2 |
2 |
3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
3 \def\xxpar#1#2{\smallskip\noindent{\bf #1} {\it #2} \smallskip} |
4 |
4 |
5 \section{$n$-categories} |
5 \section{$n$-categories} |
6 \label{sec:ncats} |
6 \label{sec:ncats} |
7 |
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8 %In order to make further progress establishing properties of the blob complex, |
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9 %we need a definition of $A_\infty$ $n$-category that is adapted to our needs. |
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10 %(Even in the case $n=1$, we need the new definition given below.) |
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11 %It turns out that the $A_\infty$ $n$-category definition and the plain $n$-category |
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12 %definition are mostly the same, so we give a new definition of plain |
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13 %$n$-categories too. |
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14 %We also define modules and tensor products for both plain and $A_\infty$ $n$-categories. |
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15 |
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16 |
7 |
17 \subsection{Definition of $n$-categories} |
8 \subsection{Definition of $n$-categories} |
18 |
9 |
19 Before proceeding, we need more appropriate definitions of $n$-categories, |
10 Before proceeding, we need more appropriate definitions of $n$-categories, |
20 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
11 $A_\infty$ $n$-categories, modules for these, and tensor products of these modules. |
327 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
318 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
328 \end{eqnarray*} |
319 \end{eqnarray*} |
329 (See Figure \ref{glue-collar}.) |
320 (See Figure \ref{glue-collar}.) |
330 \begin{figure}[!ht] |
321 \begin{figure}[!ht] |
331 \begin{equation*} |
322 \begin{equation*} |
332 \mathfig{.9}{tempkw/blah10} |
323 \begin{tikzpicture} |
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324 \def\rad{1} |
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325 \def\srad{0.75} |
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326 \def\gap{4.5} |
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327 \foreach \i in {0, 1, 2} { |
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328 \node(\i) at ($\i*(\gap,0)$) [draw, circle through = {($\i*(\gap,0)+(\rad,0)$)}] {}; |
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329 \node(\i-small) at (\i.east) [circle through={($(\i.east)+(\srad,0)$)}] {}; |
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330 \foreach \n in {1,2} { |
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331 \fill (intersection \n of \i-small and \i) node(\i-intersection-\n) {} circle (2pt); |
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332 } |
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333 } |
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334 |
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335 \begin{scope}[decoration={brace,amplitude=10,aspect=0.5}] |
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336 \draw[decorate] (0-intersection-1.east) -- (0-intersection-2.east); |
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337 \end{scope} |
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338 \node[right=1mm] at (0.east) {$a$}; |
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339 \draw[->] ($(0.east)+(0.75,0)$) -- ($(1.west)+(-0.2,0)$); |
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340 |
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341 \draw (1-small) circle (\srad); |
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342 \foreach \theta in {90, 72, ..., -90} { |
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343 \draw[blue] (1) -- ($(1)+(\rad,0)+(\theta:\srad)$); |
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344 } |
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345 \filldraw[fill=white] (1) circle (\rad); |
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346 \foreach \n in {1,2} { |
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347 \fill (intersection \n of 1-small and 1) circle (2pt); |
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348 } |
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349 \node[below] at (1-small.south) {$a \times J$}; |
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350 \draw[->] ($(1.east)+(1,0)$) -- ($(2.west)+(-0.2,0)$); |
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351 |
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352 \begin{scope} |
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353 \path[clip] (2) circle (\rad); |
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354 \draw[clip] (2.east) circle (\srad); |
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355 \foreach \y in {1, 0.86, ..., -1} { |
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356 \draw[blue] ($(2)+(-1,\y) $)-- ($(2)+(1,\y)$); |
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357 } |
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358 \end{scope} |
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359 \end{tikzpicture} |
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360 \end{equation*} |
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361 \begin{equation*} |
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362 \xymatrix@C+2cm{\cC(X) \ar[r]^(0.45){\text{glue}} & \cC(X \cup \text{collar}) \ar[r]^(0.55){\text{homeo}} & \cC(X)} |
333 \end{equation*} |
363 \end{equation*} |
334 |
364 |
335 \caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
365 \caption{Extended homeomorphism.}\label{glue-collar}\end{figure} |
336 We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
366 We say that $\psi_{Y,J}$ is {\it extended isotopic} to the identity map. |
337 \nn{bad terminology; fix it later} |
367 \nn{bad terminology; fix it later} |
343 It can be thought of as the action of the inverse of |
373 It can be thought of as the action of the inverse of |
344 a map which projects a collar neighborhood of $Y$ onto $Y$. |
374 a map which projects a collar neighborhood of $Y$ onto $Y$. |
345 |
375 |
346 The revised axiom is |
376 The revised axiom is |
347 |
377 |
348 \begin{axiom}[Extended isotopy invariance in dimension $n$] |
378 \stepcounter{axiom} |
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379 \begin{axiom-numbered}{\arabic{axiom}a}{Extended isotopy invariance in dimension $n$} |
349 \label{axiom:extended-isotopies} |
380 \label{axiom:extended-isotopies} |
350 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
381 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
351 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
382 to the identity on $\bd X$ and is extended isotopic (rel boundary) to the identity. |
352 Then $f$ acts trivially on $\cC(X)$. |
383 Then $f$ acts trivially on $\cC(X)$. |
353 \end{axiom} |
384 \end{axiom-numbered} |
354 |
385 |
355 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
386 \nn{need to rephrase this, since extended isotopies don't correspond to homeomorphisms.} |
356 |
387 |
357 \smallskip |
388 \smallskip |
358 |
389 |
359 For $A_\infty$ $n$-categories, we replace |
390 For $A_\infty$ $n$-categories, we replace |
360 isotopy invariance with the requirement that families of homeomorphisms act. |
391 isotopy invariance with the requirement that families of homeomorphisms act. |
361 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
392 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
362 |
393 |
363 \begin{axiom}[Families of homeomorphisms act in dimension $n$] |
394 \begin{axiom-numbered}{\arabic{axiom}b}{Families of homeomorphisms act in dimension $n$} |
364 For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
395 For each $n$-ball $X$ and each $c\in \cC(\bd X)$ we have a map of chain complexes |
365 \[ |
396 \[ |
366 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
397 C_*(\Homeo_\bd(X))\ot \cC(X; c) \to \cC(X; c) . |
367 \] |
398 \] |
368 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
399 Here $C_*$ means singular chains and $\Homeo_\bd(X)$ is the space of homeomorphisms of $X$ |
370 These action maps are required to be associative up to homotopy |
401 These action maps are required to be associative up to homotopy |
371 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
402 \nn{iterated homotopy?}, and also compatible with composition (gluing) in the sense that |
372 a diagram like the one in Proposition \ref{CDprop} commutes. |
403 a diagram like the one in Proposition \ref{CDprop} commutes. |
373 \nn{repeat diagram here?} |
404 \nn{repeat diagram here?} |
374 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
405 \nn{restate this with $\Homeo(X\to X')$? what about boundary fixing property?} |
375 \end{axiom} |
406 \end{axiom-numbered} |
376 |
407 |
377 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
408 We should strengthen the above axiom to apply to families of extended homeomorphisms. |
378 To do this we need to explain how extended homeomorphisms form a topological space. |
409 To do this we need to explain how extended homeomorphisms form a topological space. |
379 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
410 Roughly, the set of $n{-}1$-balls in the boundary of an $n$-ball has a natural topology, |
380 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
411 and we can replace the class of all intervals $J$ with intervals contained in $\r$. |
410 %The universal (colimit) construction becomes our generalized definition of blob homology. |
441 %The universal (colimit) construction becomes our generalized definition of blob homology. |
411 %Need to explain how it relates to the old definition.} |
442 %Need to explain how it relates to the old definition.} |
412 |
443 |
413 \medskip |
444 \medskip |
414 |
445 |
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446 \subsection{Examples of $n$-categories} |
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447 |
415 \nn{these examples need to be fleshed out a bit more} |
448 \nn{these examples need to be fleshed out a bit more} |
416 |
449 |
417 Examples of plain $n$-categories: |
450 We know describe several classes of examples of $n$-categories satisfying our axioms. |
418 \begin{itemize} |
451 |
419 |
452 \begin{example}{Maps to a space} |
420 \item Let $F$ be a closed $m$-manifold (e.g.\ a point). |
453 \label{ex:maps-to-a-space}% |
421 Let $T$ be a topological space. |
454 Fix $F$ a closed $m$-manifold (keep in mind the case where $F$ is a point). Fix a `target space' $T$, any topological space. |
422 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
455 For $X$ a $k$-ball or $k$-sphere with $k < n$, define $\cC(X)$ to be the set of |
423 all maps from $X\times F$ to $T$. |
456 all maps from $X\times F$ to $T$. |
424 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
457 For $X$ an $n$-ball define $\cC(X)$ to be maps from $X\times F$ to $T$ modulo |
425 homotopies fixed on $\bd X \times F$. |
458 homotopies fixed on $\bd X \times F$. |
426 (Note that homotopy invariance implies isotopy invariance.) |
459 (Note that homotopy invariance implies isotopy invariance.) |
427 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
460 For $a\in \cC(X)$ define the product morphism $a\times D \in \cC(X\times D)$ to |
428 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
461 be $a\circ\pi_X$, where $\pi_X : X\times D \to X$ is the projection. |
429 |
462 \end{example} |
430 \item We can linearize the above example as follows. |
463 |
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464 \begin{example}{Linearized, twisted, maps to a space} |
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465 \label{ex:linearized-maps-to-a-space}% |
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466 We can linearize the above example as follows. |
431 Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
467 Let $\alpha$ be an $(n{+}m{+}1)$-cocycle on $T$ with values in a ring $R$ |
432 (e.g.\ the trivial cocycle). |
468 (e.g.\ the trivial cocycle). |
433 For $X$ of dimension less than $n$ define $\cC(X)$ as before. |
469 For $X$ of dimension less than $n$ define $\cC(X)$ as before. |
434 For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be |
470 For $X$ an $n$-ball and $c\in \cC(\bd X)$ define $\cC(X; c)$ to be |
435 the $R$-module of finite linear combinations of maps from $X\times F$ to $T$, |
471 the $R$-module of finite linear combinations of maps from $X\times F$ to $T$, |
436 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
472 modulo the relation that if $a$ is homotopic to $b$ (rel boundary) via a homotopy |
437 $h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. |
473 $h: X\times F\times I \to T$, then $a \sim \alpha(h)b$. |
438 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
474 \nn{need to say something about fundamental classes, or choose $\alpha$ carefully} |
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475 \end{example} |
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476 |
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477 \begin{itemize} |
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478 |
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479 \item \nn{Continue converting these into examples} |
439 |
480 |
440 \item Given a traditional $n$-category $C$ (with strong duality etc.), |
481 \item Given a traditional $n$-category $C$ (with strong duality etc.), |
441 define $\cC(X)$ (with $\dim(X) < n$) |
482 define $\cC(X)$ (with $\dim(X) < n$) |
442 to be the set of all $C$-labeled sub cell complexes of $X$. |
483 to be the set of all $C$-labeled sub cell complexes of $X$. |
443 (See Subsection \ref{sec:fields}.) |
484 (See Subsection \ref{sec:fields}.) |
471 \item \nn{sphere modules; ref to below} |
512 \item \nn{sphere modules; ref to below} |
472 |
513 |
473 \end{itemize} |
514 \end{itemize} |
474 |
515 |
475 |
516 |
476 Examples of $A_\infty$ $n$-categories: |
517 We have two main examples of $A_\infty$ $n$-categories, coming from maps to a target space and from the blob complex. |
477 \begin{itemize} |
518 |
478 |
519 \begin{example}{Chains of maps to a space} |
479 \item Same as in example \nn{xxxx} above (fiber $F$, target space $T$), |
520 We can modify Example \ref{ex:maps-to-a-space} above by defining $\cC(X; c)$ for an $n$-ball $X$ to be the chain complex |
480 but we define, for an $n$-ball $X$, $\cC(X; c)$ to be the chain complex |
521 $C_*(\Maps_c(X\times F \to T))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
481 $C_*(\Maps_c(X\times F))$, where $\Maps_c$ denotes continuous maps restricting to $c$ on the boundary, |
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482 and $C_*$ denotes singular chains. |
522 and $C_*$ denotes singular chains. |
483 |
523 \end{example} |
484 \item |
524 |
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525 \begin{example}{Blob complexes of balls (with a fiber)} |
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526 Fix an $m$-dimensional manifold $F$. |
485 Given a plain $n$-category $C$, |
527 Given a plain $n$-category $C$, |
486 define $\cC(X; c) = \bc^C_*(X\times F; c)$, where $X$ is an $n$-ball |
528 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, |
487 and $\bc^C_*$ denotes the blob complex based on $C$. |
529 define $\cC(X; c) = \bc^C_*(X\times F; c)$ |
488 |
530 where $\bc^C_*$ denotes the blob complex based on $C$. |
489 \item \nn{should add $\infty$ version of bordism $n$-cat} |
531 \end{example} |
490 |
532 |
491 \end{itemize} |
533 \begin{defn} |
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534 \nn{should add $\infty$ version of bordism $n$-cat} |
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535 \end{defn} |
492 |
536 |
493 |
537 |
494 |
538 |
495 |
539 |
496 |
540 |