1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Introduction} |
3 \section{Introduction} |
4 |
4 |
5 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. This blob complex provides a simultaneous generalisation of several well-understood constructions: |
5 We construct the ``blob complex'' $\bc_*(M; \cC)$ associated to an $n$-manifold $M$ and a ``linear $n$-category with strong duality'' $\cC$. |
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6 This blob complex provides a simultaneous generalisation of several well-understood constructions: |
6 \begin{itemize} |
7 \begin{itemize} |
7 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
8 \item The vector space $H_0(\bc_*(M; \cC))$ is isomorphic to the usual topological quantum field theory invariant of $M$ associated to $\cC$. |
8 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) |
9 (See Property \ref{property:skein-modules} later in the introduction and \S \ref{sec:constructing-a-tqft}.) |
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10 \item When $n=1$ and $\cC$ is just a 1-category (e.g.\ an associative algebra), |
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11 the blob complex $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. |
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12 (See Property \ref{property:hochschild} and \S \ref{sec:hochschild}.) |
9 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
13 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
10 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
14 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
11 on the configuration space of unlabeled points in $M$. |
15 on the configuration space of unlabeled points in $M$. |
12 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
16 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
13 \end{itemize} |
17 \end{itemize} |
14 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space (replacing quotient of fields by local relations with some sort of resolution), |
18 The blob complex definition is motivated by the desire for a derived analogue of the usual TQFT Hilbert space |
15 and for a generalization of Hochschild homology to higher $n$-categories. We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
19 (replacing quotient of fields by local relations with some sort of resolution), |
16 |
20 and for a generalization of Hochschild homology to higher $n$-categories. |
17 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$, |
21 We would also like to be able to talk about $\CM{M}{T}$ when $T$ is an $n$-category rather than a manifold. |
18 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}). |
22 The blob complex gives us all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
19 |
23 |
20 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. See \S \ref{sec:future} for slightly more detail. |
24 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. |
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25 These include an action of $\CH{M}$, |
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26 extending the usual $\Homeo(M)$ action on the TQFT space $H_0$ (see Property \ref{property:evaluation}) and a gluing |
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27 formula allowing calculations by cutting manifolds into smaller parts (see Property \ref{property:gluing}). |
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28 |
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29 We expect applications of the blob complex to contact topology and Khovanov homology but do not address these in this paper. |
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30 See \S \ref{sec:future} for slightly more detail. |
21 |
31 |
22 \subsubsection{Structure of the paper} |
32 \subsubsection{Structure of the paper} |
23 The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), summarise the formal properties of the blob complex (see \S \ref{sec:properties}) and outline anticipated future directions and applications (see \S \ref{sec:future}). |
33 The three subsections of the introduction explain our motivations in defining the blob complex (see \S \ref{sec:motivations}), |
24 |
34 summarise the formal properties of the blob complex (see \S \ref{sec:properties}) |
25 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, and establishes some of its properties. There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex associated to an $n$-manifold and an $n$-dimensional system of fields. We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. |
35 and outline anticipated future directions and applications (see \S \ref{sec:future}). |
26 |
36 |
27 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, we find this situation unsatisfactory. Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) We call these ``topological $n$-categories'', to differentiate them from previous versions. Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
37 The first part of the paper (sections \S \ref{sec:fields}---\S \ref{sec:evaluation}) gives the definition of the blob complex, |
28 |
38 and establishes some of its properties. |
29 The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
39 There are many alternative definitions of $n$-categories, and part of our difficulty defining the blob complex is |
30 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
40 simply explaining what we mean by an ``$n$-category with strong duality'' as one of the inputs. |
31 |
41 At first we entirely avoid this problem by introducing the notion of a `system of fields', and define the blob complex |
32 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), in particular the `gluing formula' of Property \ref{property:gluing} below. |
42 associated to an $n$-manifold and an $n$-dimensional system of fields. |
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43 We sketch the construction of a system of fields from a $1$-category or from a pivotal $2$-category. |
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44 |
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45 Nevertheless, when we attempt to establish all of the observed properties of the blob complex, |
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46 we find this situation unsatisfactory. |
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47 Thus, in the second part of the paper (\S\S \ref{sec:ncats}-\ref{sec:ainfblob}) we give yet another |
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48 definition of an $n$-category, or rather a definition of an $n$-category with strong duality. |
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49 (It appears that removing the duality conditions from our definition would make it more complicated rather than less.) |
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50 We call these ``topological $n$-categories'', to differentiate them from previous versions. |
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51 Moreover, we find that we need analogous $A_\infty$ $n$-categories, and we define these as well following very similar axioms. |
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52 |
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53 The basic idea is that each potential definition of an $n$-category makes a choice about the `shape' of morphisms. |
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54 We try to be as lax as possible: a topological $n$-category associates a vector space to every $B$ homeomorphic to the $n$-ball. |
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55 These vector spaces glue together associatively, and we require that there is an action of the homeomorphism groupoid. |
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56 For an $A_\infty$ $n$-category, we associate a chain complex instead of a vector space to each such $B$ and ask that the action of |
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57 homeomorphisms extends to a suitably defined action of the complex of singular chains of homeomorphisms. |
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58 The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls labelled by a |
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59 topological $n$-category, and the complex $\CM{-}{T}$ of maps to a fixed target space $T$. |
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60 |
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61 In \S \ref{ss:ncat_fields} we explain how to construct a system of fields from a topological $n$-category |
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62 (using a colimit along cellulations of a manifold), and in \S \ref{sec:ainfblob} give an alternative definition |
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63 of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold (analogously, using a homotopy colimit). |
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64 Using these definitions, we show how to use the blob complex to `resolve' any topological $n$-category as an |
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65 $A_\infty$ $n$-category, and relate the first and second definitions of the blob complex. |
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66 We use the blob complex for $A_\infty$ $n$-categories to establish important properties of the blob complex (in both variants), |
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67 in particular the `gluing formula' of Property \ref{property:gluing} below. |
33 |
68 |
34 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
69 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
35 |
70 |
36 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
71 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
37 |
72 |
163 \label{sec:properties} |
205 \label{sec:properties} |
164 We now summarize the results of the paper in the following list of formal properties. |
206 We now summarize the results of the paper in the following list of formal properties. |
165 |
207 |
166 \begin{property}[Functoriality] |
208 \begin{property}[Functoriality] |
167 \label{property:functoriality}% |
209 \label{property:functoriality}% |
168 The blob complex is functorial with respect to homeomorphisms. That is, |
210 The blob complex is functorial with respect to homeomorphisms. |
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211 That is, |
169 for a fixed $n$-dimensional system of fields $\cC$, the association |
212 for a fixed $n$-dimensional system of fields $\cC$, the association |
170 \begin{equation*} |
213 \begin{equation*} |
171 X \mapsto \bc_*^{\cC}(X) |
214 X \mapsto \bc_*^{\cC}(X) |
172 \end{equation*} |
215 \end{equation*} |
173 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
216 is a functor from $n$-manifolds and homeomorphisms between them to chain |
174 \end{property} |
217 complexes and isomorphisms between them. |
175 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below. |
218 \end{property} |
176 |
219 As a consequence, there is an action of $\Homeo(X)$ on the chain complex $\bc_*^\cC(X)$; |
177 The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here. |
220 this action is extended to all of $C_*(\Homeo(X))$ in Property \ref{property:evaluation} below. |
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221 |
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222 The blob complex is also functorial (indeed, exact) with respect to $\cC$, |
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223 although we will not address this in detail here. |
178 |
224 |
179 \begin{property}[Disjoint union] |
225 \begin{property}[Disjoint union] |
180 \label{property:disjoint-union} |
226 \label{property:disjoint-union} |
181 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
227 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
182 \begin{equation*} |
228 \begin{equation*} |
183 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
229 \bc_*(X_1 \du X_2) \iso \bc_*(X_1) \tensor \bc_*(X_2) |
184 \end{equation*} |
230 \end{equation*} |
185 \end{property} |
231 \end{property} |
186 |
232 |
187 If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. Note that this includes the case of gluing two disjoint manifolds together. |
233 If an $n$-manifold $X_\text{cut}$ contains $Y \sqcup Y^\text{op}$ as a codimension $0$ submanifold of its boundary, |
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234 write $X_\text{glued} = X_\text{cut} \bigcup_{Y}\selfarrow$ for the manifold obtained by gluing together $Y$ and $Y^\text{op}$. |
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235 Note that this includes the case of gluing two disjoint manifolds together. |
188 \begin{property}[Gluing map] |
236 \begin{property}[Gluing map] |
189 \label{property:gluing-map}% |
237 \label{property:gluing-map}% |
190 %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 |
238 %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 |
191 %\begin{equation*} |
239 %\begin{equation*} |
192 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
240 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
199 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
247 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
200 \end{property} |
248 \end{property} |
201 |
249 |
202 \begin{property}[Contractibility] |
250 \begin{property}[Contractibility] |
203 \label{property:contractibility}% |
251 \label{property:contractibility}% |
204 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
252 With field coefficients, the blob complex on an $n$-ball is contractible in the sense that it is homotopic to its $0$-th homology. |
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253 Moreover, the $0$-th homology of balls can be canonically identified with the vector spaces associated by the system of fields $\cC$ to balls. |
205 \begin{equation} |
254 \begin{equation} |
206 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
255 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & H_0(\bc_*^{\cC}(B^n)) \ar[r]^(0.6)\iso & \cC(B^n)} |
207 \end{equation} |
256 \end{equation} |
208 \end{property} |
257 \end{property} |
209 |
258 |
210 \begin{property}[Skein modules] |
259 \begin{property}[Skein modules] |
211 \label{property:skein-modules}% |
260 \label{property:skein-modules}% |
212 The $0$-th blob homology of $X$ is the usual |
261 The $0$-th blob homology of $X$ is the usual |
213 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
262 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
214 by $\cC$. (See \S \ref{sec:local-relations}.) |
263 by $\cC$. |
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264 (See \S \ref{sec:local-relations}.) |
215 \begin{equation*} |
265 \begin{equation*} |
216 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
266 H_0(\bc_*^{\cC}(X)) \iso A^{\cC}(X) |
217 \end{equation*} |
267 \end{equation*} |
218 \end{property} |
268 \end{property} |
219 |
269 |
220 \todo{Somehow, the Hochschild homology thing isn't a "property". Let's move it and call it a theorem? -S} |
270 \todo{Somehow, the Hochschild homology thing isn't a "property". |
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271 Let's move it and call it a theorem? -S} |
221 \begin{property}[Hochschild homology when $X=S^1$] |
272 \begin{property}[Hochschild homology when $X=S^1$] |
222 \label{property:hochschild}% |
273 \label{property:hochschild}% |
223 The blob complex for a $1$-category $\cC$ on the circle is |
274 The blob complex for a $1$-category $\cC$ on the circle is |
224 quasi-isomorphic to the Hochschild complex. |
275 quasi-isomorphic to the Hochschild complex. |
225 \begin{equation*} |
276 \begin{equation*} |
264 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
316 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
265 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
317 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
266 for any homeomorphic pair $X$ and $Y$, |
318 for any homeomorphic pair $X$ and $Y$, |
267 satisfying corresponding conditions. |
319 satisfying corresponding conditions. |
268 |
320 |
269 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
321 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields. |
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322 Below, we talk about the blob complex associated to a topological $n$-category, implicitly passing first to the system of fields. |
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323 Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
270 |
324 |
271 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
325 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
272 \label{property:blobs-ainfty} |
326 \label{property:blobs-ainfty} |
273 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
327 Let $\cC$ be a topological $n$-category. |
274 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. |
328 Let $Y$ be an $n{-}k$-manifold. |
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329 There is an $A_\infty$ $k$-category $\bc_*(Y;\cC)$, defined on each $m$-ball $D$, for $0 \leq m < k$, |
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330 to be the set $$\bc_*(Y;\cC)(D) = \cC(Y \times D)$$ and on $k$-balls $D$ to be the set |
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331 $$\bc_*(Y;\cC)(D) = \bc_*(Y \times D; \cC).$$ |
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332 (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) |
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333 These sets have the structure of an $A_\infty$ $k$-category, with compositions coming from the gluing map in |
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334 Property \ref{property:gluing-map} and with the action of families of homeomorphisms given in Property \ref{property:evaluation}. |
275 \end{property} |
335 \end{property} |
276 \begin{rem} |
336 \begin{rem} |
277 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. We think of this $A_\infty$ $n$-category as a free resolution. |
337 Perhaps the most interesting case is when $Y$ is just a point; then we have a way of building an $A_\infty$ $n$-category from a topological $n$-category. |
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338 We think of this $A_\infty$ $n$-category as a free resolution. |
278 \end{rem} |
339 \end{rem} |
279 |
340 |
280 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
341 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
281 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
342 instead of a topological $n$-category; this is described in \S \ref{sec:ainfblob}. |
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343 The definition is in fact simpler, almost tautological, and we use a different notation, $\cl{\cC}(M)$. |
282 |
344 |
283 \begin{property}[Product formula] |
345 \begin{property}[Product formula] |
284 \label{property:product} |
346 \label{property:product} |
285 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
347 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. |
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348 Let $\cC$ be an $n$-category. |
286 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
349 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
287 Then |
350 Then |
288 \[ |
351 \[ |
289 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
352 \bc_*(Y\times W; \cC) \simeq \cl{\bc_*(Y;\cC)}(W). |
290 \] |
353 \] |
291 \end{property} |
354 \end{property} |
292 We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
355 We also give a generalization of this statement for arbitrary fibre bundles, in \S \ref{moddecss}, and a sketch of a statement for arbitrary maps. |
293 |
356 |
294 Fix a topological $n$-category $\cC$, which we'll omit from the notation. Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
357 Fix a topological $n$-category $\cC$, which we'll omit from the notation. |
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358 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category. |
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359 (See Appendix \ref{sec:comparing-A-infty} for the translation between topological $A_\infty$ $1$-categories and the usual algebraic notion of an $A_\infty$ category.) |
295 |
360 |
296 \begin{property}[Gluing formula] |
361 \begin{property}[Gluing formula] |
297 \label{property:gluing}% |
362 \label{property:gluing}% |
298 \mbox{}% <-- gets the indenting right |
363 \mbox{}% <-- gets the indenting right |
299 \begin{itemize} |
364 \begin{itemize} |
327 See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
392 See \S \ref{sec:deligne} for a full explanation of the statement, and an outline of the proof. |
328 |
393 |
329 Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in |
394 Properties \ref{property:functoriality} and \ref{property:skein-modules} will be immediate from the definition given in |
330 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. |
395 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. |
331 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
396 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
332 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
397 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} |
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398 in \S \ref{sec:evaluation}, Property \ref{property:blobs-ainfty} as Example \ref{ex:blob-complexes-of-balls} in \S \ref{sec:ncats}, |
333 and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}. |
399 and Properties \ref{property:product} and \ref{property:gluing} in \S \ref{sec:ainfblob} as consequences of Theorem \ref{product_thm}. |
334 |
400 |
335 \subsection{Future directions} |
401 \subsection{Future directions} |
336 \label{sec:future} |
402 \label{sec:future} |
337 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
403 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
338 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
404 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
339 More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). Much more could be said about other types of manifolds, in particular oriented, $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; there may be some differences for topological manifolds and smooth manifolds. |
405 We could presumably also replace many of our chain complexes with topological spaces (or indeed, work at the generality of model categories), |
340 |
406 and likely it will prove useful to think about the connections between what we do here and $(\infty,k)$-categories. |
341 The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be interesting to investigate if there is a connection with the material here. |
407 More could be said about finite characteristic |
342 |
408 (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$, for example). |
343 Many results in Hochschild homology can be understood `topologically' via the blob complex. For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details. |
409 Much more could be said about other types of manifolds, in particular oriented, |
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410 $\operatorname{Spin}$ and $\operatorname{Pin}^{\pm}$ manifolds, where boundary issues become more complicated. |
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411 (We'd recommend thinking about boundaries as germs, rather than just codimension $1$ manifolds.) |
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412 We've also take the path of least resistance by considering $\operatorname{PL}$ manifolds; |
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413 there may be some differences for topological manifolds and smooth manifolds. |
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414 |
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415 The paper ``Skein homology'' \cite{MR1624157} has similar motivations, and it may be |
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416 interesting to investigate if there is a connection with the material here. |
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417 |
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418 Many results in Hochschild homology can be understood `topologically' via the blob complex. |
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419 For example, we expect that the shuffle product on the Hochschild homology of a commutative algebra $A$ |
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420 (see \cite[\S 4.2]{MR1600246}) simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, |
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421 but haven't investigated the details. |
344 |
422 |
345 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
423 Most importantly, however, \nn{applications!} \nn{cyclic homology, $n=2$ cases, contact, Kh} |
346 |
424 |
347 |
425 |
348 \subsection{Thanks and acknowledgements} |
426 \subsection{Thanks and acknowledgements} |