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 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. |
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 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. |
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. |
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 $T$. |
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 |
31 |
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. |
32 In \S \ref{sec:ainfblob} we explain how to construct a system of fields from a topological $n$-category (using a colimit along cellulations of a manifold), and 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. |
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33 |
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34 The relationship between all these ideas is sketched in Figure \ref{fig:outline}. |
33 |
35 |
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 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
35 |
37 |
36 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
38 \tikzstyle{box} = [rectangle, rounded corners, draw,outer sep = 5pt, inner sep = 5pt, line width=0.5pt] |
37 |
39 |
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40 \begin{figure}[!ht] |
38 {\center |
41 {\center |
39 |
42 |
40 \begin{tikzpicture}[align=center,line width = 1.5pt] |
43 \begin{tikzpicture}[align=center,line width = 1.5pt] |
41 \newcommand{\xa}{2} |
44 \newcommand{\xa}{2} |
42 \newcommand{\xb}{10} |
45 \newcommand{\xb}{10} |
67 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
70 \draw[->] (FU) -- node[left] {blob complex \\ for balls} (Cs); |
68 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
71 \draw (BC) -- node[right] {$\iso$ by \\ Corollary \ref{cor:new-old}} (BCs); |
69 \end{tikzpicture} |
72 \end{tikzpicture} |
70 |
73 |
71 } |
74 } |
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75 \caption{The main gadgets and constructions of the paper.} |
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76 \label{fig:outline} |
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77 \end{figure} |
72 |
78 |
73 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 generalization 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 familiar 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. |
79 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 generalization 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 familiar 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. |
74 |
80 |
75 |
81 |
76 \nn{some more things to cover in the intro} |
82 \nn{some more things to cover in the intro} |
165 \end{equation*} |
171 \end{equation*} |
166 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
172 is a functor from $n$-manifolds and homeomorphisms between them to chain complexes and isomorphisms between them. |
167 \end{property} |
173 \end{property} |
168 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. |
174 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. |
169 |
175 |
170 The blob complex is also functorial with respect to $\cC$, although we will not address this in detail here. \todo{exact w.r.t $\cC$?} |
176 The blob complex is also functorial (indeed, exact) with respect to $\cC$, although we will not address this in detail here. |
171 |
177 |
172 \begin{property}[Disjoint union] |
178 \begin{property}[Disjoint union] |
173 \label{property:disjoint-union} |
179 \label{property:disjoint-union} |
174 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
180 The blob complex of a disjoint union is naturally the tensor product of the blob complexes. |
175 \begin{equation*} |
181 \begin{equation*} |
218 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
224 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^(0.4){\iso}_(0.4){\text{qi}} & \HC_*(\cC).} |
219 \end{equation*} |
225 \end{equation*} |
220 \end{property} |
226 \end{property} |
221 |
227 |
222 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
228 In the following $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
223 \begin{property}[$C_*(\Homeo(-))$ action] |
229 \begin{property}[$C_*(\Homeo(-))$ action]\mbox{}\\ |
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230 \vspace{-0.5cm} |
224 \label{property:evaluation}% |
231 \label{property:evaluation}% |
225 There is a chain map |
232 \begin{enumerate} |
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233 \item There is a chain map |
226 \begin{equation*} |
234 \begin{equation*} |
227 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
235 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
228 \end{equation*} |
236 \end{equation*} |
229 |
237 |
230 Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. |
238 \item Restricted to $C_0(\Homeo(X))$ this is the action of homeomorphisms described in Property \ref{property:functoriality}. |
231 \nn{should probably say something about associativity here (or not?)} |
239 |
232 |
240 \item For |
233 For |
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234 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
241 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
235 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
242 (using the gluing maps described in Property \ref{property:gluing-map}) commutes (up to homotopy). |
236 \begin{equation*} |
243 \begin{equation*} |
237 \xymatrix@C+2cm{ |
244 \xymatrix@C+2cm{ |
238 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\ |
245 \CH{X \bigcup_Y \selfarrow} \otimes \bc_*(X \bigcup_Y \selfarrow) \ar[r]^<<<<<<<<<<<<{\ev_{(X \bigcup_Y \scalebox{0.5}{\selfarrow})}} & \bc_*(X \bigcup_Y \selfarrow) \\ |
239 \CH{X} \otimes \bc_*(X) |
246 \CH{X} \otimes \bc_*(X) |
240 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
247 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
241 \bc_*(X) \ar[u]_{\gl_Y} |
248 \bc_*(X) \ar[u]_{\gl_Y} |
242 } |
249 } |
243 \end{equation*} |
250 \end{equation*} |
244 |
251 \item Any such chain map satisfying points 2. and 3. above is unique, up to an iterated homotopy. (That is, any pair of homotopies have a homotopy between them, and so on.) |
245 \nn{unique up to homotopy?} |
252 \item This map is associative, in the sense that the following diagram commutes (up to homotopy). |
246 \end{property} |
253 \begin{equation*} |
247 |
254 \xymatrix{ |
248 Since the blob complex is functorial in the manifold $X$, we can use this to build a chain map |
255 \CH{X} \tensor \CH{X} \tensor \bc_*(X) \ar[r]^<<<<<{\id \tensor \ev_X} \ar[d]^{\compose \tensor \id} & \CH{X} \tensor \bc_*(X) \ar[d]^{\ev_X} \\ |
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256 \CH{X} \tensor \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) |
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257 } |
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258 \end{equation*} |
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259 \end{enumerate} |
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260 \end{property} |
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261 |
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262 Since the blob complex is functorial in the manifold $X$, this is equivalent to having chain maps |
249 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
263 $$ev_{X \to Y} : \CH{X \to Y} \tensor \bc_*(X) \to \bc_*(Y)$$ |
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264 for any homeomorphic pair $X$ and $Y$, |
250 satisfying corresponding conditions. |
265 satisfying corresponding conditions. |
251 |
266 |
252 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 to the system of fields. Further, in \S \ref{sec:ncats} we also have the notion of an $A_\infty$ $n$-category. |
267 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. |
253 |
268 |
254 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
269 \begin{property}[Blob complexes of (products with) balls form an $A_\infty$ $n$-category] |
255 \label{property:blobs-ainfty} |
270 \label{property:blobs-ainfty} |
256 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
271 Let $\cC$ be a topological $n$-category. Let $Y$ be an $n{-}k$-manifold. |
257 There is an $A_\infty$ $k$-category $A_*(Y, \cC)$, defined 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, 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}. |
272 There is an $A_\infty$ $k$-category $A_*(Y, \cC)$, defined 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, 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}. |