255 %which are weak enough to include the basic examples and strong enough to support the proofs |
255 %which are weak enough to include the basic examples and strong enough to support the proofs |
256 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
256 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
257 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
257 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
258 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
258 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
259 |
259 |
260 We will define plain and $A_\infty$ $n$-categories simultaneously, as all but one of the axioms are identical |
260 We will define two variations simultaneously, as all but one of the axioms are identical |
261 in the two cases. |
261 in the two cases. These variations are `linear $n$-categories', where the sets associated to $n$-balls with specified boundary conditions are in fact vector spaces, and `$A_\infty$ $n$-categories', where these sets are chain complexes. |
262 |
262 |
263 |
263 |
264 There are five basic ingredients |
264 There are five basic ingredients |
265 \cite{life-of-brian} of an $n$-category definition: |
265 \cite{life-of-brian} of an $n$-category definition: |
266 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
266 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
354 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
354 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
355 to the intersection of the boundaries of $B$ and $B_i$. |
355 to the intersection of the boundaries of $B$ and $B_i$. |
356 If $k < n$, |
356 If $k < n$, |
357 or if $k=n$ and we are in the $A_\infty$ case, |
357 or if $k=n$ and we are in the $A_\infty$ case, |
358 we require that $\gl_Y$ is injective. |
358 we require that $\gl_Y$ is injective. |
359 (For $k=n$ in the plain (non-$A_\infty$) case, see below.) |
359 (For $k=n$ in the linear case, see below.) |
360 \end{axiom} |
360 \end{axiom} |
361 |
361 |
362 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
362 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
363 The gluing maps above are strictly associative. |
363 The gluing maps above are strictly associative. |
364 Given any decomposition of a ball $B$ into smaller balls |
364 Given any decomposition of a ball $B$ into smaller balls |
434 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
434 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
435 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
435 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
436 to the identity on the boundary. |
436 to the identity on the boundary. |
437 |
437 |
438 |
438 |
439 \begin{axiom}[\textup{\textbf{[plain version]}} Extended isotopy invariance in dimension $n$.] |
439 \begin{axiom}[\textup{\textbf{[linear version]}} Extended isotopy invariance in dimension $n$.] |
440 \label{axiom:extended-isotopies} |
440 \label{axiom:extended-isotopies} |
441 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
441 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
442 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
442 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
443 Then $f$ acts trivially on $\cC(X)$. |
443 Then $f$ acts trivially on $\cC(X)$. |
444 In addition, collar maps act trivially on $\cC(X)$. |
444 In addition, collar maps act trivially on $\cC(X)$. |
659 |
659 |
660 The blob complex has several important special cases. |
660 The blob complex has several important special cases. |
661 |
661 |
662 \begin{thm}[Skein modules] |
662 \begin{thm}[Skein modules] |
663 \label{thm:skein-modules} |
663 \label{thm:skein-modules} |
664 \nn{Plain n-categories only?} |
664 \nn{linear n-categories only?} |
665 The $0$-th blob homology of $X$ is the usual |
665 The $0$-th blob homology of $X$ is the usual |
666 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
666 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
667 by $\cC$. |
667 by $\cC$. |
668 \begin{equation*} |
668 \begin{equation*} |
669 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |
669 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X) |