1 %!TEX root = ../blob1.tex |
1 %!TEX root = ../blob1.tex |
2 |
2 |
3 \section{Introduction} |
3 \section{Introduction} |
4 |
4 |
5 [some things to cover in the intro] |
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: |
6 \begin{itemize} |
6 \begin{itemize} |
7 \item explain relation between old and new blob complex definitions |
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 \S \ref{sec:fields} \nn{more specific}.) |
8 \item overview of sections |
8 \item When $n=1$, $\cC$ is just an associative algebroid, and $\bc_*(S^1; \cC)$ is quasi-isomorphic to the Hochschild complex $\HC_*(\cC)$. (See \S \ref{sec:hochschild}.) |
9 \item state main properties of blob complex (already mostly done below) |
9 \item When $\cC = k[t]$, thought of as an n-category, we have $$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ (See \S \ref{sec:comm_alg}.) |
10 \item give multiple motivations/viewpoints for blob complex: (1) derived cat |
10 \end{itemize} |
|
11 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CD{M}$, extending the usual $\Diff(M)$ 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}). |
|
12 |
|
13 The blob complex definition is motivated by \nn{ continue here ...} give multiple motivations/viewpoints for blob complex: (1) derived cat |
11 version of TQFT Hilbert space; (2) generalization of Hochschild homology to higher $n$-cats; |
14 version of TQFT Hilbert space; (2) generalization of Hochschild homology to higher $n$-cats; |
12 (3) ? sort-of-obvious colimit type construction; |
15 (3) ? sort-of-obvious colimit type construction; |
13 (4) ? a generalization of $C_*(\Maps(M, T))$ to the case where $T$ is |
16 (4) ? a generalization of $C_*(\Maps(M, T))$ to the case where $T$ is |
14 a category rather than a manifold |
17 a category rather than a manifold |
15 \item hope to apply to Kh, contact, (other examples?) in the future |
18 |
|
19 We expect applications of the blob complex to \nn{ ... } but do not address these in this paper. |
|
20 \nn{hope to apply to Kh, contact, (other examples?) in the future} |
|
21 |
|
22 |
|
23 \subsubsection{Structure of the paper} |
|
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. |
|
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. When $n=1$ these reduce to the usual $A_\infty$ categories. |
|
28 |
|
29 In the third part of the paper (section \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. |
|
30 |
|
31 |
|
32 [some things to cover in the intro] |
|
33 \begin{itemize} |
|
34 \item explain relation between old and new blob complex definitions |
|
35 \item overview of sections |
16 \item ?? we have resisted the temptation |
36 \item ?? we have resisted the temptation |
17 (actually, it was not a temptation) to state things in the greatest |
37 (actually, it was not a temptation) to state things in the greatest |
18 generality possible |
38 generality possible |
19 \item related: we are being unsophisticated from a homotopy theory point of |
39 \item related: we are being unsophisticated from a homotopy theory point of |
20 view and using chain complexes in many places where we could be by with spaces |
40 view and using chain complexes in many places where we could be by with spaces |
147 \begin{equation*} |
169 \begin{equation*} |
148 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)} |
170 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)} |
149 \end{equation*} |
171 \end{equation*} |
150 \end{property} |
172 \end{property} |
151 |
173 |
152 |
174 Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$. |
153 \begin{property}[$C_*(\Diff(-))$ action] |
175 \begin{property}[$C_*(\Diff(-))$ action] |
154 \label{property:evaluation}% |
176 \label{property:evaluation}% |
155 There is a chain map |
177 There is a chain map |
156 \begin{equation*} |
178 \begin{equation*} |
157 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
179 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
158 \end{equation*} |
180 \end{equation*} |
159 (Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$.) |
|
160 |
181 |
161 Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
182 Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
162 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
183 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
163 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
184 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
164 \begin{equation*} |
185 \begin{equation*} |
168 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
189 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
169 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
190 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
170 } |
191 } |
171 \end{equation*} |
192 \end{equation*} |
172 \nn{should probably say something about associativity here (or not?)} |
193 \nn{should probably say something about associativity here (or not?)} |
173 \nn{maybe do self-gluing instead of 2 pieces case} |
194 \nn{maybe do self-gluing instead of 2 pieces case:} |
|
195 Further, for |
|
196 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
|
197 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
|
198 \begin{equation*} |
|
199 \xymatrix@C+2cm{ |
|
200 \CD{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) \\ |
|
201 \CD{X} \otimes \bc_*(X) |
|
202 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
|
203 \bc_*(X) \ar[u]_{\gl_Y} |
|
204 } |
|
205 \end{equation*} |
174 \end{property} |
206 \end{property} |
175 |
207 |
176 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
208 There is a version of the blob complex for $\cC$ an $A_\infty$ $n$-category |
177 instead of a garden variety $n$-category. |
209 instead of a garden variety $n$-category; this is described in \S \ref{sec:ainfblob}. |
178 |
210 |
179 \begin{property}[Product formula] |
211 \begin{property}[Product formula] |
180 Let $M^n = Y^{n-k}\times W^k$ and let $\cC$ be an $n$-category. |
212 Let $M^n = Y^{n-k}\times W^k$ and let $\cC$ be an $n$-category. |
181 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology, which associates to each $k$-ball $D$ the complex $A_*(Y)(D) = \bc_*(Y \times D, \cC)$. |
213 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology, which associates to each $k$-ball $D$ the complex $A_*(Y)(D) = \bc_*(Y \times D, \cC)$. |
182 Then |
214 Then |