let $X$ be a $k$-ball and $Y\subset\bd X$ be a $(k{-}1)$-ball. |
604
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422 |
Let $J$ be a 1-ball. |
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423 |
Let $Y\times_p J$ denote $Y\times J$ pinched along $(\bd Y)\times J$. |
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424 |
A collar map is an instance of the composition |
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425 |
\[ |
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426 |
\cC(X) \to \cC(X\cup_Y (Y\times_p J)) \to \cC(X) , |
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427 |
\] |
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428 |
where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
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429 |
the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
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430 |
to the identity on the boundary. |
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431 |
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432 |
|
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433 |
\begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
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434 |
\label{axiom:extended-isotopies} |
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435 |
Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
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436 |
to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
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437 |
Then $f$ acts trivially on $\cC(X)$. |
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438 |
In addition, collar maps act trivially on $\cC(X)$. |
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439 |
\end{axiom} |
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440 |
|
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441 |
\smallskip |
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442 |
|
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443 |
For $A_\infty$ $n$-categories, we replace |
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444 |
isotopy invariance with the requirement that families of homeomorphisms act. |
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445 |
For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
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446 |
Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
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447 |
$C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
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448 |
|
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449 |
|
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450 |
\begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
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451 |
\label{axiom:families} |
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452 |
For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
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453 |
\[ |
611 | 454 |
C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
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455 |
\] |
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456 |
These action maps are required to be associative up to homotopy, |
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457 |
and also compatible with composition (gluing) in the sense that |
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458 |
a diagram like the one in Theorem \ref{thm:CH} commutes. |
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459 |
\end{axiom} |
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460 |
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461 |
\subsection{Example (the fundamental $n$-groupoid)} |
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462 |
We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
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When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
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to be the set of continuous maps from $X$ to $T$. |
601
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465 |
When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
600
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Define boundary restrictions and gluing in the obvious way. |
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If $\rho:E\to X$ is a pinched product and $f:X\to T$ is a $k$-morphism, |
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define the product morphism $\rho^*(f)$ to be $f\circ\rho$. |
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469 |
|
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470 |
We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
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For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
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472 |
(if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
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473 |
|
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475 |
\subsection{Example (string diagrams)} |
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Fix a `traditional' $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
600
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477 |
Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
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478 |
that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
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If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
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480 |
Boundary restrictions and gluing are again straightforward to define. |
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481 |
Define product morphisms via product cell decompositions. |
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482 |
|
612 | 483 |
\subsection{Example (bordism)} |
484 |
When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
|
485 |
submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
|
486 |
to $\bd X$. |
|
487 |
For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
|
600
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488 |
|
612 | 489 |
There is an $A_\infty$ analogue enriched in topological spaces, where at the top level we take all such submanifolds, rather than homeomorphism classes. For each fixed $\bdy W \subset \bdy X \times \bbR^\infty$, we can topologize the set of submanifolds by ambient isotopy rel boundary. |
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\subsection{The blob complex} |
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\subsubsection{Decompositions of manifolds} |
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574 | 494 |
A \emph{ball decomposition} of $W$ is a |
573
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sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
574 | 496 |
$\du_a X_a$ and each $M_i$ is a manifold. |
497 |
If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
|
498 |
A {\it permissible decomposition} of $W$ is a map |
|
573
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499 |
\[ |
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\coprod_a X_a \to W, |
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501 |
\] |
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502 |
which can be completed to a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$. |
574 | 503 |
A permissible decomposition is weaker than a ball decomposition; we forget the order in which the balls |
504 |
are glued up to yield $W$, and just require that there is some non-pathological way to do this. |
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|
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Given permissible decompositions $x = \{X_a\}$ and $y = \{Y_b\}$ of $W$, we say that $x$ is a refinement |
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507 |
of $y$, or write $x \le y$, if there is a ball decomposition $\du_a X_a = M_0\to\cdots\to M_m = W$ |
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508 |
with $\du_b Y_b = M_i$ for some $i$. |
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509 |
|
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|
510 |
\begin{defn} |
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511 |
The poset $\cell(W)$ has objects the permissible decompositions of $W$, |
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512 |
and a unique morphism from $x$ to $y$ if and only if $x$ is a refinement of $y$. |
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513 |
See Figure \ref{partofJfig} for an example. |
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514 |
\end{defn} |
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515 |
|
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516 |
This poset in fact has more structure, since we can glue together permissible decompositions of $W_1$ and $W_2$ to obtain a permissible decomposition of $W_1 \sqcup W_2$. |
573
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517 |
|
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518 |
An $n$-category $\cC$ determines |
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519 |
a functor $\psi_{\cC;W}$ from $\cell(W)$ to the category of sets |
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520 |
(possibly with additional structure if $k=n$). |
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521 |
Each $k$-ball $X$ of a decomposition $y$ of $W$ has its boundary decomposed into $k{-}1$-balls, |
611 | 522 |
and there is a subset $\cC(X)\spl \subset \cC(X)$ of morphisms whose boundaries |
573
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523 |
are splittable along this decomposition. |
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524 |
|
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525 |
\begin{defn} |
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526 |
Define the functor $\psi_{\cC;W} : \cell(W) \to \Set$ as follows. |
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527 |
For a decomposition $x = \bigsqcup_a X_a$ in $\cell(W)$, $\psi_{\cC;W}(x)$ is the subset |
574 | 528 |
\begin{equation*} |
529 |
%\label{eq:psi-C} |
|
611 | 530 |
\psi_{\cC;W}(x) \subset \prod_a \cC(X_a)\spl |
574 | 531 |
\end{equation*} |
573
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532 |
where the restrictions to the various pieces of shared boundaries amongst the cells |
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533 |
$X_a$ all agree (this is a fibered product of all the labels of $n$-cells over the labels of $n-1$-cells). When $k=n$, the `subset' and `product' in the above formula should be interpreted in the appropriate enriching category. |
573
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534 |
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 $\cC$. |
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535 |
\end{defn} |
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536 |
|
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537 |
We will use the term `field on $W$' to refer to a point of this functor, |
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538 |
that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
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539 |
|
575
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540 |
|
632 | 541 |
\subsubsection{Colimits} |
575
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542 |
\nn{Motivation: How can we extend an $n$-category from balls to arbitrary manifolds?} |
628
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543 |
\nn{Mention that the axioms for $n$-categories can be stated in terms of decompositions of balls?} |
608 | 544 |
\nn{Explain codimension colimits here too} |
575
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545 |
|
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546 |
We can now give a straightforward but rather abstract definition of the blob complex of an $n$-manifold $W$ |
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547 |
with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$ |
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548 |
of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$. |
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549 |
|
599 | 550 |
An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as |
551 |
$$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$ |
|
552 |
where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$. |
|
598
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|
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554 |
Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the 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 differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
601
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555 |
A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
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556 |
|
605
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557 |
%When $\cC$ is a topological $n$-category, |
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558 |
%the flexibility available in the construction of a homotopy colimit allows |
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|
559 |
%us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
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560 |
%\todo{either need to explain why this is the same, or significantly rewrite this section} |
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561 |
When $\cC$ is the topological $n$-category based on string diagrams for a traditional |
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562 |
$n$-category $C$, |
610 | 563 |
one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
606 | 564 |
are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
565 |
Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
|
605
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566 |
a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
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567 |
The restriction of the string diagram to innermost blobs is required to be ``null" in the sense that |
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568 |
it evaluates to a zero $n$-morphism of $C$. |
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|
569 |
The next few paragraphs describe this in more detail. |
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570 |
|
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571 |
We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible} |
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572 |
if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that |
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573 |
each $B_i$ appears as a connected component of one of the $M_j$. Note that this allows the balls to be pairwise either disjoint or nested. Such a collection of balls cuts $W$ into pieces, the connected components of $W \setminus \bigcup \bdy B_i$. These pieces need not be manifolds, but they do automatically have permissible decompositions. |
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574 |
|
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575 |
The $k$-blob group $\bc_k(W; \cC)$ is generated by the $k$-blob diagrams. A $k$-blob diagram consists of |
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576 |
\begin{itemize} |
608 | 577 |
\item a permissible collection of $k$ embedded balls, and |
575
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578 |
\item for each resulting piece of $W$, a field, |
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579 |
\end{itemize} |
585
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580 |
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 $B$'. |
575
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581 |
|
608 | 582 |
The differential acts on a $k$-blob diagram by summing over ways to forget one of the $k$ blobs, with alternating signs. |
575
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583 |
|
598
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|
584 |
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. The differential simply forgets the ball. Thus we see that $H_0$ of the blob complex is the quotient of fields by fields which are null on some ball. |
580
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585 |
|
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586 |
For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
575
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587 |
|
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588 |
\section{Properties of the blob complex} |
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589 |
\subsection{Formal properties} |
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590 |
\label{sec:properties} |
602
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|
591 |
The blob complex enjoys the following list of formal properties. The first three are immediate from the definitions. |
572
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592 |
|
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593 |
\begin{property}[Functoriality] |
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594 |
\label{property:functoriality}% |
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|
595 |
The blob complex is functorial with respect to homeomorphisms. |
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|
596 |
That is, |
574 | 597 |
for a fixed $n$-category $\cC$, the association |
572
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|
598 |
\begin{equation*} |
574 | 599 |
X \mapsto \bc_*(X; \cC) |
572
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600 |
\end{equation*} |
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601 |
is a functor from $n$-manifolds and homeomorphisms between them to chain |
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602 |
complexes and isomorphisms between them. |
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|
603 |
\end{property} |
574 | 604 |
As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*(X; \cC)$; |
572
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605 |
this action is extended to all of $C_*(\Homeo(X))$ in Theorem \ref{thm:CH} below. |
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606 |
|
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607 |
\begin{property}[Disjoint union] |
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608 |
\label{property:disjoint-union} |
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|
609 |
The blob complex of a disjoint union is naturally isomorphic to the tensor product of the blob complexes. |
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|
610 |
\begin{equation*} |
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|
611 |
\bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
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|
612 |
\end{equation*} |
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|
613 |
\end{property} |
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614 |
|
574 | 615 |
If an $n$-manifold $X$ contains $Y \sqcup Y^\text{op}$ (we allow $Y = \eset$) as a codimension $0$ submanifold of its boundary, |
616 |
write $X \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
|
572
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617 |
\begin{property}[Gluing map] |
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|
618 |
\label{property:gluing-map}% |
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619 |
%If $X_1$ and $X_2$ are $n$-manifolds, with $Y$ a codimension $0$-submanifold of $\bdy X_1$, and $Y^{\text{op}}$ a codimension $0$-submanifold of $\bdy X_2$, there is a chain map |
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|
620 |
%\begin{equation*} |
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621 |
%\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
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|
622 |
%\end{equation*} |
607 | 623 |
Given a gluing $X \to X \bigcup_{Y}\selfarrow$, there is |
574 | 624 |
a map |
572
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|
625 |
\[ |
574 | 626 |
\bc_*(X) \to \bc_*(X \bigcup_{Y}\selfarrow), |
572
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627 |
\] |
574 | 628 |
natural with respect to homeomorphisms, and associative with respect to iterated gluings. |
572
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|
629 |
\end{property} |
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630 |
|
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|
631 |
\begin{property}[Contractibility] |
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|
632 |
\label{property:contractibility}% |
589
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633 |
The blob complex on an $n$-ball is contractible in the sense |
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634 |
that it is homotopic to its $0$-th homology, and this is just the vector space associated to the ball by the $n$-category. |
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635 |
\begin{equation*} |
589
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|
636 |
\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)} |
572
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|
637 |
\end{equation*} |
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|
638 |
\end{property} |
627
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|
639 |
%\nn{maybe should say something about the $A_\infty$ case} |
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|
640 |
|
583 | 641 |
\begin{proof}(Sketch) |
642 |
For $k\ge 1$, the contracting homotopy sends a $k$-blob diagram to the $(k{+}1)$-blob diagram |
|
643 |
obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
|
644 |
For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
|
645 |
$x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
|
646 |
\end{proof} |
|
572
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|
647 |
|
627
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|
648 |
If $\cC$ is an $A-\infty$ $n$-category then $\bc_*(B^n;\cC)$ is still homotopy equivalent to $\cC(B^n)$, |
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|
649 |
but this is no longer concentrated in degree zero. |
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650 |
|
572
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651 |
\subsection{Specializations} |
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652 |
\label{sec:specializations} |
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653 |
|
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654 |
The blob complex has several important special cases. |
572
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655 |
|
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656 |
\begin{thm}[Skein modules] |
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657 |
\label{thm:skein-modules} |
589
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658 |
\nn{Plain n-categories only?} |
572
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659 |
The $0$-th blob homology of $X$ is the usual |
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660 |
(dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
589
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661 |
by $\cC$. |
572
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662 |
\begin{equation*} |
589
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663 |
H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
572
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664 |
\end{equation*} |
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665 |
\end{thm} |
599 | 666 |
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. |
572
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667 |
|
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668 |
\begin{thm}[Hochschild homology when $X=S^1$] |
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669 |
\label{thm:hochschild} |
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670 |
The blob complex for a $1$-category $\cC$ on the circle is |
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671 |
quasi-isomorphic to the Hochschild complex. |
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672 |
\begin{equation*} |
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673 |
\xymatrix{\bc_*(S^1;\cC) \ar[r]^(0.47){\iso}_(0.47){\text{qi}} & \HC_*(\cC).} |
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674 |
\end{equation*} |
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675 |
\end{thm} |
628
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676 |
This theorem is established by extending the statement to bimodules as well as categories, then verifying that the universal properties of Hochschild homology also hold for $\bc_*(S^1; -)$. |
572
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677 |
|
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678 |
\begin{thm}[Mapping spaces] |
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679 |
\label{thm:map-recon} |
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680 |
Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
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681 |
$B^n \to T$. |
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682 |
(The case $n=1$ is the usual $A_\infty$-category of paths in $T$.) |
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683 |
Then |
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684 |
$$\bc_*(X; \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
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685 |
\end{thm} |
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686 |
|
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687 |
This says that we can recover (up to homotopy) the space of maps to $T$ via blob homology from local data. |
628
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688 |
Note that there is no restriction on the connectivity of $T$ as there is for the corresponding result in topological chiral homology \cite[Theorem 3.8.6]{0911.0018}. The result was proved in \cite[\S 7.3]{1009.5025}. |
615
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689 |
|
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|
690 |
\subsection{Structure of the blob complex} |
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|
691 |
\label{sec:structure} |
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|
692 |
|
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|
693 |
In the following $\CH{X} = C_*(\Homeo(X))$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
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|
694 |
|
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|
695 |
\begin{thm} |
572
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|
696 |
\label{thm:CH}\label{thm:evaluation} |
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|
697 |
There is a chain map |
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|
698 |
\begin{equation*} |
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|
699 |
e_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X) |
572
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|
700 |
\end{equation*} |
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|
701 |
such that |
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|
702 |
\begin{enumerate} |
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|
703 |
\item Restricted to $CH_0(X)$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
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|
704 |
|
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|
705 |
\item For |
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|
706 |
any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
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|
707 |
(using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
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|
708 |
\begin{equation*} |
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|
709 |
\xymatrix@C+0.3cm{ |
611 | 710 |
\CH{X} \tensor \bc_*(X) |
711 |
\ar[r]_{e_{X}} \ar[d]^{\gl^{\Homeo}_Y \tensor \gl_Y} & |
|
572
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|
712 |
\bc_*(X) \ar[d]_{\gl_Y} \\ |
611 | 713 |
\CH{X \bigcup_Y \selfarrow} \tensor \bc_*(X \bigcup_Y \selfarrow) \ar[r]_<<<<<<<{e_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) |
572
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|
714 |
} |
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|
715 |
\end{equation*} |
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|
716 |
\end{enumerate} |
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|
717 |
|
609 | 718 |
Further, this map is associative, in the sense that the following diagram commutes (up to homotopy). |
572
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|
719 |
\begin{equation*} |
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|
720 |
\xymatrix{ |
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|
721 |
\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} \\ |
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|
722 |
\CH{X} \tensor \bc_*(X) \ar[r]^{e_X} & \bc_*(X) |
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|
723 |
} |
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|
724 |
\end{equation*} |
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|
725 |
\end{thm} |
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|
726 |
|
609 | 727 |
\begin{proof}(Sketch.) |
622
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|
728 |
We introduce yet another homotopy equivalent version of |
609 | 729 |
the blob complex, $\cB\cT_*(X)$. |
730 |
Blob diagrams have a natural topology, which is ignored by $\bc_*(X)$. |
|
731 |
In $\cB\cT_*(X)$ we take this topology into account, treating the blob diagrams as something |
|
732 |
analogous to a simplicial space (but with cone-product polyhedra replacing simplices). |
|
622
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|
733 |
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 homeomorphism can be localized to at most $k$ small sets. |
609 | 734 |
|
735 |
With this alternate version in hand, it is straightforward to prove the theorem. |
|
736 |
The evaluation map $\Homeo(X)\times BD_j(X)\to BD_j(X)$ |
|
614 | 737 |
induces a chain map $\CH{X}\tensor C_*(BD_j(X))\to C_*(BD_j(X))$ |
738 |
and hence a map $e_X: \CH{X} \tensor \cB\cT_*(X) \to \cB\cT_*(X)$. |
|
609 | 739 |
It is easy to check that $e_X$ thus defined has the desired properties. |
740 |
\end{proof} |
|
575
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|
741 |
|
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|
742 |
\begin{thm} |
572
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|
743 |
\label{thm:blobs-ainfty} |
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|
744 |
Let $\cC$ be a topological $n$-category. |
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|
745 |
Let $Y$ be an $n{-}k$-manifold. |
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|
746 |
There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
610 | 747 |
to be the set $$\bc_*(Y;\cC)(D) = \cl{\cC}(Y \times D)$$ and on $k$-balls $D$ to be the set |
572
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|
748 |
$$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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|
749 |
(When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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|
750 |
These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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|
751 |
Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Theorem \ref{thm:evaluation}. |
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|
752 |
\end{thm} |
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|
753 |
\begin{rem} |
610 | 754 |
When $Y$ is a point this produces an $A_\infty$ $n$-category from a topological $n$-category, |
755 |
which can be thought of as a free resolution. |
|
572
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|
756 |
\end{rem} |
610 | 757 |
This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
572
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|
758 |
|
618 | 759 |
Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
760 |
Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
|
572
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|
761 |
|
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|
762 |
\begin{thm}[Gluing formula] |
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|
763 |
\label{thm:gluing} |
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|
764 |
\mbox{}% <-- gets the indenting right |
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|
765 |
\begin{itemize} |
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|
766 |
\item For any $n$-manifold $X$, with $Y$ a codimension $0$-submanifold of its boundary, the blob complex of $X$ is naturally an |
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|
767 |
$A_\infty$ module for $\bc_*(Y)$. |
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|
768 |
|
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|
769 |
\item The blob complex of a glued manifold $X\bigcup_Y \selfarrow$ is the $A_\infty$ self-tensor product of |
572
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|
770 |
$\bc_*(X)$ as an $\bc_*(Y)$-bimodule: |
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|
771 |
\begin{equation*} |
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|
772 |
\bc_*(X\bigcup_Y \selfarrow) \simeq \bc_*(X) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y)}} \selfarrow |
572
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|
773 |
\end{equation*} |
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|
774 |
\end{itemize} |
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|
775 |
\end{thm} |
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|
776 |
|
618 | 777 |
\begin{proof} (Sketch.) |
620
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|
778 |
The $A_\infty$ action of $\bc_*(Y)$ follows from the naturality of the blob complex with respect to gluing |
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|
779 |
and the $C_*(\Homeo(-))$ action of Theorem \ref{thm:evaluation}. |
618 | 780 |
|
620
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|
781 |
Let $T_*$ denote the self tensor product of $\bc_*(X)$, which is a homotopy colimit. |
622
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|
782 |
There is a tautological map from the 0-simplices of $T_*$ to $\bc_*(X\bigcup_Y \selfarrow)$, |
620
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|
783 |
and this map can be extended to a chain map on all of $T_*$ by sending the higher simplices to zero. |
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|
784 |
Constructing a homotopy inverse to this natural map invloves making various choices, but one can show that the |
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|
785 |
choices form contractible subcomplexes and apply the acyclic models theorem. |
618 | 786 |
\end{proof} |
610 | 787 |
|
788 |
We next describe the blob complex for product manifolds, in terms of the $A_\infty$ blob complex of the $A_\infty$ $n$-categories constructed as above. |
|
789 |
||
790 |
\begin{thm}[Product formula] |
|
791 |
\label{thm:product} |
|
792 |
Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
|
793 |
Let $\cC$ be an $n$-category. |
|
794 |
Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
|
795 |
Then |
|
796 |
\[ |
|
797 |
\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
|
798 |
\] |
|
799 |
\end{thm} |
|
800 |
The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps |
|
801 |
(see \cite[\S7.1]{1009.5025}). |
|
802 |
||
620
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|
803 |
\begin{proof} (Sketch.) |
623 | 804 |
The proof is similar to that of the second part of Theorem \ref{thm:gluing}. |
805 |
There is a natural map from the 0-simplices of $\clh{\bc_*(Y;\cC)}(W)$ to $\bc_*(Y\times W; \cC)$, |
|
806 |
given by reinterpreting a decomposition of $W$ labeled by $(n{-}k)$-morphisms of $\bc_*(Y; \cC)$ as a blob |
|
807 |
diagram on $W\times Y$. |
|
808 |
This map can be extended to all of $\clh{\bc_*(Y;\cC)}(W)$ by sending higher simplices to zero. |
|
620
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|
809 |
|
623 | 810 |
To construct the homotopy inverse of the above map one first shows that |
811 |
$\bc_*(Y\times W; \cC)$ is homotopy equivalent to the subcomplex generated by blob diagrams which |
|
812 |
are small with respect any fixed open cover of $Y\times W$. |
|
813 |
For a sufficiently fine open cover the generators of this ``small" blob complex are in the image of the map |
|
814 |
of the previous paragraph, and furthermore the preimage in $\clh{\bc_*(Y;\cC)}(W)$ of such small diagrams |
|
815 |
lie in contractible subcomplexes. |
|
816 |
A standard acyclic models argument now constructs the homotopy inverse. |
|
620
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|
817 |
\end{proof} |
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|
818 |
|
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|
819 |
%\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
572
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|
820 |
|
623 | 821 |
\section{Deligne conjecture for $n$-categories} |
572
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|
822 |
\label{sec:applications} |
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|
823 |
|
625
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|
824 |
Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
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changeset
|
825 |
Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
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changeset
|
826 |
acts on $\bc_*(M)$ and $\bc_*(N)$. |
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changeset
|
827 |
Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
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changeset
|
828 |
from $\bc_*(M)$ to $\bc_*(N)$. |
c6d069b8f931
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changeset
|
829 |
Let $R$ be another $n$-manifold with boundary $-E$. |
c6d069b8f931
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changeset
|
830 |
There is a chain map |
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changeset
|
831 |
\[ |
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changeset
|
832 |
\hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
c6d069b8f931
starting on Deligne section
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parents:
624
diff
changeset
|
833 |
\] |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
834 |
We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
835 |
replaces it with $N$, yielding $N\cup_E R$. |
626 | 836 |
(This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds |
837 |
which share a common boundary.) |
|
627
b0ed73b141d8
finish deligne section; misc
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parents:
626
diff
changeset
|
838 |
In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains". |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
839 |
|
626 | 840 |
Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
841 |
blob complex. |
626 | 842 |
An $n$-dimensional surgery cylinder is |
843 |
defined to be a sequence of mapping cylinders and surgeries (Figure \ref{delfig2}), |
|
844 |
modulo changing the order of distant surgeries, and conjugating a submanifold not modified in a surgery by a homeomorphism. |
|
845 |
One can associated to this data an $(n{+}1)$-manifold with a foliation by intervals, |
|
846 |
and the relations we impose correspond to homeomorphisms of the $(n{+}1)$-manifolds |
|
847 |
which preserve the foliation. |
|
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
848 |
|
626 | 849 |
Surgery cylinders form an operad, by gluing the outer boundary of one cylinder into an inner boundary of another. |
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
850 |
|
572
e0f5ec582725
incorporating statements of results in PNAS article
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parents:
571
diff
changeset
|
851 |
\begin{thm}[Higher dimensional Deligne conjecture] |
e0f5ec582725
incorporating statements of results in PNAS article
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parents:
571
diff
changeset
|
852 |
\label{thm:deligne} |
e0f5ec582725
incorporating statements of results in PNAS article
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parents:
571
diff
changeset
|
853 |
The singular chains of the $n$-dimensional surgery cylinder operad act on blob cochains. |
e0f5ec582725
incorporating statements of results in PNAS article
Scott Morrison <scott@tqft.net>
parents:
571
diff
changeset
|
854 |
\end{thm} |
577
9a60488cd2fc
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Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
855 |
|
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
856 |
More specifically, let $M_0, N_0, \ldots, M_k, N_k$ be $n$-manifolds and let $SC^n_{\overline{M}, \overline{N}}$ |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
857 |
denote the component of the operad with outer boundary $M_0\cup N_0$ and inner boundaries |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
858 |
$M_1\cup N_1,\ldots, M_k\cup N_k$. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
859 |
Then there is a collection of chain maps |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
860 |
\begin{multline*} |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
861 |
C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots \\ |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
862 |
\otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
863 |
\end{multline*} |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
864 |
which satisfy the operad compatibility conditions. |
577
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
575
diff
changeset
|
865 |
|
595
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
866 |
\begin{proof} |
610 | 867 |
We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
868 |
and the action of surgeries is just composition of maps of $A_\infty$-modules. |
|
869 |
We only need to check that the relations of the $n$-SC operad are satisfied. |
|
870 |
This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
|
595
9c708975b61b
making pinched products axioms terser, and writing a short proof of the higher deligne conjecture
Scott Morrison <scott@tqft.net>
parents:
594
diff
changeset
|
871 |
\end{proof} |
577
9a60488cd2fc
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575
diff
changeset
|
872 |
|
627
b0ed73b141d8
finish deligne section; misc
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parents:
626
diff
changeset
|
873 |
Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are 1-balls. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
874 |
We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
875 |
disks operad and $\hom(\bc_*(M_i), \bc_*(N_i))$ is homotopy equivalent to Hochschild cochains. |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
876 |
This special case is just the usual Deligne conjecture |
628
4cce595ae1d3
adding Gerstenhaber-Voronov, explicitly not proving the mapping spaces result, and slight tweaks
Scott Morrison <scott@tqft.net>
parents:
627
diff
changeset
|
877 |
(see \cite{hep-th/9403055, MR1328534, MR1805894, MR1805923, MR2064592}). |
627
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
878 |
|
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
879 |
The general case when $n=1$ goes beyond the original Deligne conjecture, as the $M_i$'s and $N_i$'s |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
880 |
could be disjoint unions of 1-balls and circles, and the surgery cylinders could be high genus surfaces. |
b0ed73b141d8
finish deligne section; misc
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parents:
626
diff
changeset
|
881 |
|
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
882 |
If all of the $M_i$'s and $N_i$'s are $n$-balls, then $SC^n_{\overline{M}, \overline{N}}$ |
b0ed73b141d8
finish deligne section; misc
Kevin Walker <kevin@canyon23.net>
parents:
626
diff
changeset
|
883 |
contains a copy of the little $(n{+}1)$-balls operad. |
b0ed73b141d8
finish deligne section; misc
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parents:
626
diff
changeset
|
884 |
Thus the little $(n{+}1)$-balls operad acts on blob cochains of the $n$-ball. |
b0ed73b141d8
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parents:
626
diff
changeset
|
885 |
|
577
9a60488cd2fc
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parents:
575
diff
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|
886 |
|
566 | 887 |
|
888 |
%% == end of paper: |
|
889 |
||
890 |
%% Optional Materials and Methods Section |
|
891 |
%% The Materials and Methods section header will be added automatically. |
|
892 |
||
893 |
%% Enter any subheads and the Materials and Methods text below. |
|
894 |
%\begin{materials} |
|
895 |
% Materials text |
|
896 |
%\end{materials} |
|
897 |
||
898 |
||
899 |
%% Optional Appendix or Appendices |
|
900 |
%% \appendix Appendix text... |
|
901 |
%% or, for appendix with title, use square brackets: |
|
902 |
%% \appendix[Appendix Title] |
|
903 |
||
904 |
\begin{acknowledgments} |
|
610 | 905 |
It is a pleasure to acknowledge helpful conversations with |
906 |
Kevin Costello, |
|
625
c6d069b8f931
starting on Deligne section
Kevin Walker <kevin@canyon23.net>
parents:
624
diff
changeset
|
907 |
Michael Freedman, |
610 | 908 |
Justin Roberts, |
909 |
and |
|
910 |
Peter Teichner. |
|
911 |
We also thank the Aspen Center for Physics for providing a pleasant and productive |
|
912 |
environment during the last stages of this project. |
|
566 | 913 |
\end{acknowledgments} |
914 |
||
915 |
%% PNAS does not support submission of supporting .tex files such as BibTeX. |
|
916 |
%% Instead all references must be included in the article .tex document. |
|
917 |
%% If you currently use BibTeX, your bibliography is formed because the |
|
918 |
%% command \verb+\bibliography{}+ brings the <filename>.bbl file into your |
|
919 |
%% .tex document. To conform to PNAS requirements, copy the reference listings |
|
920 |
%% from your .bbl file and add them to the article .tex file, using the |
|
921 |
%% bibliography environment described above. |
|
922 |
||
923 |
%% Contact pnas@nas.edu if you need assistance with your |
|
924 |
%% bibliography. |
|
925 |
||
926 |
% Sample bibliography item in PNAS format: |
|
927 |
%% \bibitem{in-text reference} comma-separated author names up to 5, |
|
928 |
%% for more than 5 authors use first author last name et al. (year published) |
|
929 |
%% article title {\it Journal Name} volume #: start page-end page. |
|
930 |
%% ie, |
|
931 |
% \bibitem{Neuhaus} Neuhaus J-M, Sitcher L, Meins F, Jr, Boller T (1991) |
|
932 |
% A short C-terminal sequence is necessary and sufficient for the |
|
933 |
% targeting of chitinases to the plant vacuole. |
|
934 |
% {\it Proc Natl Acad Sci USA} 88:10362-10366. |
|
935 |
||
936 |
||
937 |
%% Enter the largest bibliography number in the facing curly brackets |
|
938 |
%% following \begin{thebibliography} |
|
939 |
||
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|
940 |
%%%% BIBTEX |
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|
941 |
\bibliographystyle{alpha} |
e0f5ec582725
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diff
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|
942 |
\bibliography{../bibliography/bibliography} |
566 | 943 |
|
572
e0f5ec582725
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|
944 |
%%%% non-BIBTEX |
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|
945 |
%\begin{thebibliography}{} |
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|
946 |
% |
e0f5ec582725
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|
947 |
%\end{thebibliography} |
e0f5ec582725
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diff
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|
948 |
|
566 | 949 |
|
950 |
\end{article} |
|
951 |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
952 |
||
953 |
%% Adding Figure and Table References |
|
954 |
%% Be sure to add figures and tables after \end{article} |
|
955 |
%% and before \end{document} |
|
956 |
||
957 |
%% For figures, put the caption below the illustration. |
|
958 |
%% |
|
959 |
%% \begin{figure} |
|
960 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
961 |
%% \end{figure} |
|
962 |
||
573
8378e03d3c7f
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diff
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|
963 |
|
8378e03d3c7f
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Scott Morrison <scott@tqft.net>
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diff
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|
964 |
\begin{figure} |
594
6945422bed13
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Scott Morrison <scott@tqft.net>
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diff
changeset
|
965 |
\centering |
6945422bed13
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Scott Morrison <scott@tqft.net>
parents:
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diff
changeset
|
966 |
\begin{tikzpicture}[%every label/.style={green} |
6945422bed13
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Scott Morrison <scott@tqft.net>
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diff
changeset
|
967 |
] |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
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diff
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|
968 |
\node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; |
6945422bed13
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diff
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|
969 |
\node[fill=black, circle, label=above:$E$, inner sep=1.5pt](N) at (0,2) {}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
970 |
\draw (S) arc (-90:90:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
971 |
\draw (N) arc (90:270:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
972 |
\node[left] at (-1,1) {$B_1$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
973 |
\node[right] at (1,1) {$B_2$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
974 |
\end{tikzpicture} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
975 |
\caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
976 |
|
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
977 |
\begin{figure} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
978 |
\centering |
6945422bed13
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Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
979 |
\begin{tikzpicture}[%every label/.style={green}, |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
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diff
changeset
|
980 |
x=1.5cm,y=1.5cm] |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
981 |
\node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
982 |
\node[fill=black, circle, label=above:$E$, inner sep=2pt](N) at (0,2) {}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
983 |
\draw (S) arc (-90:90:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
984 |
\draw (N) arc (90:270:1); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
985 |
\draw (N) -- (S); |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
986 |
\node[left] at (-1/4,1) {$B_1$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
987 |
\node[right] at (1/4,1) {$B_2$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
988 |
\node at (1/6,3/2) {$Y$}; |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
989 |
\end{tikzpicture} |
6945422bed13
adding some figures for the axioms
Scott Morrison <scott@tqft.net>
parents:
591
diff
changeset
|
990 |
\caption{From two balls to one ball.}\label{blah5}\end{figure} |
6945422bed13
adding some figures for the axioms
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parents:
591
diff
changeset
|
991 |
|
6945422bed13
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Scott Morrison <scott@tqft.net>
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diff
changeset
|
992 |
\begin{figure} |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
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572
diff
changeset
|
993 |
\begin{equation*} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
994 |
\mathfig{.23}{ncat/zz2} |
8378e03d3c7f
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572
diff
changeset
|
995 |
\end{equation*} |
594
6945422bed13
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diff
changeset
|
996 |
\caption{A small part of $\cell(W)$.} |
573
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
997 |
\label{partofJfig} |
8378e03d3c7f
starting on cell decompositions
Scott Morrison <scott@tqft.net>
parents:
572
diff
changeset
|
998 |
\end{figure} |
8378e03d3c7f
starting on cell decompositions
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parents:
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diff
changeset
|
999 |
|
577
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|
1000 |
\begin{figure} |
626 | 1001 |
%$$\mathfig{.4}{deligne/manifolds}$$ |
1002 |
$$\mathfig{.4}{deligne/mapping-cylinders}$$ |
|
594
6945422bed13
adding some figures for the axioms
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diff
changeset
|
1003 |
\caption{An $n$-dimensional surgery cylinder.}\label{delfig2} |
577
9a60488cd2fc
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changeset
|
1004 |
\end{figure} |
9a60488cd2fc
out of battery. writing a little about the deligne conjecture
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parents:
575
diff
changeset
|
1005 |
|
573
8378e03d3c7f
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|
1006 |
|
566 | 1007 |
%% For Tables, put caption above table |
1008 |
%% |
|
1009 |
%% Table caption should start with a capital letter, continue with lower case |
|
1010 |
%% and not have a period at the end |
|
1011 |
%% Using @{\vrule height ?? depth ?? width0pt} in the tabular preamble will |
|
1012 |
%% keep that much space between every line in the table. |
|
1013 |
||
1014 |
%% \begin{table} |
|
1015 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
1016 |
%% \begin{tabular}{@{\vrule height 10.5pt depth4pt width0pt}lrcccc} |
|
1017 |
%% table text |
|
1018 |
%% \end{tabular} |
|
1019 |
%% \end{table} |
|
1020 |
||
1021 |
%% For two column figures and tables, use the following: |
|
1022 |
||
1023 |
%% \begin{figure*} |
|
1024 |
%% \caption{Almost Sharp Front}\label{afoto} |
|
1025 |
%% \end{figure*} |
|
1026 |
||
1027 |
%% \begin{table*} |
|
1028 |
%% \caption{Repeat length of longer allele by age of onset class} |
|
1029 |
%% \begin{tabular}{ccc} |
|
1030 |
%% table text |
|
1031 |
%% \end{tabular} |
|
1032 |
%% \end{table*} |
|
1033 |
||
1034 |
\end{document} |
|
1035 |