144 What we really mean is that there exists a functor which interacts with the other data of $\cC$ as specified |
144 What we really mean is that there exists a functor which interacts with the other data of $\cC$ as specified |
145 in the axioms below. |
145 in the axioms below. |
146 |
146 |
147 |
147 |
148 \begin{axiom}[Boundaries]\label{nca-boundary} |
148 \begin{axiom}[Boundaries]\label{nca-boundary} |
149 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
149 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \colimit{\cC}_{k-1}(\bd X)$. |
150 These maps, for various $X$, comprise a natural transformation of functors. |
150 These maps, for various $X$, comprise a natural transformation of functors. |
151 \end{axiom} |
151 \end{axiom} |
152 |
152 |
153 Note that the first ``$\bd$" above is part of the data for the category, |
153 Note that the first ``$\bd$" above is part of the data for the category, |
154 while the second is the ordinary boundary of manifolds. |
154 while the second is the ordinary boundary of manifolds. |
155 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
155 Given $c\in\colimit{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
156 |
156 |
157 \medskip |
157 \medskip |
158 |
158 |
159 In order to simplify the exposition we have concentrated on the case of |
159 In order to simplify the exposition we have concentrated on the case of |
160 unoriented PL manifolds and avoided the question of what exactly we mean by |
160 unoriented PL manifolds and avoided the question of what exactly we mean by |
174 |
174 |
175 We have just argued that the boundary of a morphism has no preferred splitting into |
175 We have just argued that the boundary of a morphism has no preferred splitting into |
176 domain and range, but the converse meets with our approval. |
176 domain and range, but the converse meets with our approval. |
177 That is, given compatible domain and range, we should be able to combine them into |
177 That is, given compatible domain and range, we should be able to combine them into |
178 the full boundary of a morphism. |
178 the full boundary of a morphism. |
179 The following lemma will follow from the colimit construction used to define $\cl{\cC}_{k-1}$ |
179 The following lemma will follow from the colimit construction used to define $\colimit{\cC}_{k-1}$ |
180 on spheres. |
180 on spheres. |
181 |
181 |
182 \begin{lem}[Boundary from domain and range] |
182 \begin{lem}[Boundary from domain and range] |
183 \label{lem:domain-and-range} |
183 \label{lem:domain-and-range} |
184 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
184 Let $S = B_1 \cup_E B_2$, where $S$ is a $k{-}1$-sphere $(1\le k\le n)$, |
185 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
185 $B_i$ is a $k{-}1$-ball, and $E = B_1\cap B_2$ is a $k{-}2$-sphere (Figure \ref{blah3}). |
186 Let $\cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
186 Let $\cC(B_1) \times_{\colimit{\cC}(E)} \cC(B_2)$ denote the fibered product of the |
187 two maps $\bd: \cC(B_i)\to \cl{\cC}(E)$. |
187 two maps $\bd: \cC(B_i)\to \colimit{\cC}(E)$. |
188 Then we have an injective map |
188 Then we have an injective map |
189 \[ |
189 \[ |
190 \gl_E : \cC(B_1) \times_{\cl{\cC}(E)} \cC(B_2) \into \cl{\cC}(S) |
190 \gl_E : \cC(B_1) \times_{\colimit{\cC}(E)} \cC(B_2) \into \colimit{\cC}(S) |
191 \] |
191 \] |
192 which is natural with respect to the actions of homeomorphisms. |
192 which is natural with respect to the actions of homeomorphisms. |
193 (When $k=1$ we stipulate that $\cl{\cC}(E)$ is a point, so that the above fibered product |
193 (When $k=1$ we stipulate that $\colimit{\cC}(E)$ is a point, so that the above fibered product |
194 becomes a normal product.) |
194 becomes a normal product.) |
195 \end{lem} |
195 \end{lem} |
196 |
196 |
197 \begin{figure}[t] \centering |
197 \begin{figure}[t] \centering |
198 \begin{tikzpicture}[%every label/.style={green} |
198 \begin{tikzpicture}[%every label/.style={green} |
215 If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is |
215 If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is |
216 in the image of the gluing map precisely when the cell complex is in general position |
216 in the image of the gluing map precisely when the cell complex is in general position |
217 with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective. |
217 with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective. |
218 |
218 |
219 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
219 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
220 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
220 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\colimit{\cC}(S)$ can be identified |
221 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
221 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
222 |
222 |
223 Let $\cl{\cC}(S)\trans E$ denote the image of $\gl_E$. |
223 Let $\colimit{\cC}(S)\trans E$ denote the image of $\gl_E$. |
224 We will refer to elements of $\cl{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
224 We will refer to elements of $\colimit{\cC}(S)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
225 When the gluing map is surjective every such element is splittable. |
225 When the gluing map is surjective every such element is splittable. |
226 |
226 |
227 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
227 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
228 as above, then we define $\cC(X)\trans E = \bd^{-1}(\cl{\cC}(\bd X)\trans E)$. |
228 as above, then we define $\cC(X)\trans E = \bd^{-1}(\colimit{\cC}(\bd X)\trans E)$. |
229 |
229 |
230 We will call the projection $\cl{\cC}(S)\trans E \to \cC(B_i)$ given by the composition |
230 We will call the projection $\colimit{\cC}(S)\trans E \to \cC(B_i)$ given by the composition |
231 $$\cl{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$ |
231 $$\colimit{\cC}(S)\trans E \xrightarrow{\gl^{-1}} \cC(B_1) \times \cC(B_2) \xrightarrow{\pr_i} \cC(B_i)$$ |
232 a {\it restriction} map and write $\res_{B_i}(a)$ |
232 a {\it restriction} map and write $\res_{B_i}(a)$ |
233 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)\trans E$. |
233 (or simply $\res(a)$ when there is no ambiguity), for $a\in \colimit{\cC}(S)\trans E$. |
234 More generally, we also include under the rubric ``restriction map" |
234 More generally, we also include under the rubric ``restriction map" |
235 the boundary maps of Axiom \ref{nca-boundary} above, |
235 the boundary maps of Axiom \ref{nca-boundary} above, |
236 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
236 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
237 of restriction maps. |
237 of restriction maps. |
238 In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$ |
238 In particular, we have restriction maps $\cC(X)\trans E \to \cC(B_i)$ |
239 defined as the composition of the boundary with the first restriction map described above: |
239 defined as the composition of the boundary with the first restriction map described above: |
240 $$ |
240 $$ |
241 \cC(X) \trans E \xrightarrow{\bdy} \cl{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i) |
241 \cC(X) \trans E \xrightarrow{\bdy} \colimit{\cC}(\bdy X)\trans E \xrightarrow{\res} \cC(B_i) |
242 .$$ |
242 .$$ |
243 These restriction maps can be thought of as |
243 These restriction maps can be thought of as |
244 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
244 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
245 %%%% the next sentence makes no sense to me, even though I'm probably the one who wrote it -- KW |
245 %%%% the next sentence makes no sense to me, even though I'm probably the one who wrote it -- KW |
246 \noop{These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, |
246 \noop{These restriction maps in fact have their image in the subset $\cC(B_i)\trans E$, |
315 the smaller balls to $X$. |
315 the smaller balls to $X$. |
316 We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
316 We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
317 In situations where the splitting is notationally anonymous, we will write |
317 In situations where the splitting is notationally anonymous, we will write |
318 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
318 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
319 the unnamed splitting. |
319 the unnamed splitting. |
320 If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
320 If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\colimit{\cC}(\bd X)_\beta)$; |
321 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
321 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
322 decomposition of $\bd X$ and no competing splitting of $X$. |
322 decomposition of $\bd X$ and no competing splitting of $X$. |
323 |
323 |
324 The above two composition axioms are equivalent to the following one, |
324 The above two composition axioms are equivalent to the following one, |
325 which we state in slightly vague form. |
325 which we state in slightly vague form. |
1019 |
1019 |
1020 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
1020 Before stating the revised axioms for an $n$-category enriched in a distributive monoidal category, |
1021 we need a preliminary definition. |
1021 we need a preliminary definition. |
1022 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
1022 Once we have the above $n$-category axioms for $n{-}1$-morphisms, we can define the |
1023 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
1023 category $\bbc$ of {\it $n$-balls with boundary conditions}. |
1024 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \cl\cC(\bd X)$ is the ``boundary condition". |
1024 Its objects are pairs $(X, c)$, where $X$ is an $n$-ball and $c \in \colimit{\cC}(\bd X)$ is the ``boundary condition". |
1025 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X; c \to X'; c')$, are |
1025 The morphisms from $(X, c)$ to $(X', c')$, denoted $\Homeo(X; c \to X'; c')$, are |
1026 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
1026 homeomorphisms $f:X\to X'$ such that $f|_{\bd X}(c) = c'$. |
1027 %Let $\pi_0(\bbc)$ denote |
1027 %Let $\pi_0(\bbc)$ denote |
1028 |
1028 |
1029 \begin{axiom}[Enriched $n$-categories] |
1029 \begin{axiom}[Enriched $n$-categories] |
1036 %[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
1036 %[already said this above. ack] Furthermore, $\cC_n(f)$ depends only on the path component of a homeomorphism $f: (X, c) \to (X', c')$. |
1037 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
1037 %In particular, homeomorphisms which are isotopic to the identity rel boundary act trivially |
1038 \item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}. |
1038 \item Composition. Let $B = B_1\cup_Y B_2$ as in Axiom \ref{axiom:composition}. |
1039 Let $Y_i = \bd B_i \setmin Y$. |
1039 Let $Y_i = \bd B_i \setmin Y$. |
1040 Note that $\bd B = Y_1\cup Y_2$. |
1040 Note that $\bd B = Y_1\cup Y_2$. |
1041 Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \cl\cC(E)$. |
1041 Let $c_i \in \cC(Y_i)$ with $\bd c_1 = \bd c_2 = d \in \colimit{\cC}(E)$. |
1042 Then we have a map |
1042 Then we have a map |
1043 \[ |
1043 \[ |
1044 \gl_Y : \bigoplus_c \cC(B_1; c_1 \bullet c) \otimes \cC(B_2; c_2\bullet c) \to \cC(B; c_1\bullet c_2), |
1044 \gl_Y : \bigoplus_c \cC(B_1; c_1 \bullet c) \otimes \cC(B_2; c_2\bullet c) \to \cC(B; c_1\bullet c_2), |
1045 \] |
1045 \] |
1046 where the sum is over $c\in\cC(Y)$ such that $\bd c = d$. |
1046 where the sum is over $c\in\cC(Y)$ such that $\bd c = d$. |
1182 before we can describe the data for $k$-morphisms. |
1182 before we can describe the data for $k$-morphisms. |
1183 |
1183 |
1184 An $n$-category consists of the following data: |
1184 An $n$-category consists of the following data: |
1185 \begin{itemize} |
1185 \begin{itemize} |
1186 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
1186 \item functors $\cC_k$ from $k$-balls to sets, $0\le k\le n$ (Axiom \ref{axiom:morphisms}); |
1187 \item boundary natural transformations $\cC_k \to \cl{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
1187 \item boundary natural transformations $\cC_k \to \colimit{\cC}_{k-1} \circ \bd$ (Axiom \ref{nca-boundary}); |
1188 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
1188 \item ``composition'' or ``gluing'' maps $\gl_Y : \cC(B_1)\trans E \times_{\cC(Y)} \cC(B_2)\trans E \to \cC(B_1\cup_Y B_2)\trans E$ (Axiom \ref{axiom:composition}); |
1189 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
1189 \item ``product'' or ``identity'' maps $\pi^*:\cC(X)\to \cC(E)$ for each pinched product $\pi:E\to X$ (Axiom \ref{axiom:product}); |
1190 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}); |
1190 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}); |
1191 %\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
1191 %\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
1192 \item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions |
1192 \item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions |
1282 \rm |
1282 \rm |
1283 \label{ex:traditional-n-categories} |
1283 \label{ex:traditional-n-categories} |
1284 Given a ``traditional $n$-category with strong duality" $C$ |
1284 Given a ``traditional $n$-category with strong duality" $C$ |
1285 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
1285 define $\cC(X)$, for $X$ a $k$-ball with $k < n$, |
1286 to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
1286 to be the set of all $C$-labeled embedded cell complexes of $X$ (c.f. \S \ref{sec:fields}). |
1287 For $X$ an $n$-ball and $c\in \cl{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
1287 For $X$ an $n$-ball and $c\in \colimit{\cC}(\bd X)$, define $\cC(X; c)$ to be finite linear |
1288 combinations of $C$-labeled embedded cell complexes of $X$ |
1288 combinations of $C$-labeled embedded cell complexes of $X$ |
1289 modulo the kernel of the evaluation map. |
1289 modulo the kernel of the evaluation map. |
1290 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
1290 Define a product morphism $a\times D$, for $D$ an $m$-ball, to be the product of the cell complex of $a$ with $D$, |
1291 with each cell labelled according to the corresponding cell for $a$. |
1291 with each cell labelled according to the corresponding cell for $a$. |
1292 (These two cells have the same codimension.) |
1292 (These two cells have the same codimension.) |
1457 |
1457 |
1458 |
1458 |
1459 \subsection{From balls to manifolds} |
1459 \subsection{From balls to manifolds} |
1460 \label{ss:ncat_fields} \label{ss:ncat-coend} |
1460 \label{ss:ncat_fields} \label{ss:ncat-coend} |
1461 In this section we show how to extend an $n$-category $\cC$ as described above |
1461 In this section we show how to extend an $n$-category $\cC$ as described above |
1462 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\cl{\cC}$. |
1462 (of either the ordinary or $A_\infty$ variety) to an invariant of manifolds, which we denote by $\colimit{\cC}$. |
1463 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
1463 This extension is a certain colimit, and the arrow in the notation is intended as a reminder of this. |
1464 |
1464 |
1465 In the case of ordinary $n$-categories, this construction factors into a construction of a |
1465 In the case of ordinary $n$-categories, this construction factors into a construction of a |
1466 system of fields and local relations, followed by the usual TQFT definition of a |
1466 system of fields and local relations, followed by the usual TQFT definition of a |
1467 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1467 vector space invariant of manifolds given as Definition \ref{defn:TQFT-invariant}. |
1468 For an $A_\infty$ $n$-category, $\cl{\cC}$ is defined using a homotopy colimit instead. |
1468 For an $A_\infty$ $n$-category, $\colimit{\cC}$ is defined using a homotopy colimit instead. |
1469 Recall that we can take an ordinary $n$-category $\cC$ and pass to the ``free resolution", |
1469 Recall that we can take an ordinary $n$-category $\cC$ and pass to the ``free resolution", |
1470 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1470 an $A_\infty$ $n$-category $\bc_*(\cC)$, by computing the blob complex of balls |
1471 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1471 (recall Example \ref{ex:blob-complexes-of-balls} above). |
1472 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1472 We will show in Corollary \ref{cor:new-old} below that the homotopy colimit invariant |
1473 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
1473 for a manifold $M$ associated to this $A_\infty$ $n$-category is actually the |
1474 same as the original blob complex for $M$ with coefficients in $\cC$. |
1474 same as the original blob complex for $M$ with coefficients in $\cC$. |
1475 |
1475 |
1476 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
1476 Recall that we've already anticipated this construction Subsection \ref{ss:n-cat-def}, |
1477 inductively defining $\cl{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1477 inductively defining $\colimit{\cC}$ on $k$-spheres in terms of $\cC$ on $k$-balls, |
1478 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1478 so that we can state the boundary axiom for $\cC$ on $k+1$-balls. |
1479 |
1479 |
1480 \medskip |
1480 \medskip |
1481 |
1481 |
1482 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1482 We will first define the {\it decomposition poset} $\cell(W)$ for any $k$-manifold $W$, for $1 \leq k \leq n$. |
1483 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1483 An $n$-category $\cC$ provides a functor from this poset to the category of sets, |
1484 and we will define $\cl{\cC}(W)$ as a suitable colimit |
1484 and we will define $\colimit{\cC}(W)$ as a suitable colimit |
1485 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1485 (or homotopy colimit in the $A_\infty$ case) of this functor. |
1486 We'll later give a more explicit description of this colimit. |
1486 We'll later give a more explicit description of this colimit. |
1487 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
1487 In the case that the $n$-category $\cC$ is enriched (e.g. associates vector spaces or chain |
1488 complexes to $n$-balls with boundary data), |
1488 complexes to $n$-balls with boundary data), |
1489 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
1489 then the resulting colimit is also enriched, that is, the set associated to $W$ splits into |
1544 $\cC(X_a)$ and $\cC(X_b)$ agree whenever some part of $\bd X_a$ is glued to some part of $\bd X_b$. |
1544 $\cC(X_a)$ and $\cC(X_b)$ agree whenever some part of $\bd X_a$ is glued to some part of $\bd X_b$. |
1545 (Keep in mind that perhaps $a=b$.) |
1545 (Keep in mind that perhaps $a=b$.) |
1546 Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy |
1546 Since we allow decompositions in which the intersection of $X_a$ and $X_b$ might be messy |
1547 (see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way. |
1547 (see Example \ref{sin1x-example}), we must define $\psi_{\cC;W}(x)$ in a more roundabout way. |
1548 |
1548 |
1549 Inductively, we may assume that we have already defined the colimit $\cl\cC(M)$ for $k{-}1$-manifolds $M$. |
1549 Inductively, we may assume that we have already defined the colimit $\colimit\cC(M)$ for $k{-}1$-manifolds $M$. |
1550 (To start the induction, we define $\cl\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is |
1550 (To start the induction, we define $\colimit\cC(M)$, where $M = \du_a P_a$ is a 0-manifold and each $P_a$ is |
1551 a 0-ball, to be $\prod_a \cC(P_a)$.) |
1551 a 0-ball, to be $\prod_a \cC(P_a)$.) |
1552 We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds. |
1552 We also assume, inductively, that we have gluing and restriction maps for colimits of $k{-}1$-manifolds. |
1553 Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection. |
1553 Gluing and restriction maps for colimits of $k$-manifolds will be defined later in this subsection. |
1554 |
1554 |
1555 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1555 Let $\du_a X_a = M_0\to\cdots\to M_m = W$ be a ball decomposition compatible with $x$. |
1556 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1556 Let $\bd M_i = N_i \cup Y_i \cup Y'_i$, where $Y_i$ and $Y'_i$ are glued together to produce $M_{i+1}$. |
1557 We will define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1557 We will define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies a series of conditions |
1558 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1558 related to the gluings $M_{i-1} \to M_i$, $1\le i \le m$. |
1559 By Axiom \ref{nca-boundary}, we have a map |
1559 By Axiom \ref{nca-boundary}, we have a map |
1560 \[ |
1560 \[ |
1561 \prod_a \cC(X_a) \to \cl\cC(\bd M_0) . |
1561 \prod_a \cC(X_a) \to \colimit\cC(\bd M_0) . |
1562 \] |
1562 \] |
1563 The first condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_0)$ is splittable |
1563 The first condition is that the image of $\psi_{\cC;W}(x)$ in $\colimit\cC(\bd M_0)$ is splittable |
1564 along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\cl\cC(Y_0)$ and $\cl\cC(Y'_0)$ agree |
1564 along $\bd Y_0$ and $\bd Y'_0$, and that the restrictions to $\colimit\cC(Y_0)$ and $\colimit\cC(Y'_0)$ agree |
1565 (with respect to the identification of $Y_0$ and $Y'_0$ provided by the gluing map). |
1565 (with respect to the identification of $Y_0$ and $Y'_0$ provided by the gluing map). |
1566 |
1566 |
1567 On the subset of $\prod_a \cC(X_a)$ which satisfies the first condition above, we have a restriction |
1567 On the subset of $\prod_a \cC(X_a)$ which satisfies the first condition above, we have a restriction |
1568 map to $\cl\cC(N_0)$ which we can compose with the gluing map |
1568 map to $\colimit\cC(N_0)$ which we can compose with the gluing map |
1569 $\cl\cC(N_0) \to \cl\cC(\bd M_1)$. |
1569 $\colimit\cC(N_0) \to \colimit\cC(\bd M_1)$. |
1570 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\cl\cC(\bd M_1)$ is splittable |
1570 The second condition is that the image of $\psi_{\cC;W}(x)$ in $\colimit\cC(\bd M_1)$ is splittable |
1571 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\cl\cC(Y_1)$ and $\cl\cC(Y'_1)$ agree |
1571 along $\bd Y_1$ and $\bd Y'_1$, and that the restrictions to $\colimit\cC(Y_1)$ and $\colimit\cC(Y'_1)$ agree |
1572 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1572 (with respect to the identification of $Y_1$ and $Y'_1$ provided by the gluing map). |
1573 The $i$-th condition is defined similarly. |
1573 The $i$-th condition is defined similarly. |
1574 Note that these conditions depend only on the boundaries of elements of $\prod_a \cC(X_a)$. |
1574 Note that these conditions depend only on the boundaries of elements of $\prod_a \cC(X_a)$. |
1575 |
1575 |
1576 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1576 We define $\psi_{\cC;W}(x)$ to be the subset of $\prod_a \cC(X_a)$ which satisfies the |
1597 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agrees with $\beta$. |
1597 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agrees with $\beta$. |
1598 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
1598 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
1599 $\cS$ and the coproduct and product in the above expression should be replaced by the appropriate |
1599 $\cS$ and the coproduct and product in the above expression should be replaced by the appropriate |
1600 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
1600 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
1601 |
1601 |
1602 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1602 Finally, we construct $\colimit{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1603 |
1603 |
1604 \begin{defn}[System of fields functor] |
1604 \begin{defn}[System of fields functor] |
1605 \label{def:colim-fields} |
1605 \label{def:colim-fields} |
1606 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1606 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\colimit{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1607 That is, for each decomposition $x$ there is a map |
1607 That is, for each decomposition $x$ there is a map |
1608 $\psi_{\cC;W}(x)\to \cl{\cC}(W)$, these maps are compatible with the refinement maps |
1608 $\psi_{\cC;W}(x)\to \colimit{\cC}(W)$, these maps are compatible with the refinement maps |
1609 above, and $\cl{\cC}(W)$ is universal with respect to these properties. |
1609 above, and $\colimit{\cC}(W)$ is universal with respect to these properties. |
1610 \end{defn} |
1610 \end{defn} |
1611 |
1611 |
1612 \begin{defn}[System of fields functor, $A_\infty$ case] |
1612 \begin{defn}[System of fields functor, $A_\infty$ case] |
1613 When $\cC$ is an $A_\infty$ $n$-category, $\cl{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1613 When $\cC$ is an $A_\infty$ $n$-category, $\colimit{\cC}(W)$ for $W$ a $k$-manifold with $k < n$ |
1614 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1614 is defined as above, as the colimit of $\psi_{\cC;W}$. |
1615 When $W$ is an $n$-manifold, the chain complex $\cl{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1615 When $W$ is an $n$-manifold, the chain complex $\colimit{\cC}(W)$ is the homotopy colimit of the functor $\psi_{\cC;W}$. |
1616 \end{defn} |
1616 \end{defn} |
1617 |
1617 |
1618 %We can specify boundary data $c \in \cl{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1618 %We can specify boundary data $c \in \colimit{\cC}(\bdy W)$, and define functors $\psi_{\cC;W,c}$ |
1619 %with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
1619 %with values the subsets of those of $\psi_{\cC;W}$ which agree with $c$ on the boundary of $W$. |
1620 |
1620 |
1621 \medskip |
1621 \medskip |
1622 |
1622 |
1623 We must now define restriction maps $\bd : \cl{\cC}(W) \to \cl{\cC}(\bd W)$ and gluing maps. |
1623 We must now define restriction maps $\bd : \colimit{\cC}(W) \to \colimit{\cC}(\bd W)$ and gluing maps. |
1624 |
1624 |
1625 Let $y\in \cl{\cC}(W)$. |
1625 Let $y\in \colimit{\cC}(W)$. |
1626 Choose a representative of $y$ in the colimit: a permissible decomposition $\du_a X_a \to W$ and elements |
1626 Choose a representative of $y$ in the colimit: a permissible decomposition $\du_a X_a \to W$ and elements |
1627 $y_a \in \cC(X_a)$. |
1627 $y_a \in \cC(X_a)$. |
1628 By assumption, $\du_a (X_a \cap \bd W) \to \bd W$ can be completed to a decomposition of $\bd W$. |
1628 By assumption, $\du_a (X_a \cap \bd W) \to \bd W$ can be completed to a decomposition of $\bd W$. |
1629 Let $r(y_a) \in \cl\cC(X_a \cap \bd W)$ be the restriction. |
1629 Let $r(y_a) \in \colimit\cC(X_a \cap \bd W)$ be the restriction. |
1630 Choose a representative of $r(y_a)$ in the colimit $\cl\cC(X_a \cap \bd W)$: a permissible decomposition |
1630 Choose a representative of $r(y_a)$ in the colimit $\colimit\cC(X_a \cap \bd W)$: a permissible decomposition |
1631 $\du_b Q_{ab} \to X_a \cap \bd W$ and elements $z_{ab} \in \cC(Q_{ab})$. |
1631 $\du_b Q_{ab} \to X_a \cap \bd W$ and elements $z_{ab} \in \cC(Q_{ab})$. |
1632 Then $\du_{ab} Q_{ab} \to \bd W$ is a permissible decomposition of $\bd W$ and $\{z_{ab}\}$ represents |
1632 Then $\du_{ab} Q_{ab} \to \bd W$ is a permissible decomposition of $\bd W$ and $\{z_{ab}\}$ represents |
1633 an element of $\cl{\cC}(\bd W)$. Define $\bd y$ to be this element. |
1633 an element of $\colimit{\cC}(\bd W)$. Define $\bd y$ to be this element. |
1634 It is not hard to see that it is independent of the various choices involved. |
1634 It is not hard to see that it is independent of the various choices involved. |
1635 |
1635 |
1636 Note that since we have already (inductively) defined gluing maps for colimits of $k{-}1$-manifolds, |
1636 Note that since we have already (inductively) defined gluing maps for colimits of $k{-}1$-manifolds, |
1637 we can also define restriction maps from $\cl{\cC}(W)\trans{}$ to $\cl{\cC}(Y)$ where $Y$ is a codimension 0 |
1637 we can also define restriction maps from $\colimit{\cC}(W)\trans{}$ to $\colimit{\cC}(Y)$ where $Y$ is a codimension 0 |
1638 submanifold of $\bd W$. |
1638 submanifold of $\bd W$. |
1639 |
1639 |
1640 Next we define gluing maps for colimits of $k$-manifolds. |
1640 Next we define gluing maps for colimits of $k$-manifolds. |
1641 Let $W = W_1 \cup_Y W_2$. |
1641 Let $W = W_1 \cup_Y W_2$. |
1642 Let $y_i \in \cl\cC(W_i)$ and assume that the restrictions of $y_1$ and $y_2$ to $\cl\cC(Y)$ agree. |
1642 Let $y_i \in \colimit\cC(W_i)$ and assume that the restrictions of $y_1$ and $y_2$ to $\colimit\cC(Y)$ agree. |
1643 We want to define $y_1\bullet y_2 \in \cl\cC(W)$. |
1643 We want to define $y_1\bullet y_2 \in \colimit\cC(W)$. |
1644 Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements |
1644 Choose a permissible decomposition $\du_a X_{ia} \to W_i$ and elements |
1645 $y_{ia} \in \cC(X_{ia})$ representing $y_i$. |
1645 $y_{ia} \in \cC(X_{ia})$ representing $y_i$. |
1646 It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$, |
1646 It might not be the case that $\du_{ia} X_{ia} \to W$ is a permissible decomposition of $W$, |
1647 since intersections of the pieces with $\bd W$ might not be well-behaved. |
1647 since intersections of the pieces with $\bd W$ might not be well-behaved. |
1648 However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:splittings}, |
1648 However, using the fact that $\bd y_i$ splits along $\bd Y$ and applying Axiom \ref{axiom:splittings}, |
1658 We now give more concrete descriptions of the above colimits. |
1658 We now give more concrete descriptions of the above colimits. |
1659 |
1659 |
1660 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
1660 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
1661 the colimit is |
1661 the colimit is |
1662 \[ |
1662 \[ |
1663 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
1663 \colimit{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
1664 \] |
1664 \] |
1665 where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation |
1665 where $x$ runs through decompositions of $W$, and $\sim$ is the obvious equivalence relation |
1666 induced by refinement and gluing. |
1666 induced by refinement and gluing. |
1667 If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold, |
1667 If $\cC$ is enriched over, for example, vector spaces and $W$ is an $n$-manifold, |
1668 we can take |
1668 we can take |
1669 \begin{equation*} |
1669 \begin{equation*} |
1670 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
1670 \colimit{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
1671 \end{equation*} |
1671 \end{equation*} |
1672 where $K$ is the vector space spanned by elements $a - g(a)$, with |
1672 where $K$ is the vector space spanned by elements $a - g(a)$, with |
1673 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1673 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1674 \to \psi_{\cC;W,c}(y)$ is the value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1674 \to \psi_{\cC;W,c}(y)$ is the value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1675 |
1675 |
1682 construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties |
1682 construction of \S \ref{sec:blob-definition} and takes advantage of local (gluing) properties |
1683 of the indexing category $\cell(W)$. |
1683 of the indexing category $\cell(W)$. |
1684 |
1684 |
1685 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
1685 Define an $m$-sequence in $W$ to be a sequence $x_0 \le x_1 \le \dots \le x_m$ of permissible decompositions of $W$. |
1686 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1686 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1687 Define $\cl{\cC}(W)$ as a vector space via |
1687 Define $\colimit{\cC}(W)$ as a vector space via |
1688 \[ |
1688 \[ |
1689 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1689 \colimit{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1690 \] |
1690 \] |
1691 where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$. |
1691 where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$. |
1692 Elements of a summand indexed by an $m$-sequence will be call $m$-simplices. |
1692 Elements of a summand indexed by an $m$-sequence will be call $m$-simplices. |
1693 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1693 We endow $\colimit{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1694 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1694 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1695 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1695 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1696 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1696 summand of $\colimit{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1697 \[ |
1697 \[ |
1698 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1698 \bd (a, \bar{x}) = (\bd a, \bar{x}) + (-1)^{\deg{a}} (g(a), d_0(\bar{x})) + (-1)^{\deg{a}} \sum_{j=1}^k (-1)^{j} (a, d_j(\bar{x})) , |
1699 \] |
1699 \] |
1700 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1700 where $d_j(\bar{x}) = (x_0,\dots,x_{j-1},x_{j+1},\dots,x_k)$ and $g: \psi_\cC(x_0)\to \psi_\cC(x_1)$ |
1701 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1701 is the usual gluing map coming from the antirefinement $x_0 \le x_1$. |
1726 One can show that the usual hocolimit and the local hocolimit are homotopy equivalent using an |
1726 One can show that the usual hocolimit and the local hocolimit are homotopy equivalent using an |
1727 Eilenberg-Zilber type subdivision argument. |
1727 Eilenberg-Zilber type subdivision argument. |
1728 |
1728 |
1729 \medskip |
1729 \medskip |
1730 |
1730 |
1731 $\cl{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
1731 $\colimit{\cC}(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
1732 Restricting to $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1732 Restricting to $k$-spheres, we have now proved Lemma \ref{lem:spheres}. |
1733 |
1733 |
1734 \begin{lem} |
1734 \begin{lem} |
1735 \label{lem:colim-injective} |
1735 \label{lem:colim-injective} |
1736 Let $W$ be a manifold of dimension $j<n$. Then for each |
1736 Let $W$ be a manifold of dimension $j<n$. Then for each |
1737 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \cl{\cC}(W)$ is injective. |
1737 decomposition $x$ of $W$ the natural map $\psi_{\cC;W}(x)\to \colimit{\cC}(W)$ is injective. |
1738 \end{lem} |
1738 \end{lem} |
1739 \begin{proof} |
1739 \begin{proof} |
1740 $\cl{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
1740 $\colimit{\cC}(W)$ is a colimit of a diagram of sets, and each of the arrows in the diagram is |
1741 injective. |
1741 injective. |
1742 Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$), |
1742 Concretely, the colimit is the disjoint union of the sets (one for each decomposition of $W$), |
1743 modulo the relation which identifies the domain of each of the injective maps |
1743 modulo the relation which identifies the domain of each of the injective maps |
1744 with its image. |
1744 with its image. |
1745 |
1745 |
1746 To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$. |
1746 To save ink and electrons we will simplify notation and write $\psi(x)$ for $\psi_{\cC;W}(x)$. |
1747 |
1747 |
1748 Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\cl{\cC}(W)$ but $a\ne \hat{a}$. |
1748 Suppose $a, \hat{a}\in \psi(x)$ have the same image in $\colimit{\cC}(W)$ but $a\ne \hat{a}$. |
1749 Then there exist |
1749 Then there exist |
1750 \begin{itemize} |
1750 \begin{itemize} |
1751 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
1751 \item decompositions $x = x_0, x_1, \ldots , x_{k-1}, x_k = x$ and $v_1,\ldots, v_k$ of $W$; |
1752 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
1752 \item anti-refinements $v_i\to x_i$ and $v_i\to x_{i-1}$; and |
1753 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
1753 \item elements $a_i\in \psi(x_i)$ and $b_i\in \psi(v_i)$, with $a_0 = a$ and $a_k = \hat{a}$, |
1838 We call it a hemisphere instead of a ball because it plays a role analogous |
1838 We call it a hemisphere instead of a ball because it plays a role analogous |
1839 to the $k{-}1$-spheres in the $n$-category definition.) |
1839 to the $k{-}1$-spheres in the $n$-category definition.) |
1840 |
1840 |
1841 \begin{lem} |
1841 \begin{lem} |
1842 \label{lem:hemispheres} |
1842 \label{lem:hemispheres} |
1843 {For each $1 \le k \le n$, we have a functor $\cl\cM_{k-1}$ from |
1843 {For each $1 \le k \le n$, we have a functor $\colimit\cM_{k-1}$ from |
1844 the category of marked $k$-hemispheres and |
1844 the category of marked $k$-hemispheres and |
1845 homeomorphisms to the category of sets and bijections.} |
1845 homeomorphisms to the category of sets and bijections.} |
1846 \end{lem} |
1846 \end{lem} |
1847 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1847 The proof is exactly analogous to that of Lemma \ref{lem:spheres}, and we omit the details. |
1848 We use the same type of colimit construction. |
1848 We use the same type of colimit construction. |
1849 |
1849 |
1850 In our example, $\cl\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
1850 In our example, $\colimit\cM(H) = \cD(H\times\bd W \cup \bd H\times W)$. |
1851 |
1851 |
1852 \begin{module-axiom}[Module boundaries] |
1852 \begin{module-axiom}[Module boundaries] |
1853 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \cl\cM(\bd M)$. |
1853 {For each marked $k$-ball $M$ we have a map of sets $\bd: \cM(M)\to \colimit\cM(\bd M)$. |
1854 These maps, for various $M$, comprise a natural transformation of functors.} |
1854 These maps, for various $M$, comprise a natural transformation of functors.} |
1855 \end{module-axiom} |
1855 \end{module-axiom} |
1856 |
1856 |
1857 Given $c\in\cl\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1857 Given $c\in\colimit\cM(\bd M)$, let $\cM(M; c) \deq \bd^{-1}(c)$. |
1858 |
1858 |
1859 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1859 If the $n$-category $\cC$ is enriched over some other category (e.g.\ vector spaces), |
1860 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
1860 then for each marked $n$-ball $M=(B,N)$ and $c\in \cC(\bd B \setminus N)$, the set $\cM(M; c)$ should be an object in that category. |
1861 |
1861 |
1862 \begin{lem}[Boundary from domain and range] |
1862 \begin{lem}[Boundary from domain and range] |
1863 \label{lem:module-boundary} |
1863 \label{lem:module-boundary} |
1864 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1864 {Let $H = M_1 \cup_E M_2$, where $H$ is a marked $k{-}1$-hemisphere ($1\le k\le n$), |
1865 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1865 $M_i$ is a marked $k{-}1$-ball, and $E = M_1\cap M_2$ is a marked $k{-}2$-hemisphere. |
1866 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1866 Let $\cM(M_1) \times_{\cM(E)} \cM(M_2)$ denote the fibered product of the |
1867 two maps $\bd: \cM(M_i)\to \cl\cM(E)$. |
1867 two maps $\bd: \cM(M_i)\to \colimit\cM(E)$. |
1868 Then we have an injective map |
1868 Then we have an injective map |
1869 \[ |
1869 \[ |
1870 \gl_E : \cM(M_1) \times_{\cl\cM(E)} \cM(M_2) \hookrightarrow \cl\cM(H) |
1870 \gl_E : \cM(M_1) \times_{\colimit\cM(E)} \cM(M_2) \hookrightarrow \colimit\cM(H) |
1871 \] |
1871 \] |
1872 which is natural with respect to the actions of homeomorphisms.} |
1872 which is natural with respect to the actions of homeomorphisms.} |
1873 \end{lem} |
1873 \end{lem} |
1874 This is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
1874 This is in exact analogy with Lemma \ref{lem:domain-and-range}, and illustrated in Figure \ref{fig:module-boundary}. |
1875 \begin{figure}[t] |
1875 \begin{figure}[t] |
1894 \end{tikzpicture} |
1894 \end{tikzpicture} |
1895 \end{equation*}\caption{The marked hemispheres and marked balls from Lemma \ref{lem:module-boundary}.} |
1895 \end{equation*}\caption{The marked hemispheres and marked balls from Lemma \ref{lem:module-boundary}.} |
1896 \label{fig:module-boundary} |
1896 \label{fig:module-boundary} |
1897 \end{figure} |
1897 \end{figure} |
1898 |
1898 |
1899 Let $\cl\cM(H)\trans E$ denote the image of $\gl_E$. |
1899 Let $\colimit\cM(H)\trans E$ denote the image of $\gl_E$. |
1900 We will refer to elements of $\cl\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
1900 We will refer to elements of $\colimit\cM(H)\trans E$ as ``splittable along $E$" or ``transverse to $E$". |
1901 |
1901 |
1902 \noop{ %%%%%%% |
1902 \noop{ %%%%%%% |
1903 \begin{lem}[Module to category restrictions] |
1903 \begin{lem}[Module to category restrictions] |
1904 {For each marked $k$-hemisphere $H$ there is a restriction map |
1904 {For each marked $k$-hemisphere $H$ there is a restriction map |
1905 $\cl\cM(H)\to \cC(H)$. |
1905 $\colimit\cM(H)\to \cC(H)$. |
1906 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1906 ($\cC(H)$ means apply $\cC$ to the underlying $k$-ball of $H$.) |
1907 These maps comprise a natural transformation of functors.} |
1907 These maps comprise a natural transformation of functors.} |
1908 \end{lem} |
1908 \end{lem} |
1909 } %%%%%%% end \noop |
1909 } %%%%%%% end \noop |
1910 |
1910 |
1911 It follows from the definition of the colimit $\cl\cM(H)$ that |
1911 It follows from the definition of the colimit $\colimit\cM(H)$ that |
1912 given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map |
1912 given any (unmarked) $k{-}1$-ball $Y$ in the interior of $H$ there is a restriction map |
1913 from a subset $\cl\cM(H)_{\trans{\bdy Y}}$ of $\cl\cM(H)$ to $\cC(Y)$. |
1913 from a subset $\colimit\cM(H)_{\trans{\bdy Y}}$ of $\colimit\cM(H)$ to $\cC(Y)$. |
1914 Combining this with the boundary map $\cM(B,N) \to \cl\cM(\bd(B,N))$, we also have a restriction |
1914 Combining this with the boundary map $\cM(B,N) \to \colimit\cM(\bd(B,N))$, we also have a restriction |
1915 map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$. |
1915 map from a subset $\cM(B,N)_{\trans{\bdy Y}}$ of $\cM(B,N)$ to $\cC(Y)$ whenever $Y$ is in the interior of $\bd B \setmin N$. |
1916 This fact will be used below. |
1916 This fact will be used below. |
1917 |
1917 |
1918 \noop{ %%%% |
1918 \noop{ %%%% |
1919 Note that combining the various boundary and restriction maps above |
1919 Note that combining the various boundary and restriction maps above |
2392 We will be mainly interested in the case $n=1$ and enriched over chain complexes, |
2392 We will be mainly interested in the case $n=1$ and enriched over chain complexes, |
2393 since this is the case that's relevant to the generalized Deligne conjecture of \S\ref{sec:deligne}. |
2393 since this is the case that's relevant to the generalized Deligne conjecture of \S\ref{sec:deligne}. |
2394 So we treat this case in more detail. |
2394 So we treat this case in more detail. |
2395 |
2395 |
2396 First we explain the remark about derived hom above. |
2396 First we explain the remark about derived hom above. |
2397 Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction |
2397 Let $L$ be a marked 1-ball and let $\colimit{\cX}(L)$ denote the local homotopy colimit construction |
2398 associated to $L$ by $\cX$ and $\cC$. |
2398 associated to $L$ by $\cX$ and $\cC$. |
2399 (See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) |
2399 (See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) |
2400 Define $\cl{\cY}(L)$ similarly. |
2400 Define $\colimit{\cY}(L)$ similarly. |
2401 For $K$ an unmarked 1-ball let $\cl{\cC}(K)$ denote the local homotopy colimit |
2401 For $K$ an unmarked 1-ball let $\colimit{\cC}(K)$ denote the local homotopy colimit |
2402 construction associated to $K$ by $\cC$. |
2402 construction associated to $K$ by $\cC$. |
2403 Then we have an injective gluing map |
2403 Then we have an injective gluing map |
2404 \[ |
2404 \[ |
2405 \gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K) |
2405 \gl: \colimit{\cX}(L) \ot \colimit{\cC}(K) \to \colimit{\cX}(L\cup K) |
2406 \] |
2406 \] |
2407 which is also a chain map. |
2407 which is also a chain map. |
2408 (For simplicity we are suppressing mention of boundary conditions on the unmarked |
2408 (For simplicity we are suppressing mention of boundary conditions on the unmarked |
2409 boundary components of the 1-balls.) |
2409 boundary components of the 1-balls.) |
2410 We define $\hom_\cC(\cX \to \cY)$ to be a collection of (graded linear) natural transformations |
2410 We define $\hom_\cC(\cX \to \cY)$ to be a collection of (graded linear) natural transformations |
2411 $g: \cl{\cX}(L)\to \cl{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$: |
2411 $g: \colimit{\cX}(L)\to \colimit{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$: |
2412 \[ \xymatrix{ |
2412 \[ \xymatrix{ |
2413 \cl{\cX}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \cl{\cX}(L\cup K) \ar[d]^{g}\\ |
2413 \colimit{\cX}(L) \ot \colimit{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \colimit{\cX}(L\cup K) \ar[d]^{g}\\ |
2414 \cl{\cY}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} & \cl{\cY}(L\cup K) |
2414 \colimit{\cY}(L) \ot \colimit{\cC}(K) \ar[r]^{\gl} & \colimit{\cY}(L\cup K) |
2415 } \] |
2415 } \] |
2416 |
2416 |
2417 The usual differential on graded linear maps between chain complexes induces a differential |
2417 The usual differential on graded linear maps between chain complexes induces a differential |
2418 on $\hom_\cC(\cX \to \cY)$, giving it the structure of a chain complex. |
2418 on $\hom_\cC(\cX \to \cY)$, giving it the structure of a chain complex. |
2419 |
2419 |
2426 Recall that the tensor product $\cX \ot_\cC \cZ$ depends on a choice of interval $J$, labeled |
2426 Recall that the tensor product $\cX \ot_\cC \cZ$ depends on a choice of interval $J$, labeled |
2427 by $\cX$ on one boundary component and $\cZ$ on the other. |
2427 by $\cX$ on one boundary component and $\cZ$ on the other. |
2428 Because we are using the {\it local} homotopy colimit, any generator |
2428 Because we are using the {\it local} homotopy colimit, any generator |
2429 $D\ot x\ot \bar{c}\ot z$ of $\cX \ot_\cC \cZ$ can be written (perhaps non-uniquely) as a gluing |
2429 $D\ot x\ot \bar{c}\ot z$ of $\cX \ot_\cC \cZ$ can be written (perhaps non-uniquely) as a gluing |
2430 $(D'\ot x \ot \bar{c}') \bullet (D''\ot \bar{c}''\ot z)$, for some decomposition $J = L'\cup L''$ |
2430 $(D'\ot x \ot \bar{c}') \bullet (D''\ot \bar{c}''\ot z)$, for some decomposition $J = L'\cup L''$ |
2431 and with $D'\ot x \ot \bar{c}'$ a generator of $\cl{\cX}(L')$ and |
2431 and with $D'\ot x \ot \bar{c}'$ a generator of $\colimit{\cX}(L')$ and |
2432 $D''\ot \bar{c}''\ot z$ a generator of $\cl{\cZ}(L'')$. |
2432 $D''\ot \bar{c}''\ot z$ a generator of $\colimit{\cZ}(L'')$. |
2433 (Such a splitting exists because the blob diagram $D$ can be split into left and right halves, |
2433 (Such a splitting exists because the blob diagram $D$ can be split into left and right halves, |
2434 since no blob can include both the leftmost and rightmost intervals in the underlying decomposition. |
2434 since no blob can include both the leftmost and rightmost intervals in the underlying decomposition. |
2435 This step would fail if we were using the usual hocolimit instead of the local hocolimit.) |
2435 This step would fail if we were using the usual hocolimit instead of the local hocolimit.) |
2436 We now define |
2436 We now define |
2437 \[ |
2437 \[ |