22 \subsubsection{Structure of the paper} |
22 \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}). |
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}). |
24 |
24 |
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. |
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. |
26 |
26 |
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 (section \S \ref{sec:ncats}) we pause and give yet another definition of an $n$-category, or rather a definition of an $n$-category with strong duality. (It's not clear that we could remove the duality conditions from our definition, even if we wanted to.) 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. |
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 (section \S \ref{sec:ncats}) we pause and 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. |
28 |
28 |
29 \nn{Not sure that the next para is appropriate here} |
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 group. |
30 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. This vector spaces glue together associatively. For an $A_\infty$ $n$-category, we instead associate a chain complex to each such $B$. We require that homeomorphisms (or the complex of singular chains of homeomorphisms in the $A_\infty$ case) act. The axioms for an $A_\infty$ $n$-category are designed to capture two main examples: the blob complexes of $n$-balls, using a topological $n$-category, and the complex $\CM{-}{X}$ of maps to a fixed target space $X$. |
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 $T$. |
31 \nn{we might want to make our choice of notation here ($B$, $X$) consistent with later sections ($X$, $T$), or vice versa} |
31 |
32 |
32 In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $A_\infty$ $n$-category on an $n$-manifold. 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. |
33 In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category, and give an alternative definition of the blob complex for an $n$-manifold and an $A_\infty$ $n$-category. 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. |
|
34 |
33 |
35 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
34 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
36 |
35 |
37 Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a generalisation of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
36 Finally, later sections address other topics. Section \S \ref{sec:comm_alg} describes the blob complex when $\cC$ is a commutative algebra, thought of as a topological $n$-category, in terms of the topology of $M$. Section \S \ref{sec:deligne} states (and in a later edition of this paper, hopefully proves) a higher dimensional generalisation of the Deligne conjecture (that the little discs operad acts on Hochschild cohomology) in terms of the blob complex. The appendixes prove technical results about $\CH{M}$, and make connections between our definitions of $n$-categories and familar definitions for $n=1$ and $n=2$, as well as relating the $n=1$ case of our $A_\infty$ $n$-categories with usual $A_\infty$ algebras. |
38 |
37 |
39 |
38 |
40 \nn{some more things to cover in the intro} |
39 \nn{some more things to cover in the intro} |
41 \begin{itemize} |
40 \begin{itemize} |
42 \item related: we are being unsophisticated from a homotopy theory point of |
41 \item related: we are being unsophisticated from a homotopy theory point of |
121 We now summarize the results of the paper in the following list of formal properties. |
120 We now summarize the results of the paper in the following list of formal properties. |
122 |
121 |
123 \begin{property}[Functoriality] |
122 \begin{property}[Functoriality] |
124 \label{property:functoriality}% |
123 \label{property:functoriality}% |
125 The blob complex is functorial with respect to homeomorphisms. That is, |
124 The blob complex is functorial with respect to homeomorphisms. That is, |
126 for fixed $n$-category / fields $\cC$, the association |
125 for a fixed $n$-dimensional system of fields $\cC$, the association |
127 \begin{equation*} |
126 \begin{equation*} |
128 X \mapsto \bc_*^{\cC}(X) |
127 X \mapsto \bc_*^{\cC}(X) |
129 \end{equation*} |
128 \end{equation*} |
130 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
129 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
131 \end{property} |
130 \end{property} |
148 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
147 %\gl_Y: \bc_*(X_1) \tensor \bc_*(X_2) \to \bc_*(X_1 \cup_Y X_2). |
149 %\end{equation*} |
148 %\end{equation*} |
150 Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is |
149 Given a gluing $X_\mathrm{cut} \to X_\mathrm{glued}$, there is |
151 a natural map |
150 a natural map |
152 \[ |
151 \[ |
153 \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) . |
152 \bc_*(X_\mathrm{cut}) \to \bc_*(X_\mathrm{glued}) |
154 \] |
153 \] |
155 (Natural with respect to homeomorphisms, and also associative with respect to iterated gluings.) |
154 (natural with respect to homeomorphisms, and also associative with respect to iterated gluings). |
156 \end{property} |
155 \end{property} |
157 |
156 |
158 \begin{property}[Contractibility] |
157 \begin{property}[Contractibility] |
159 \label{property:contractibility}% |
158 \label{property:contractibility}% |
160 \nn{this holds with field coefficients, or more generally when |
159 \nn{this holds with field coefficients, or more generally when |
161 the map to 0-th homology has a splitting; need to fix statement} |
160 the map to 0-th homology has a splitting; need to fix statement} |
162 The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$. |
161 The blob complex on an $n$-ball is contractible in the sense that it is quasi-isomorphic to its $0$-th homology. Moreover, the $0$-th homology of balls can be canonically identified with the original $n$-category $\cC$. |
163 \begin{equation} |
162 \begin{equation} |
164 \xymatrix{\bc_*^{\cC}(B^n) \ar[r]^{\iso}_{\text{qi}} & H_0(\bc_*^{\cC}(B^n))} |
163 \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)} |
165 \end{equation} |
164 \end{equation} |
166 \end{property} |
165 \end{property} |
167 |
166 |
168 \begin{property}[Skein modules] |
167 \begin{property}[Skein modules] |
169 \label{property:skein-modules}% |
168 \label{property:skein-modules}% |
178 \begin{property}[Hochschild homology when $X=S^1$] |
177 \begin{property}[Hochschild homology when $X=S^1$] |
179 \label{property:hochschild}% |
178 \label{property:hochschild}% |
180 The blob complex for a $1$-category $\cC$ on the circle is |
179 The blob complex for a $1$-category $\cC$ on the circle is |
181 quasi-isomorphic to the Hochschild complex. |
180 quasi-isomorphic to the Hochschild complex. |
182 \begin{equation*} |
181 \begin{equation*} |
183 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)} |
182 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
184 \end{equation*} |
183 \end{equation*} |
185 \end{property} |
184 \end{property} |
186 |
185 |
187 Here $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
186 Here $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
188 \begin{property}[$C_*(\Homeo(-))$ action] |
187 \begin{property}[$C_*(\Homeo(-))$ action] |
218 \end{equation*} |
216 \end{equation*} |
219 \end{property} |
217 \end{property} |
220 |
218 |
221 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category. |
219 In \S \ref{sec:ncats} we introduce the notion of topological $n$-categories, from which we can construct systems of fields, as well as the notion of an $A_\infty$ $n$-category. |
222 |
220 |
223 \begin{property}[Blob complexes of balls form an $A_\infty$ $n$-category] |
221 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
224 \label{property:blobs-ainfty} |
222 \label{property:blobs-ainfty} |
225 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
223 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
226 Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m \leq k$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category. |
224 Define $A_*(Y, \cC)$ on each $m$-ball $D$, for $0 \leq m < k$, to be the set $$A_*(Y,\cC)(D) = A^\cC(Y \times D)$$ and on $k$-balls $D$ to be the set $$A_*(Y, \cC)(D) = \bc_*(Y \times D, \cC).$$ (When $m=k$ the subsets with fixed boundary conditions form a chain complex.) These sets have the structure of an $A_\infty$ $k$-category. |
227 \nn{the subscript * is only appropriate when $m=k$. } |
|
228 \end{property} |
225 \end{property} |
229 \begin{rem} |
226 \begin{rem} |
230 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. |
227 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. |
231 \end{rem} |
228 \end{rem} |
232 |
229 |
233 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
230 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
234 instead of a garden variety $n$-category; this is described in \S \ref{sec:ainfblob}. |
231 instead of a garden variety $n$-category; this is described in \S \ref{sec:ainfblob}. |
235 |
232 |
236 \begin{property}[Product formula] |
233 \begin{property}[Product formula] |
237 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
234 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
238 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
235 Let $A_*(Y,\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
239 Then |
236 Then |
240 \[ |
237 \[ |
241 \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y)) . |
238 \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y,\cC)) . |
242 \] |
239 \] |
243 Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories. |
240 Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories. |
244 \end{property} |
241 \end{property} |
245 It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework as such a statement. |
242 It seems reasonable to expect a generalization describing an arbitrary fibre bundle. See in particular \S \ref{moddecss} for the framework for such a statement. |
246 |
243 |
247 \begin{property}[Gluing formula] |
244 \begin{property}[Gluing formula] |
248 \label{property:gluing}% |
245 \label{property:gluing}% |
249 \mbox{}% <-- gets the indenting right |
246 \mbox{}% <-- gets the indenting right |
250 \begin{itemize} |
247 \begin{itemize} |
260 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow |
257 \bc_*(X_\text{glued}) \simeq \bc_*(X_\text{cut}) \Tensor^{A_\infty}_{\mathclap{\bc_*(Y \times I)}} \selfarrow |
261 \end{equation*} |
258 \end{equation*} |
262 \end{itemize} |
259 \end{itemize} |
263 \end{property} |
260 \end{property} |
264 |
261 |
265 |
262 Finally, we state two more properties, which we will not prove in this paper. |
266 |
263 |
267 \begin{property}[Mapping spaces] |
264 \begin{property}[Mapping spaces] |
268 Let $\pi^\infty_{\le n}(W)$ denote the $A_\infty$ $n$-category based on maps |
265 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps |
269 $B^n \to W$. |
266 $B^n \to T$. |
270 (The case $n=1$ is the usual $A_\infty$ category of paths in $W$.) |
267 (The case $n=1$ is the usual $A_\infty$ category of paths in $T$.) |
271 Then |
268 Then |
272 $$\bc_*(M, \pi^\infty_{\le n}(W) \simeq \CM{M}{W}.$$ |
269 $$\bc_*(X, \pi^\infty_{\le n}(T)) \simeq \CM{X}{T}.$$ |
273 \end{property} |
270 \end{property} |
|
271 |
|
272 This says that we can recover the (homotopic) space of maps to $T$ via blob homology from local data. |
274 |
273 |
275 \begin{property}[Higher dimensional Deligne conjecture] |
274 \begin{property}[Higher dimensional Deligne conjecture] |
276 \label{property:deligne} |
275 \label{property:deligne} |
277 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
276 The singular chains of the $n$-dimensional fat graph operad act on blob cochains. |
278 \end{property} |
277 \end{property} |
279 See \S \ref{sec:deligne} for an explanation of the terms appearing here, and (in a later edition of this paper) the proof. |
278 See \S \ref{sec:deligne} for an explanation of the terms appearing here. The proof will appear elsewhere. |
280 |
|
281 |
279 |
282 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
280 Properties \ref{property:functoriality}, \ref{property:gluing-map} and \ref{property:skein-modules} will be immediate from the definition given in |
283 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
281 \S \ref{sec:blob-definition}, and we'll recall them at the appropriate points there. \todo{Make sure this gets done.} |
284 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
282 Properties \ref{property:disjoint-union} and \ref{property:contractibility} are established in \S \ref{sec:basic-properties}. |
285 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |
283 Property \ref{property:hochschild} is established in \S \ref{sec:hochschild}, Property \ref{property:evaluation} in \S \ref{sec:evaluation}, |