8 |
8 |
9 \subsection{Definition of $n$-categories} |
9 \subsection{Definition of $n$-categories} |
10 \label{ss:n-cat-def} |
10 \label{ss:n-cat-def} |
11 |
11 |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
12 Before proceeding, we need more appropriate definitions of $n$-categories, |
13 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
13 $A_\infty$ $n$-categories, as well as modules for these, and tensor products of these modules. |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
14 (As is the case throughout this paper, by ``$n$-category" we mean some notion of |
15 a ``weak" $n$-category with ``strong duality".) |
15 a ``weak" $n$-category with ``strong duality".) |
16 |
16 |
17 The definitions presented below tie the categories more closely to the topology |
17 The definitions presented below tie the categories more closely to the topology |
18 and avoid combinatorial questions about, for example, the minimal sufficient |
18 and avoid combinatorial questions about, for example, the minimal sufficient |
21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
21 (e.g.\ categories whose morphisms are maps into spaces or decorated balls), |
22 it is easy to show that they |
22 it is easy to show that they |
23 satisfy our axioms. |
23 satisfy our axioms. |
24 For examples of a more purely algebraic origin, one would typically need the combinatorial |
24 For examples of a more purely algebraic origin, one would typically need the combinatorial |
25 results that we have avoided here. |
25 results that we have avoided here. |
|
26 |
|
27 \nn{Say something explicit about Lurie's work here? It seems like this was something that Dan Freed wanted explaining when we talked to him in Aspen} |
26 |
28 |
27 \medskip |
29 \medskip |
28 |
30 |
29 There are many existing definitions of $n$-categories, with various intended uses. |
31 There are many existing definitions of $n$-categories, with various intended uses. |
30 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
32 In any such definition, there are sets of $k$-morphisms for each $0 \leq k \leq n$. |
68 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
70 (If smooth, ``homeomorphism" should be read ``diffeomorphism", and we would need |
69 to be fussier about corners and boundaries.) |
71 to be fussier about corners and boundaries.) |
70 For each flavor of manifold there is a corresponding flavor of $n$-category. |
72 For each flavor of manifold there is a corresponding flavor of $n$-category. |
71 For simplicity, we will concentrate on the case of PL unoriented manifolds. |
73 For simplicity, we will concentrate on the case of PL unoriented manifolds. |
72 |
74 |
73 (The ambitious reader may want to keep in mind two other classes of balls. |
75 An ambitious reader may want to keep in mind two other classes of balls. |
74 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
76 The first is balls equipped with a map to some other space $Y$ (c.f. \cite{MR2079378}). |
75 This will be used below to describe the blob complex of a fiber bundle with |
77 This will be used below (see the end of \S \ref{ss:product-formula}) to describe the blob complex of a fiber bundle with |
76 base space $Y$. |
78 base space $Y$. |
77 The second is balls equipped with a section of the tangent bundle, or the frame |
79 The second is balls equipped with a section of the tangent bundle, or the frame |
78 bundle (i.e.\ framed balls), or more generally some flag bundle associated to the tangent bundle. |
80 bundle (i.e.\ framed balls), or more generally some partial flag bundle associated to the tangent bundle. |
79 These can be used to define categories with less than the ``strong" duality we assume here, |
81 These can be used to define categories with less than the ``strong" duality we assume here, |
80 though we will not develop that idea fully in this paper.) |
82 though we will not develop that idea fully in this paper. |
81 |
83 |
82 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
84 Next we consider domains and ranges of morphisms (or, as we prefer to say, boundaries |
83 of morphisms). |
85 of morphisms). |
84 The 0-sphere is unusual among spheres in that it is disconnected. |
86 The 0-sphere is unusual among spheres in that it is disconnected. |
85 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
87 Correspondingly, for 1-morphisms it makes sense to distinguish between domain and range. |
86 (Actually, this is only true in the oriented case, with 1-morphisms parameterized |
88 (Actually, this is only true in the oriented case, with 1-morphisms parameterized |
87 by {\it oriented} 1-balls.) |
89 by {\it oriented} 1-balls.) |
88 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
90 For $k>1$ and in the presence of strong duality the division into domain and range makes less sense. |
89 For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
91 For example, in a pivotal tensor category, there are natural isomorphisms $\Hom{}{A}{B \tensor C} \isoto \Hom{}{B^* \tensor A}{C}$, etc. |
90 (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
92 (sometimes called ``Frobenius reciprocity''), which canonically identify all the morphism spaces which have the same boundary. |
91 We prefer to not make the distinction in the first place. |
93 We prefer not to make the distinction in the first place. |
92 |
94 |
93 Instead, we will combine the domain and range into a single entity which we call the |
95 Instead, we will combine the domain and range into a single entity which we call the |
94 boundary of a morphism. |
96 boundary of a morphism. |
95 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
97 Morphisms are modeled on balls, so their boundaries are modeled on spheres. |
96 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
98 In other words, we need to extend the functors $\cC_{k-1}$ from balls to spheres, for |
116 \begin{axiom}[Boundaries]\label{nca-boundary} |
118 \begin{axiom}[Boundaries]\label{nca-boundary} |
117 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
119 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
118 These maps, for various $X$, comprise a natural transformation of functors. |
120 These maps, for various $X$, comprise a natural transformation of functors. |
119 \end{axiom} |
121 \end{axiom} |
120 |
122 |
121 (Note that the first ``$\bd$" above is part of the data for the category, |
123 Note that the first ``$\bd$" above is part of the data for the category, |
122 while the second is the ordinary boundary of manifolds.) |
124 while the second is the ordinary boundary of manifolds. |
123 |
|
124 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
125 Given $c\in\cl{\cC}(\bd(X))$, we will write $\cC(X; c)$ for $\bd^{-1}(c)$, those morphisms with specified boundary $c$. |
125 |
126 |
126 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
127 Most of the examples of $n$-categories we are interested in are enriched in the following sense. |
127 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
128 The various sets of $n$-morphisms $\cC(X; c)$, for all $n$-balls $X$ and |
128 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
129 all $c\in \cl{\cC}(\bd X)$, have the structure of an object in some auxiliary symmetric monoidal category |
129 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
130 (e.g.\ vector spaces, or modules over some ring, or chain complexes), |
130 \nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} |
131 \nn{actually, need both disj-union/sub and product/tensor-product; what's the name for this sort of cat?} |
131 and all the structure maps of the $n$-category should be compatible with the auxiliary |
132 and all the structure maps of the $n$-category should be compatible with the auxiliary |
132 category structure. |
133 category structure. |
133 Note that this auxiliary structure is only in dimension $n$; |
134 Note that this auxiliary structure is only in dimension $n$; if $\dim(Y) < n$ then |
134 $\cC(Y; c)$ is just a plain set if $\dim(Y) < n$. |
135 $\cC(Y; c)$ is just a plain set. |
135 |
136 |
136 \medskip |
137 \medskip |
137 |
138 |
138 (In order to simplify the exposition we have concentrated on the case of |
139 In order to simplify the exposition we have concentrated on the case of |
139 unoriented PL manifolds and avoided the question of what exactly we mean by |
140 unoriented PL manifolds and avoided the question of what exactly we mean by |
140 the boundary a manifold with extra structure, such as an oriented manifold. |
141 the boundary of a manifold with extra structure, such as an oriented manifold. |
141 In general, all manifolds of dimension less than $n$ should be equipped with the germ |
142 In general, all manifolds of dimension less than $n$ should be equipped with the germ |
142 of a thickening to dimension $n$, and this germ should carry whatever structure we have |
143 of a thickening to dimension $n$, and this germ should carry whatever structure we have |
143 on $n$-manifolds. |
144 on $n$-manifolds. |
144 In addition, lower dimensional manifolds should be equipped with a framing |
145 In addition, lower dimensional manifolds should be equipped with a framing |
145 of their normal bundle in the thickening; the framing keeps track of which |
146 of their normal bundle in the thickening; the framing keeps track of which |
146 side (iterated) bounded manifolds lie on. |
147 side (iterated) bounded manifolds lie on. |
147 For example, the boundary of an oriented $n$-ball |
148 For example, the boundary of an oriented $n$-ball |
148 should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent |
149 should be an $n{-}1$-sphere equipped with an orientation of its once stabilized tangent |
149 bundle and a choice of direction in this bundle indicating |
150 bundle and a choice of direction in this bundle indicating |
150 which side the $n$-ball lies on.) |
151 which side the $n$-ball lies on. |
151 |
152 |
152 \medskip |
153 \medskip |
153 |
154 |
154 We have just argued that the boundary of a morphism has no preferred splitting into |
155 We have just argued that the boundary of a morphism has no preferred splitting into |
155 domain and range, but the converse meets with our approval. |
156 domain and range, but the converse meets with our approval. |
186 \end{tikzpicture} |
187 \end{tikzpicture} |
187 $$ |
188 $$ |
188 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
189 \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} |
189 |
190 |
190 Note that we insist on injectivity above. |
191 Note that we insist on injectivity above. |
191 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. |
192 The lemma follows from Definition \ref{def:colim-fields} and Lemma \ref{lem:colim-injective}. \nn{we might want a more official looking proof...} |
192 |
193 |
193 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
194 Let $\cl{\cC}(S)_E$ denote the image of $\gl_E$. |
194 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
195 We will refer to elements of $\cl{\cC}(S)_E$ as ``splittable along $E$" or ``transverse to $E$". |
195 |
196 |
196 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
197 If $X$ is a $k$-ball and $E \sub \bd X$ splits $\bd X$ into two $k{-}1$-balls $B_1$ and $B_2$ |
197 as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
198 as above, then we define $\cC(X)_E = \bd^{-1}(\cl{\cC}(\bd X)_E)$. |
198 |
199 |
199 We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
200 We will call the projection $\cl{\cC}(S)_E \to \cC(B_i)$ |
200 a {\it restriction} map and write $\res_{B_i}(a)$ |
201 a {\it restriction} map and write $\res_{B_i}(a)$ |
201 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. |
202 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cl{\cC}(S)_E$. |
202 More generally, we also include under the rubric ``restriction map" the |
203 More generally, we also include under the rubric ``restriction map" |
203 the boundary maps of Axiom \ref{nca-boundary} above, |
204 the boundary maps of Axiom \ref{nca-boundary} above, |
204 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
205 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
205 of restriction maps. |
206 of restriction maps. |
206 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
207 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
207 ($i = 1, 2$, notation from previous paragraph). |
208 ($i = 1, 2$, notation from previous paragraph). |
249 $$ |
250 $$ |
250 \caption{From two balls to one ball.}\label{blah5}\end{figure} |
251 \caption{From two balls to one ball.}\label{blah5}\end{figure} |
251 |
252 |
252 \begin{axiom}[Strict associativity] \label{nca-assoc} |
253 \begin{axiom}[Strict associativity] \label{nca-assoc} |
253 The composition (gluing) maps above are strictly associative. |
254 The composition (gluing) maps above are strictly associative. |
254 Given any splitting of a ball $B$ into smaller balls $B_1,\ldots,B_m$, |
255 Given any splitting of a ball $B$ into smaller balls |
255 any sequence of gluings of the smaller balls yields the same result. |
256 $$\bigsqcup B_i \to B,$$ |
|
257 any sequence of gluings (in the sense of Definition \ref{defn:gluing-decomposition}, where all the intermediate steps are also disjoint unions of balls) yields the same result. |
256 \end{axiom} |
258 \end{axiom} |
257 |
259 |
258 \begin{figure}[!ht] |
260 \begin{figure}[!ht] |
259 $$\mathfig{.65}{ncat/strict-associativity}$$ |
261 $$\mathfig{.65}{ncat/strict-associativity}$$ |
260 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
262 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
261 |
263 |
262 We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$. |
264 We'll use the notation $a\bullet b$ for the glued together field $\gl_Y(a, b)$. |
263 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
265 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
264 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
266 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
265 %Compositions of boundary and restriction maps will also be called restriction maps. |
267 %Compositions of boundary and restriction maps will also be called restriction maps. |
266 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
268 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
267 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
269 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
269 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
271 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
270 We will call elements of $\cC(B)_Y$ morphisms which are |
272 We will call elements of $\cC(B)_Y$ morphisms which are |
271 ``splittable along $Y$'' or ``transverse to $Y$''. |
273 ``splittable along $Y$'' or ``transverse to $Y$''. |
272 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
274 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
273 |
275 |
274 More generally, let $\alpha$ be a subdivision of a ball $X$ into smaller balls. |
276 More generally, let $\alpha$ be a splitting of $X$ into smaller balls. |
275 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
277 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
276 the smaller balls to $X$. |
278 the smaller balls to $X$. |
277 We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
279 We say that elements of $\cC(X)_\alpha$ are morphisms which are ``splittable along $\alpha$". |
278 In situations where the subdivision is notationally anonymous, we will write |
280 In situations where the splitting is notationally anonymous, we will write |
279 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
281 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
280 the unnamed subdivision. |
282 the unnamed splitting. |
281 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
283 If $\beta$ is a ball decomposition of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cl{\cC}(\bd X)_\beta)$; |
282 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
284 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
283 subdivision of $\bd X$ and no competing subdivision of $X$. |
285 decomposition of $\bd X$ and no competing splitting of $X$. |
284 |
286 |
285 The above two composition axioms are equivalent to the following one, |
287 The above two composition axioms are equivalent to the following one, |
286 which we state in slightly vague form. |
288 which we state in slightly vague form. |
287 |
289 |
288 \xxpar{Multi-composition:} |
290 \xxpar{Multi-composition:} |
289 {Given any decomposition $B = B_1\cup\cdots\cup B_m$ of a $k$-ball |
291 {Given any splitting $B_1 \sqcup \cdots \sqcup B_m \to B$ of a $k$-ball |
290 into small $k$-balls, there is a |
292 into small $k$-balls, there is a |
291 map from an appropriate subset (like a fibered product) |
293 map from an appropriate subset (like a fibered product) |
292 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
294 of $\cC(B_1)\spl\times\cdots\times\cC(B_m)\spl$ to $\cC(B)\spl$, |
293 and these various $m$-fold composition maps satisfy an |
295 and these various $m$-fold composition maps satisfy an |
294 operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
296 operad-type strict associativity condition (Figure \ref{fig:operad-composition}).} |
382 d: \Delta^{k+m}\to\Delta^k . |
384 d: \Delta^{k+m}\to\Delta^k . |
383 \] |
385 \] |
384 (We thank Kevin Costello for suggesting this approach.) |
386 (We thank Kevin Costello for suggesting this approach.) |
385 |
387 |
386 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
388 Note that for each interior point $x\in X$, $\pi\inv(x)$ is an $m$-ball, |
387 and for for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
389 and for each boundary point $x\in\bd X$, $\pi\inv(x)$ is a ball of dimension |
388 $l \le m$, with $l$ depending on $x$. |
390 $l \le m$, with $l$ depending on $x$. |
389 |
|
390 It is easy to see that a composition of pinched products is again a pinched product. |
391 It is easy to see that a composition of pinched products is again a pinched product. |
391 |
|
392 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
392 A {\it sub pinched product} is a sub-$m$-ball $E'\sub E$ such that the restriction |
393 $\pi:E'\to \pi(E')$ is again a pinched product. |
393 $\pi:E'\to \pi(E')$ is again a pinched product. |
394 A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
394 A {union} of pinched products is a decomposition $E = \cup_i E_i$ |
395 such that each $E_i\sub E$ is a sub pinched product. |
395 such that each $E_i\sub E$ is a sub pinched product. |
396 (See Figure \ref{pinched_prod_unions}.) |
396 (See Figure \ref{pinched_prod_unions}.) |
521 We start with the plain $n$-category case. |
521 We start with the plain $n$-category case. |
522 |
522 |
523 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
523 \begin{axiom}[\textup{\textbf{[preliminary]}} Isotopy invariance in dimension $n$] |
524 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
524 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
525 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
525 to the identity on $\bd X$ and is isotopic (rel boundary) to the identity. |
526 Then $f$ acts trivially on $\cC(X)$; $f(a) = a$ for all $a\in \cC(X)$. |
526 Then $f$ acts trivially on $\cC(X)$; that is $f(a) = a$ for all $a\in \cC(X)$. |
527 \end{axiom} |
527 \end{axiom} |
528 |
528 |
529 This axiom needs to be strengthened to force product morphisms to act as the identity. |
529 This axiom needs to be strengthened to force product morphisms to act as the identity. |
530 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
530 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
531 Let $J$ be a 1-ball (interval). |
531 Let $J$ be a 1-ball (interval). |
532 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
532 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
533 (Here we use the ``pinched" version of $Y\times J$. |
533 (Here we use $Y\times J$ with boundary entirely pinched.) |
534 \nn{do we need notation for this?}) |
|
535 We define a map |
534 We define a map |
536 \begin{eqnarray*} |
535 \begin{eqnarray*} |
537 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
536 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
538 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
537 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
539 \end{eqnarray*} |
538 \end{eqnarray*} |
623 a diagram like the one in Theorem \ref{thm:CH} commutes. |
622 a diagram like the one in Theorem \ref{thm:CH} commutes. |
624 %\nn{repeat diagram here?} |
623 %\nn{repeat diagram here?} |
625 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
624 %\nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
626 \end{axiom} |
625 \end{axiom} |
627 |
626 |
628 We should strengthen the above axiom to apply to families of collar maps. |
627 We should strengthen the above $A_\infty$ axiom to apply to families of collar maps. |
629 To do this we need to explain how collar maps form a topological space. |
628 To do this we need to explain how collar maps form a topological space. |
630 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
629 Roughly, the set of collared $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
631 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
630 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
632 Having chains on the space of collar maps act gives rise to coherence maps involving |
631 Having chains on the space of collar maps act gives rise to coherence maps involving |
633 weak identities. |
632 weak identities. |
650 (and their boundaries), while for fields we consider all manifolds. |
649 (and their boundaries), while for fields we consider all manifolds. |
651 Second, in category definition we directly impose isotopy |
650 Second, in category definition we directly impose isotopy |
652 invariance in dimension $n$, while in the fields definition we |
651 invariance in dimension $n$, while in the fields definition we |
653 instead remember a subspace of local relations which contain differences of isotopic fields. |
652 instead remember a subspace of local relations which contain differences of isotopic fields. |
654 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
653 (Recall that the compensation for this complication is that we can demand that the gluing map for fields is injective.) |
655 Thus a system of fields and local relations $(\cF,\cU)$ determines an $n$-category $\cC_ {\cF,\cU}$ simply by restricting our attention to |
654 Thus a system of fields and local relations $(\cF,U)$ determines an $n$-category $\cC_ {\cF,U}$ simply by restricting our attention to |
656 balls and, at level $n$, quotienting out by the local relations: |
655 balls and, at level $n$, quotienting out by the local relations: |
657 \begin{align*} |
656 \begin{align*} |
658 \cC_{\cF,\cU}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / \cU(B) & \text{when $k=n$.}\end{cases} |
657 \cC_{\cF,U}(B^k) & = \begin{cases}\cF(B) & \text{when $k<n$,} \\ \cF(B) / U(B) & \text{when $k=n$.}\end{cases} |
659 \end{align*} |
658 \end{align*} |
660 This $n$-category can be thought of as the local part of the fields. |
659 This $n$-category can be thought of as the local part of the fields. |
661 Conversely, given a topological $n$-category we can construct a system of fields via |
660 Conversely, given a topological $n$-category we can construct a system of fields via |
662 a colimit construction; see \S \ref{ss:ncat_fields} below. |
661 a colimit construction; see \S \ref{ss:ncat_fields} below. |
663 |
662 |
805 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, |
804 Alternatively, if we take the $n$-morphisms to be simply $\Maps_c(X\times F \to T)$, |
806 we get an $A_\infty$ $n$-category enriched over spaces. |
805 we get an $A_\infty$ $n$-category enriched over spaces. |
807 \end{example} |
806 \end{example} |
808 |
807 |
809 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
808 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to |
810 homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
809 homotopy as the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
811 |
810 |
812 \begin{example}[Blob complexes of balls (with a fiber)] |
811 \begin{example}[Blob complexes of balls (with a fiber)] |
813 \rm |
812 \rm |
814 \label{ex:blob-complexes-of-balls} |
813 \label{ex:blob-complexes-of-balls} |
815 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
814 Fix an $n{-}k$-dimensional manifold $F$ and an $n$-dimensional system of fields $\cE$. |
994 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
993 means the subset of $\cC(X_a)$ whose restriction to $\bd X_a$ agress with $\beta$. |
995 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
994 If we are enriching over $\cS$ and $k=n$, then $\cC(X_a; \beta)$ is an object in |
996 $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
995 $\cS$ and the coproduct and product in Equation \ref{eq:psi-CC} should be replaced by the approriate |
997 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
996 operations in $\cS$ (e.g. direct sum and tensor product if $\cS$ is Vect). |
998 |
997 |
999 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$. |
998 Finally, we construct $\cl{\cC}(W)$ as the appropriate colimit of $\psi_{\cC;W}$: |
1000 |
999 |
1001 \begin{defn}[System of fields functor] |
1000 \begin{defn}[System of fields functor] |
1002 \label{def:colim-fields} |
1001 \label{def:colim-fields} |
1003 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1002 If $\cC$ is an $n$-category enriched in sets or vector spaces, $\cl{\cC}(W)$ is the usual colimit of the functor $\psi_{\cC;W}$. |
1004 That is, for each decomposition $x$ there is a map |
1003 That is, for each decomposition $x$ there is a map |
1018 We now give more concrete descriptions of the above colimits. |
1017 We now give more concrete descriptions of the above colimits. |
1019 |
1018 |
1020 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
1019 In the non-enriched case (e.g.\ $k<n$), where each $\cC(X_a; \beta)$ is just a set, |
1021 the colimit is |
1020 the colimit is |
1022 \[ |
1021 \[ |
1023 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) / \sim , |
1022 \cl{\cC}(W,c) = \left( \coprod_x \coprod_\beta \prod_a \cC(X_a; \beta) \right) \Bigg/ \sim , |
1024 \] |
1023 \] |
1025 where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation |
1024 where $x$ runs through decomposition of $W$, and $\sim$ is the obvious equivalence relation |
1026 induced by refinement and gluing. |
1025 induced by refinement and gluing. |
1027 If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
1026 If $\cC$ is enriched over vector spaces and $W$ is an $n$-manifold, |
1028 we can take |
1027 we can take |
1029 \begin{equation*} |
1028 \begin{equation*} |
1030 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) / K |
1029 \cl{\cC}(W,c) = \left( \bigoplus_x \bigoplus_\beta \bigotimes_a \cC(X_a; \beta) \right) \Bigg/ K, |
1031 \end{equation*} |
1030 \end{equation*} |
1032 where $K$ is the vector space spanned by elements $a - g(a)$, with |
1031 where $K$ is the vector space spanned by elements $a - g(a)$, with |
1033 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1032 $a\in \psi_{\cC;W,c}(x)$ for some decomposition $x$, and $g: \psi_{\cC;W,c}(x) |
1034 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1033 \to \psi_{\cC;W,c}(y)$ is value of $\psi_{\cC;W,c}$ on some antirefinement $x \leq y$. |
1035 |
1034 |
1039 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1038 Such sequences (for all $m$) form a simplicial set in $\cell(W)$. |
1040 Define $\cl{\cC}(W)$ as a vector space via |
1039 Define $\cl{\cC}(W)$ as a vector space via |
1041 \[ |
1040 \[ |
1042 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1041 \cl{\cC}(W) = \bigoplus_{(x_i)} \psi_{\cC;W}(x_0)[m] , |
1043 \] |
1042 \] |
1044 where the sum is over all $m$-sequences $(x_i)$ and all $m$, and each summand is degree shifted by $m$. |
1043 where the sum is over all $m$ and all $m$-sequences $(x_i)$, and each summand is degree shifted by $m$. |
1045 Elements of a summand indexed by an $m$-sequence will be call $m$-simplices. |
1044 Elements of a summand indexed by an $m$-sequence will be call $m$-simplices. |
1046 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1045 We endow $\cl{\cC}(W)$ with a differential which is the sum of the differential of the $\psi_{\cC;W}(x_0)$ |
1047 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1046 summands plus another term using the differential of the simplicial set of $m$-sequences. |
1048 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1047 More specifically, if $(a, \bar{x})$ denotes an element in the $\bar{x}$ |
1049 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1048 summand of $\cl{\cC}(W)$ (with $\bar{x} = (x_0,\dots,x_k)$), define |
1093 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1092 Such a $W$ gives rise to a module for the $n$-category associated to $\bd W$. |
1094 This will be explained in more detail as we present the axioms. |
1093 This will be explained in more detail as we present the axioms. |
1095 |
1094 |
1096 Throughout, we fix an $n$-category $\cC$. |
1095 Throughout, we fix an $n$-category $\cC$. |
1097 For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
1096 For all but one axiom, it doesn't matter whether $\cC$ is a topological $n$-category or an $A_\infty$ $n$-category. |
1098 We state the final axiom, on actions of homeomorphisms, differently in the two cases. |
1097 We state the final axiom, regarding actions of homeomorphisms, differently in the two cases. |
1099 |
1098 |
1100 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1099 Define a {\it marked $k$-ball} to be a pair $(B, N)$ homeomorphic to the pair |
1101 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1100 $$(\text{standard $k$-ball}, \text{northern hemisphere in boundary of standard $k$-ball}).$$ |
1102 We call $B$ the ball and $N$ the marking. |
1101 We call $B$ the ball and $N$ the marking. |
1103 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
1102 A homeomorphism between marked $k$-balls is a homeomorphism of balls which |
1120 (The union is along $N\times \bd W$.) |
1119 (The union is along $N\times \bd W$.) |
1121 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1120 %(If $\cD$ were a general TQFT, we would define $\cM(B, N)$ to be |
1122 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1121 %the subset of $\cD((B\times \bd W)\cup (N\times W))$ which is splittable along $N\times \bd W$.) |
1123 |
1122 |
1124 \begin{figure}[!ht] |
1123 \begin{figure}[!ht] |
1125 $$\mathfig{.8}{ncat/boundary-collar}$$ |
1124 $$\mathfig{.55}{ncat/boundary-collar}$$ |
1126 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1125 \caption{From manifold with boundary collar to marked ball}\label{blah15}\end{figure} |
1127 |
1126 |
1128 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1127 Define the boundary of a marked $k$-ball $(B, N)$ to be the pair $(\bd B \setmin N, \bd N)$. |
1129 Call such a thing a {marked $k{-}1$-hemisphere}. |
1128 Call such a thing a {marked $k{-}1$-hemisphere}. |
1130 |
1129 |
1256 \end{figure} |
1255 \end{figure} |
1257 |
1256 |
1258 |
1257 |
1259 The above three axioms are equivalent to the following axiom, |
1258 The above three axioms are equivalent to the following axiom, |
1260 which we state in slightly vague form. |
1259 which we state in slightly vague form. |
1261 \nn{need figure for this} |
|
1262 |
1260 |
1263 \xxpar{Module multi-composition:} |
1261 \xxpar{Module multi-composition:} |
1264 {Given any decomposition |
1262 {Given any splitting |
1265 \[ |
1263 \[ |
1266 M = X_1 \cup\cdots\cup X_p \cup M_1\cup\cdots\cup M_q |
1264 X_1 \sqcup\cdots\sqcup X_p \sqcup M_1\sqcup\cdots\sqcup M_q \to M |
1267 \] |
1265 \] |
1268 of a marked $k$-ball $M$ |
1266 of a marked $k$-ball $M$ |
1269 into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
1267 into small (marked and plain) $k$-balls $M_i$ and $X_j$, there is a |
1270 map from an appropriate subset (like a fibered product) |
1268 map from an appropriate subset (like a fibered product) |
1271 of |
1269 of |
1447 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
1445 Fix a topological $n$-category or $A_\infty$ $n$-category $\cC$. |
1448 Let $W$ be a $k$-manifold ($k\le n$), |
1446 Let $W$ be a $k$-manifold ($k\le n$), |
1449 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1447 let $\{Y_i\}$ be a collection of disjoint codimension 0 submanifolds of $\bd W$, |
1450 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1448 and let $\cN = (\cN_i)$ be an assignment of a $\cC$ module $\cN_i$ to $Y_i$. |
1451 |
1449 |
1452 We will define a set $\cC(W, \cN)$ using a colimit construction similar to |
1450 We will define a set $\cC(W, \cN)$ using a colimit construction very similar to |
1453 the one appearing in \S \ref{ss:ncat_fields} above. |
1451 the one appearing in \S \ref{ss:ncat_fields} above. |
1454 (If $k = n$ and our $n$-categories are enriched, then |
1452 (If $k = n$ and our $n$-categories are enriched, then |
1455 $\cC(W, \cN)$ will have additional structure; see below.) |
1453 $\cC(W, \cN)$ will have additional structure; see below.) |
1456 |
1454 |
1457 Define a permissible decomposition of $W$ to be a decomposition |
1455 Define a permissible decomposition of $W$ to be a map |
1458 \[ |
1456 \[ |
1459 W = \left(\bigcup_a X_a\right) \cup \left(\bigcup_{i,b} M_{ib}\right) , |
1457 \left(\bigsqcup_a X_a\right) \sqcup \left(\bigsqcup_{i,b} M_{ib}\right) \to W, |
1460 \] |
1458 \] |
1461 where each $X_a$ is a plain $k$-ball (disjoint from $\cup Y_i$) and |
1459 where each $X_a$ is a plain $k$-ball disjoint, in $W$, from $\cup Y_i$, and |
1462 each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$, |
1460 each $M_{ib}$ is a marked $k$-ball intersecting $Y_i$ (once mapped into $W$), |
1463 with $M_{ib}\cap Y_i$ being the marking. |
1461 with $M_{ib}\cap Y_i$ being the marking, which extends to a ball decomposition in the sense of Definition \ref{defn:gluing-decomposition}. |
1464 (See Figure \ref{mblabel}.) |
1462 (See Figure \ref{mblabel}.) |
1465 \begin{figure}[t] |
1463 \begin{figure}[t] |
1466 \begin{equation*} |
1464 \begin{equation*} |
1467 \mathfig{.4}{ncat/mblabel} |
1465 \mathfig{.4}{ncat/mblabel} |
1468 \end{equation*} |
1466 \end{equation*} |