173 We will call the projection $\cC(S)_E \to \cC(B_i)$ |
173 We will call the projection $\cC(S)_E \to \cC(B_i)$ |
174 a {\it restriction} map and write $\res_{B_i}(a)$ |
174 a {\it restriction} map and write $\res_{B_i}(a)$ |
175 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
175 (or simply $\res(a)$ when there is no ambiguity), for $a\in \cC(S)_E$. |
176 More generally, we also include under the rubric ``restriction map" the |
176 More generally, we also include under the rubric ``restriction map" the |
177 the boundary maps of Axiom \ref{nca-boundary} above, |
177 the boundary maps of Axiom \ref{nca-boundary} above, |
178 another calss of maps introduced after Axion \ref{nca-assoc} below, as well as any composition |
178 another class of maps introduced after Axiom \ref{nca-assoc} below, as well as any composition |
179 of restriction maps (inductive definition). |
179 of restriction maps. |
180 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
180 In particular, we have restriction maps $\cC(X)_E \to \cC(B_i)$ |
181 ($i = 1, 2$, notation from previous paragraph). |
181 ($i = 1, 2$, notation from previous paragraph). |
182 These restriction maps can be thought of as |
182 These restriction maps can be thought of as |
183 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
183 domain and range maps, relative to the choice of splitting $\bd X = B_1 \cup_E B_2$. |
184 |
184 |
195 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
195 and $Y = B_1\cap B_2$ is a $k{-}1$-ball (Figure \ref{blah5}). |
196 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
196 Let $E = \bd Y$, which is a $k{-}2$-sphere. |
197 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
197 Note that each of $B$, $B_1$ and $B_2$ has its boundary split into two $k{-}1$-balls by $E$. |
198 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
198 We have restriction (domain or range) maps $\cC(B_i)_E \to \cC(Y)$. |
199 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
199 Let $\cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E$ denote the fibered product of these two maps. |
200 Then (axiom) we have a map |
200 We have a map |
201 \[ |
201 \[ |
202 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
202 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
203 \] |
203 \] |
204 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
204 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
205 to the intersection of the boundaries of $B$ and $B_i$. |
205 to the intersection of the boundaries of $B$ and $B_i$. |
219 \node[left] at (-1/4,1) {$B_1$}; |
219 \node[left] at (-1/4,1) {$B_1$}; |
220 \node[right] at (1/4,1) {$B_2$}; |
220 \node[right] at (1/4,1) {$B_2$}; |
221 \node at (1/6,3/2) {$Y$}; |
221 \node at (1/6,3/2) {$Y$}; |
222 \end{tikzpicture} |
222 \end{tikzpicture} |
223 $$ |
223 $$ |
224 $$\mathfig{.4}{tempkw/blah5}$$ |
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225 \caption{From two balls to one ball.}\label{blah5}\end{figure} |
224 \caption{From two balls to one ball.}\label{blah5}\end{figure} |
226 |
225 |
227 \begin{axiom}[Strict associativity] \label{nca-assoc} |
226 \begin{axiom}[Strict associativity] \label{nca-assoc} |
228 The composition (gluing) maps above are strictly associative. |
227 The composition (gluing) maps above are strictly associative. |
229 \end{axiom} |
228 \end{axiom} |
230 |
229 |
231 \begin{figure}[!ht] |
230 \begin{figure}[!ht] |
232 $$\mathfig{.65}{tempkw/blah6}$$ |
231 $$\mathfig{.65}{ncat/strict-associativity}$$ |
233 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
232 \caption{An example of strict associativity.}\label{blah6}\end{figure} |
234 |
233 |
235 \nn{figure \ref{blah6} (blah6) needs a dotted line in the south split ball} |
234 We'll use the notations $a\bullet b$ as well as $a \cup b$ for the glued together field $\gl_Y(a, b)$. |
236 |
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237 Notation: $a\bullet b \deq \gl_Y(a, b)$ and/or $a\cup b \deq \gl_Y(a, b)$. |
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238 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
235 In the other direction, we will call the projection from $\cC(B)_E$ to $\cC(B_i)_E$ |
239 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
236 a restriction map (one of many types of map so called) and write $\res_{B_i}(a)$ for $a\in \cC(B)_E$. |
240 %Compositions of boundary and restriction maps will also be called restriction maps. |
237 %Compositions of boundary and restriction maps will also be called restriction maps. |
241 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
238 %For example, if $B$ is a $k$-ball and $Y\sub \bd B$ is a $k{-}1$-ball, there is a |
242 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
239 %restriction map from $\cC(B)_{\bd Y}$ to $\cC(Y)$. |
243 |
240 |
244 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
241 We will write $\cC(B)_Y$ for the image of $\gl_Y$ in $\cC(B)$. |
245 We will call $\cC(B)_Y$ morphisms which are splittable along $Y$ or transverse to $Y$. |
242 We will call elements of $\cC(B)_Y$ morphisms which are `splittable along $Y$' or `transverse to $Y$'. |
246 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
243 We have $\cC(B)_Y \sub \cC(B)_E \sub \cC(B)$. |
247 |
244 |
248 More generally, let $\alpha$ be a subdivision of a ball (or sphere) $X$ into smaller balls. |
245 More generally, let $\alpha$ be a subdivision of a ball (or sphere) $X$ into smaller balls. |
249 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
246 Let $\cC(X)_\alpha \sub \cC(X)$ denote the image of the iterated gluing maps from |
250 the smaller balls to $X$. |
247 the smaller balls to $X$. |
251 We will also say that $\cC(X)_\alpha$ are morphisms which are splittable along $\alpha$. |
248 We say that elements of $\cC(X)_\alpha$ are morphisms which are `splittable along $\alpha$'. |
252 In situations where the subdivision is notationally anonymous, we will write |
249 In situations where the subdivision is notationally anonymous, we will write |
253 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
250 $\cC(X)\spl$ for the morphisms which are splittable along (a.k.a.\ transverse to) |
254 the unnamed subdivision. |
251 the unnamed subdivision. |
255 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$; |
252 If $\beta$ is a subdivision of $\bd X$, we define $\cC(X)_\beta \deq \bd\inv(\cC(\bd X)_\beta)$; |
256 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
253 this can also be denoted $\cC(X)\spl$ if the context contains an anonymous |
410 It can be thought of as the action of the inverse of |
407 It can be thought of as the action of the inverse of |
411 a map which projects a collar neighborhood of $Y$ onto $Y$. |
408 a map which projects a collar neighborhood of $Y$ onto $Y$. |
412 |
409 |
413 The revised axiom is |
410 The revised axiom is |
414 |
411 |
415 \stepcounter{axiom} |
412 \begin{axiom}{\textup{\textbf{[topological version]}} Extended isotopy invariance in dimension $n$} |
416 \begin{axiom-numbered}{\arabic{axiom}a}{Extended isotopy invariance in dimension $n$} |
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417 \label{axiom:extended-isotopies} |
413 \label{axiom:extended-isotopies} |
418 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
414 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
419 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
415 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
420 Then $f$ acts trivially on $\cC(X)$. |
416 Then $f$ acts trivially on $\cC(X)$. |
421 \end{axiom-numbered} |
417 \end{axiom} |
422 |
418 |
423 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
419 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
424 |
420 |
425 \smallskip |
421 \smallskip |
426 |
422 |
427 For $A_\infty$ $n$-categories, we replace |
423 For $A_\infty$ $n$-categories, we replace |
428 isotopy invariance with the requirement that families of homeomorphisms act. |
424 isotopy invariance with the requirement that families of homeomorphisms act. |
429 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
425 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
430 |
426 |
431 \begin{axiom-numbered}{\arabic{axiom}b}{Families of homeomorphisms act in dimension $n$} |
427 \addtocounter{axiom}{-1} |
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428 \begin{axiom}{\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$} |
432 For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
429 For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
433 \[ |
430 \[ |
434 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
431 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
435 \] |
432 \] |
436 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
433 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
438 These action maps are required to be associative up to homotopy |
435 These action maps are required to be associative up to homotopy |
439 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
436 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
440 a diagram like the one in Proposition \ref{CHprop} commutes. |
437 a diagram like the one in Proposition \ref{CHprop} commutes. |
441 \nn{repeat diagram here?} |
438 \nn{repeat diagram here?} |
442 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
439 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
443 \end{axiom-numbered} |
440 \end{axiom} |
444 |
441 |
445 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
442 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
446 To do this we need to explain how extended homeomorphisms form a topological space. |
443 To do this we need to explain how extended homeomorphisms form a topological space. |
447 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
444 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
448 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
445 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
450 \nn{this paragraph needs work.} |
447 \nn{this paragraph needs work.} |
451 |
448 |
452 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
449 Note that if we take homology of chain complexes, we turn an $A_\infty$ $n$-category |
453 into a plain $n$-category (enriched over graded groups). |
450 into a plain $n$-category (enriched over graded groups). |
454 \nn{say more here?} |
451 \nn{say more here?} |
455 In the other direction, if we enrich over topological spaces instead of chain complexes, |
452 In a different direction, if we enrich over topological spaces instead of chain complexes, |
456 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
453 we get a space version of an $A_\infty$ $n$-category, with $\Homeo_\bd(X)$ acting |
457 instead of $C_*(\Homeo_\bd(X))$. |
454 instead of $C_*(\Homeo_\bd(X))$. |
458 Taking singular chains converts a space-type $A_\infty$ $n$-category into a chain complex |
455 Taking singular chains converts such a space type $A_\infty$ $n$-category into a chain complex |
459 type $A_\infty$ $n$-category. |
456 type $A_\infty$ $n$-category. |
460 |
457 |
461 \medskip |
458 \medskip |
462 |
459 |
463 The alert reader will have already noticed that our definition of (plain) $n$-category |
460 The alert reader will have already noticed that our definition of (plain) $n$-category |
571 |
568 |
572 \begin{example}[Chains of maps to a space] |
569 \begin{example}[Chains of maps to a space] |
573 \rm |
570 \rm |
574 \label{ex:chains-of-maps-to-a-space} |
571 \label{ex:chains-of-maps-to-a-space} |
575 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
572 We can modify Example \ref{ex:maps-to-a-space} above to define the fundamental $A_\infty$ $n$-category $\pi^\infty_{\le n}(T)$ of a topological space $T$. |
576 For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps{X \to T}$. |
573 For $k$-balls and $k$-spheres $X$, with $k < n$, the sets $\pi^\infty_{\leq n}(T)(X)$ are just $\Maps(X \to T)$. |
577 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
574 Define $\pi^\infty_{\leq n}(T)(X; c)$ for an $n$-ball $X$ and $c \in \pi^\infty_{\leq n}(T)(\bdy X)$ to be the chain complex |
578 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
575 $$C_*(\Maps_c(X\times F \to T)),$$ where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
579 and $C_*$ denotes singular chains. |
576 and $C_*$ denotes singular chains. |
580 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
577 \nn{maybe should also mention version where we enrich over spaces rather than chain complexes} |
581 \end{example} |
578 \end{example} |
582 |
579 |
583 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps{M \to T})$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
580 See also Theorem \ref{thm:map-recon} below, recovering $C_*(\Maps(M \to T))$ up to homotopy the blob complex of $M$ with coefficients in $\pi^\infty_{\le n}(T)$. |
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581 |
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582 \todo{Yikes! We can't do this yet, because we haven't explained how to produce a system of fields and local relations from a topological $n$-category.} |
584 |
583 |
585 \begin{example}[Blob complexes of balls (with a fiber)] |
584 \begin{example}[Blob complexes of balls (with a fiber)] |
586 \rm |
585 \rm |
587 \label{ex:blob-complexes-of-balls} |
586 \label{ex:blob-complexes-of-balls} |
588 Fix an $m$-dimensional manifold $F$. We will define an $A_\infty$ $n-m$-category $\cC$. |
587 Fix an $m$-dimensional manifold $F$. We will define an $A_\infty$ $(n-m)$-category $\cC$. |
589 Given a plain $n$-category $C$, |
588 Given a plain $n$-category $C$, |
590 when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball, |
589 when $X$ is a $k$-ball or $k$-sphere, with $k<n-m$, define $\cC(X) = C(X)$. When $X$ is an $(n-m)$-ball, |
591 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
590 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
592 where $\bc^C_*$ denotes the blob complex based on $C$. |
591 where $\bc^C_*$ denotes the blob complex based on $C$. |
593 \end{example} |
592 \end{example} |
594 |
593 |
595 This example will be essential for Theorem \ref{product_thm} below, which relates ... |
594 This example will be essential for Theorem \ref{product_thm} below, which allows us to compute the blob complex of a product. Notice that with $F$ a point, the above example is a construction turning a topological $n$-category into an $A_\infty$ $n$-category. We think of this as providing a `free resolution' of the topological $n$-category. \todo{Say more here!} In fact, there is also a trivial way to do this: we can think of each vector space associated to an $n$-ball as a chain complex concentrated in degree $0$, and take $\CD{B}$ to act trivially. |
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595 |
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596 Be careful that the `free resolution' of the topological $n$-category $\pi_{\leq n}(T)$ is not the $A_\infty$ $n$-category $\pi^\infty_{\leq n}(T)$. It's easy to see that with $n=0$, the corresponding system of fields is just linear combinations of connected components of $T$, and the local relations are trivial. There's no way for the blob complex to magically recover all the data of $\pi^\infty_{\leq 0}(T) \iso C_* T$. |
596 |
597 |
597 \begin{example} |
598 \begin{example} |
598 \nn{should add $\infty$ version of bordism $n$-cat} |
599 \nn{should add $\infty$ version of bordism $n$-cat} |
599 \end{example} |
600 \end{example} |
600 |
601 |