194 |
194 |
195 There are five basic ingredients of an $n$-category definition: |
195 There are five basic ingredients of an $n$-category definition: |
196 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
196 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
197 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
197 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
198 in some auxiliary category, or strict associativity instead of weak associativity). |
198 in some auxiliary category, or strict associativity instead of weak associativity). |
199 We will treat each of these it turn. |
199 We will treat each of these in turn. |
200 |
200 |
201 To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
201 To motivate our morphism axiom, consider the venerable notion of the Moore loop space |
202 \nn{need citation}. |
202 \nn{need citation -- \S 2.2 of Adams' ``Infinite Loop Spaces''?}. |
203 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
203 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$, |
204 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
204 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation |
205 of higher associativity relations. |
205 of higher associativity relations. |
206 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
206 While this proliferation is manageable for 1-categories (and indeed leads to an elegant theory |
207 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
207 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
221 For each $0 \le k \le n$, we have a functor $\cC_k$ from |
222 the category of $k$-balls and |
222 the category of $k$-balls and |
223 homeomorphisms to the category of sets and bijections. |
223 homeomorphisms to the category of sets and bijections. |
224 \end{axiom} |
224 \end{axiom} |
225 |
225 |
226 Note that the functoriality in the above axiom allows us to operate via |
226 Note that the functoriality in the above axiom allows us to operate via \nn{fragment?} |
227 |
227 |
228 Next we consider domains and ranges of $k$-morphisms. |
228 Next we consider domains and ranges of $k$-morphisms. |
229 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
229 Because we assume strong duality, it doesn't make much sense to subdivide the boundary of a morphism |
230 into domain and range --- the duality operations can convert domain to range and vice-versa. |
230 into domain and range --- the duality operations can convert domain to range and vice-versa. |
231 Instead, we will use a unified domain/range, which we will call a ``boundary". |
231 Instead, we will use a unified domain/range, which we will call a ``boundary". |
281 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
281 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
282 \end{axiom} |
282 \end{axiom} |
283 |
283 |
284 \begin{axiom}[Strict associativity] \label{nca-assoc} |
284 \begin{axiom}[Strict associativity] \label{nca-assoc} |
285 The composition (gluing) maps above are strictly associative. |
285 The composition (gluing) maps above are strictly associative. |
286 Given any splitting of a ball $B$ into smaller balls |
286 Given any decomposition of a ball $B$ into smaller balls |
287 $$\bigsqcup B_i \to B,$$ |
287 $$\bigsqcup B_i \to B,$$ |
288 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. |
288 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
289 \end{axiom} |
289 \end{axiom} |
290 \begin{axiom}[Product (identity) morphisms] |
290 \begin{axiom}[Product (identity) morphisms] |
291 \label{axiom:product} |
291 \label{axiom:product} |
292 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
292 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
293 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
293 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
372 \subsubsection{Decompositions of manifolds} |
372 \subsubsection{Decompositions of manifolds} |
373 |
373 |
374 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
374 \nn{KW: I'm inclined to suppress all discussion of the subtleties of decompositions. |
375 Maybe just a single remark that we are omitting some details which appear in our |
375 Maybe just a single remark that we are omitting some details which appear in our |
376 longer paper.} |
376 longer paper.} |
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377 \nn{SM: for now I disagree: the space expense is pretty minor, and it always us to be "in principle" complete. Let's see how we go for length.} |
377 |
378 |
378 A \emph{ball decomposition} of $W$ is a |
379 A \emph{ball decomposition} of $W$ is a |
379 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
380 sequence of gluings $M_0\to M_1\to\cdots\to M_m = W$ such that $M_0$ is a disjoint union of balls |
380 $\du_a X_a$ and each $M_i$ is a manifold. |
381 $\du_a X_a$ and each $M_i$ is a manifold. |
381 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
382 If $X_a$ is some component of $M_0$, its image in $W$ need not be a ball; $\bd X_a$ may have been glued to itself. |
454 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
455 For $k=2$, we have a two types of generators; they each consists of a field $f$ on $W$, and two balls $B_1$ and $B_2$. In the first case, the balls are disjoint, and $f$ restricted to either of the $B_i$ is a null field. In the second case, the balls are properly nested, say $B_1 \subset B_2$, and $f$ restricted to $B_1$ is null. Note that this implies that $f$ restricted to $B_2$ is also null, by the associativity of the gluing operation. This ensures that the differential is well-defined. |
455 |
456 |
456 \section{Properties of the blob complex} |
457 \section{Properties of the blob complex} |
457 \subsection{Formal properties} |
458 \subsection{Formal properties} |
458 \label{sec:properties} |
459 \label{sec:properties} |
459 The blob complex enjoys the following list of formal properties. |
460 The blob complex enjoys the following list of formal properties. The first three properties are immediate from the definitions. |
460 |
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461 The proofs of the first three properties are immediate from the definitions. |
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462 |
461 |
463 \begin{property}[Functoriality] |
462 \begin{property}[Functoriality] |
464 \label{property:functoriality}% |
463 \label{property:functoriality}% |
465 The blob complex is functorial with respect to homeomorphisms. |
464 The blob complex is functorial with respect to homeomorphisms. |
466 That is, |
465 That is, |
514 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
513 obtained by adding an outer $(k{+}1)$-st blob consisting of all $B^n$. |
515 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
514 For $k=0$ we choose a splitting $s: H_0(\bc_*(B^n)) \to \bc_0(B^n)$ and send |
516 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
515 $x\in \bc_0(B^n)$ to $x - s([x])$, where $[x]$ denotes the image of $x$ in $H_0(\bc_*(B^n))$. |
517 \end{proof} |
516 \end{proof} |
518 |
517 |
519 \nn{Properties \ref{property:functoriality} will be immediate from the definition given in |
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520 \S \ref{sec:blob-definition}, and we'll recall it at the appropriate point there. |
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521 Properties \ref{property:disjoint-union}, \ref{property:gluing-map} and |
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522 \ref{property:contractibility} are established in \S \ref{sec:basic-properties}.} |
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523 |
518 |
524 \subsection{Specializations} |
519 \subsection{Specializations} |
525 \label{sec:specializations} |
520 \label{sec:specializations} |
526 |
521 |
527 The blob complex has two important special cases. |
522 The blob complex has two important special cases. |