9 \item When $\cC$ is the polynomial algebra $k[t]$, thought of as an n-category (see \S \ref{sec:comm_alg}), we have |
9 \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 |
10 that $\bc_*(M; k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M), k)$, the singular chains |
11 on the configurations space of unlabeled points in $M$. |
11 on the configurations space of unlabeled points in $M$. |
12 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
12 %$$H_*(\bc_*(M; k[t])) = H^{\text{sing}}_*(\Delta^\infty(M), k).$$ |
13 \end{itemize} |
13 \end{itemize} |
14 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CD{M}$, |
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), |
15 \nn{maybe replace Diff with Homeo?} |
|
16 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}). |
|
17 |
|
18 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'), |
|
19 \nn{are the quotes around `derived' and `resolution' necessary?} |
|
20 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 allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
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 allows us to do all of these! More detailed motivations are described in \S \ref{sec:motivations}. |
|
16 |
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17 The blob complex has good formal properties, summarized in \S \ref{sec:properties}. These include an action of $\CH{M}$, |
|
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}). |
21 |
19 |
22 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. |
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. |
23 |
21 |
24 \subsubsection{Structure of the paper} |
22 \subsubsection{Structure of the paper} |
25 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}). |
27 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. |
28 |
26 |
29 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'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. |
30 |
28 |
31 \nn{Not sure that the next para is appropriate here} |
29 \nn{Not sure that the next para is appropriate here} |
32 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 $B$ diffeomorphic 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 diffeomorphisms (or the complex of singular chains of diffeomorphisms 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 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 $X$. |
33 \nn{we might want to make our choice of notation here ($B$, $X$) consistent with later sections ($X$, $T$), or vice versa} |
31 \nn{we might want to make our choice of notation here ($B$, $X$) consistent with later sections ($X$, $T$), or vice versa} |
34 |
32 |
35 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. |
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. |
36 |
34 |
37 \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 \nn{KW: the previous two paragraphs seem a little awkward to me, but I don't presently have a good idea for fixing them.} |
38 |
36 |
39 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 $\CD{M}$, 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. |
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. |
40 |
38 |
41 |
39 |
42 \nn{some more things to cover in the intro} |
40 \nn{some more things to cover in the intro} |
43 \begin{itemize} |
41 \begin{itemize} |
44 \item related: we are being unsophisticated from a homotopy theory point of |
42 \item related: we are being unsophisticated from a homotopy theory point of |
56 We will briefly sketch our original motivation for defining the blob complex. |
54 We will briefly sketch our original motivation for defining the blob complex. |
57 \nn{this is adapted from an old draft of the intro; it needs further modification |
55 \nn{this is adapted from an old draft of the intro; it needs further modification |
58 in order to better integrate it into the current intro.} |
56 in order to better integrate it into the current intro.} |
59 |
57 |
60 As a starting point, consider TQFTs constructed via fields and local relations. |
58 As a starting point, consider TQFTs constructed via fields and local relations. |
61 (See Section \ref{sec:tqftsviafields} or \cite{kwtqft}.) |
59 (See Section \ref{sec:tqftsviafields} or \cite{kw:tqft}.) |
62 This gives a satisfactory treatment for semisimple TQFTs |
60 This gives a satisfactory treatment for semisimple TQFTs |
63 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
61 (i.e.\ TQFTs for which the cylinder 1-category associated to an |
64 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
62 $n{-}1$-manifold $Y$ is semisimple for all $Y$). |
65 |
63 |
66 For non-semiemple TQFTs, this approach is less satisfactory. |
64 For non-semi-simple TQFTs, this approach is less satisfactory. |
67 Our main motivating example (though we will not develop it in this paper) |
65 Our main motivating example (though we will not develop it in this paper) |
68 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
66 is the (decapitated) $4{+}1$-dimensional TQFT associated to Khovanov homology. |
69 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
67 It associates a bigraded vector space $A_{Kh}(W^4, L)$ to a 4-manifold $W$ together |
70 with a link $L \subset \bd W$. |
68 with a link $L \subset \bd W$. |
71 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
69 The original Khovanov homology of a link in $S^3$ is recovered as $A_{Kh}(B^4, L)$. |
72 |
70 |
73 How would we go about computing $A_{Kh}(W^4, L)$? |
71 How would we go about computing $A_{Kh}(W^4, L)$? |
74 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
72 For $A_{Kh}(B^4, L)$, the main tool is the exact triangle (long exact sequence) |
75 \nn{... $L_1, L_2, L_3$}. |
73 relating resolutions of a crossing. |
76 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
74 Unfortunately, the exactness breaks if we glue $B^4$ to itself and attempt |
77 to compute $A_{Kh}(S^1\times B^3, L)$. |
75 to compute $A_{Kh}(S^1\times B^3, L)$. |
78 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
76 According to the gluing theorem for TQFTs-via-fields, gluing along $B^3 \subset \bd B^4$ |
79 corresponds to taking a coend (self tensor product) over the cylinder category |
77 corresponds to taking a coend (self tensor product) over the cylinder category |
80 associated to $B^3$ (with appropriate boundary conditions). |
78 associated to $B^3$ (with appropriate boundary conditions). |
111 $\bc_1$ is linear combinations of local relations on $W$, |
109 $\bc_1$ is linear combinations of local relations on $W$, |
112 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
110 $\bc_2$ is linear combinations of relations amongst relations on $W$, |
113 and so on. |
111 and so on. |
114 |
112 |
115 None of the above ideas depend on the details of the Khovanov homology example, |
113 None of the above ideas depend on the details of the Khovanov homology example, |
116 so we develop the general theory in the paper and postpone specific applications |
114 so we develop the general theory in this paper and postpone specific applications |
117 to later papers. |
115 to later papers. |
118 |
116 |
119 |
117 |
120 |
118 |
121 \subsection{Formal properties} |
119 \subsection{Formal properties} |
184 \begin{equation*} |
182 \begin{equation*} |
185 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)} |
183 \xymatrix{\bc_*^{\cC}(S^1) \ar[r]^{\iso}_{\text{qi}} & \HC_*(\cC)} |
186 \end{equation*} |
184 \end{equation*} |
187 \end{property} |
185 \end{property} |
188 |
186 |
189 Here $\CD{X}$ is the singular chain complex of the space of diffeomorphisms of $X$, fixed on $\bdy X$. |
187 Here $\CH{X}$ is the singular chain complex of the space of homeomorphisms of $X$, fixed on $\bdy X$. |
190 \begin{property}[$C_*(\Diff(-))$ action] |
188 \begin{property}[$C_*(\Homeo(-))$ action] |
191 \label{property:evaluation}% |
189 \label{property:evaluation}% |
192 There is a chain map |
190 There is a chain map |
193 \begin{equation*} |
191 \begin{equation*} |
194 \ev_X: \CD{X} \tensor \bc_*(X) \to \bc_*(X). |
192 \ev_X: \CH{X} \tensor \bc_*(X) \to \bc_*(X). |
195 \end{equation*} |
193 \end{equation*} |
196 |
194 |
197 Restricted to $C_0(\Diff(X))$ this is just the action of diffeomorphisms described in Property \ref{property:functoriality}. Further, for |
195 Restricted to $C_0(\Homeo(X))$ this is just the action of homeomorphisms described in Property \ref{property:functoriality}. Further, for |
198 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
196 any codimension $1$-submanifold $Y \subset X$ dividing $X$ into $X_1 \cup_Y X_2$, the following diagram |
199 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
197 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
200 \begin{equation*} |
198 \begin{equation*} |
201 \xymatrix{ |
199 \xymatrix{ |
202 \CD{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
200 \CH{X} \otimes \bc_*(X) \ar[r]^{\ev_X} & \bc_*(X) \\ |
203 \CD{X_1} \otimes \CD{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
201 \CH{X_1} \otimes \CH{X_2} \otimes \bc_*(X_1) \otimes \bc_*(X_2) |
204 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
202 \ar@/_4ex/[r]_{\ev_{X_1} \otimes \ev_{X_2}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
205 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
203 \bc_*(X_1) \otimes \bc_*(X_2) \ar[u]_{\gl_Y} |
206 } |
204 } |
207 \end{equation*} |
205 \end{equation*} |
208 \nn{should probably say something about associativity here (or not?)} |
206 \nn{should probably say something about associativity here (or not?)} |
209 \nn{maybe do self-gluing instead of 2 pieces case:} |
207 \nn{maybe do self-gluing instead of 2 pieces case:} |
210 Further, for |
208 Further, for |
211 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
209 any codimension $0$-submanifold $Y \sqcup Y^\text{op} \subset \bdy X$ the following diagram |
212 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
210 (using the gluing maps described in Property \ref{property:gluing-map}) commutes. |
213 \begin{equation*} |
211 \begin{equation*} |
214 \xymatrix@C+2cm{ |
212 \xymatrix@C+2cm{ |
215 \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) \\ |
213 \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) \\ |
216 \CD{X} \otimes \bc_*(X) |
214 \CH{X} \otimes \bc_*(X) |
217 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Diff}_Y \otimes \gl_Y} & |
215 \ar[r]_{\ev_{X}} \ar[u]^{\gl^{\Homeo}_Y \otimes \gl_Y} & |
218 \bc_*(X) \ar[u]_{\gl_Y} |
216 \bc_*(X) \ar[u]_{\gl_Y} |
219 } |
217 } |
220 \end{equation*} |
218 \end{equation*} |
221 \end{property} |
219 \end{property} |
222 |
220 |
238 \begin{property}[Product formula] |
236 \begin{property}[Product formula] |
239 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
237 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold. Let $\cC$ be an $n$-category. |
240 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
238 Let $A_*(Y)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology (see Property \ref{property:blobs-ainfty}). |
241 Then |
239 Then |
242 \[ |
240 \[ |
243 \bc_*(Y^{n-k}\times W^k, \cC) \simeq \bc_*(W, A_*(Y)) . |
241 \bc_*(Y\times W, \cC) \simeq \bc_*(W, A_*(Y)) . |
244 \] |
242 \] |
245 Note on the right here we have the version of the blob complex for $A_\infty$ $n$-categories. |
243 Note on the right hand side we have the version of the blob complex for $A_\infty$ $n$-categories. |
246 \end{property} |
244 \end{property} |
247 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. |
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. |
248 |
246 |
249 \begin{property}[Gluing formula] |
247 \begin{property}[Gluing formula] |
250 \label{property:gluing}% |
248 \label{property:gluing}% |
291 \subsection{Future directions} |
289 \subsection{Future directions} |
292 \label{sec:future} |
290 \label{sec:future} |
293 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
291 Throughout, we have resisted the temptation to work in the greatest generality possible (don't worry, it wasn't that hard). |
294 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
292 In most of the places where we say ``set" or ``vector space", any symmetric monoidal category would do. |
295 \nn{maybe make similar remark about chain complexes and $(\infty, 0)$-categories} |
293 \nn{maybe make similar remark about chain complexes and $(\infty, 0)$-categories} |
296 More could be said about finite characteristic (there appears in be $2$-torsion in $\bc_1(S^2, \cC)$ for any spherical $2$-category $\cC$). 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. |
294 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. |
297 |
295 |
298 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 $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details. |
296 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 $A$ simply corresponds to the gluing operation on $\bc_*(S^1 \times [0,1], A)$, but haven't investigated the details. |
299 |
297 |
300 Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh} |
298 Most importantly, however, \nn{applications!} \nn{$n=2$ cases, contact, Kh} |
301 |
299 |