268 %which are weak enough to include the basic examples and strong enough to support the proofs |
268 %which are weak enough to include the basic examples and strong enough to support the proofs |
269 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
269 %of the main theorems; and (2) specify a minimal set of generators and/or axioms. |
270 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
270 %We separate these two tasks, and address only the first, which becomes much easier when not burdened by the second. |
271 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
271 %More specifically, life is easier when working with maximal, rather than minimal, collections of axioms.} |
272 |
272 |
273 We will define two variations simultaneously, as all but one of the axioms are identical |
273 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
274 in the two cases. These variations are `linear $n$-categories', where the sets associated to |
274 These variations are `isotopy $n$-categories', where homeomorphisms fixing the boundary |
275 $n$-balls with specified boundary conditions are in fact vector spaces, and `$A_\infty$ $n$-categories', |
275 act trivially on the sets associated to $n$-balls |
276 where these sets are chain complexes. |
276 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
277 |
277 and `$A_\infty$ $n$-categories', where there is a homotopy action of |
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278 $k$-parameter families of homeomorphisms on these sets |
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279 (which are usually chain complexes or topological spaces). |
278 |
280 |
279 There are five basic ingredients |
281 There are five basic ingredients |
280 \cite{life-of-brian} of an $n$-category definition: |
282 \cite{life-of-brian} of an $n$-category definition: |
281 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
283 $k$-morphisms (for $0\le k \le n$), domain and range, composition, |
282 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
284 identity morphisms, and special behavior in dimension $n$ (e.g. enrichment |
372 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
374 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
373 to the intersection of the boundaries of $B$ and $B_i$. |
375 to the intersection of the boundaries of $B$ and $B_i$. |
374 If $k < n$, |
376 If $k < n$, |
375 or if $k=n$ and we are in the $A_\infty$ case, |
377 or if $k=n$ and we are in the $A_\infty$ case, |
376 we require that $\gl_Y$ is injective. |
378 we require that $\gl_Y$ is injective. |
377 (For $k=n$ in the linear case, see below.) |
379 (For $k=n$ in the isotopy $n$-category case, see below.) |
378 \end{axiom} |
380 \end{axiom} |
379 |
381 |
380 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
382 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
381 The gluing maps above are strictly associative. |
383 The gluing maps above are strictly associative. |
382 Given any decomposition of a ball $B$ into smaller balls |
384 Given any decomposition of a ball $B$ into smaller balls |
453 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
455 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
454 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
456 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
455 to the identity on the boundary. |
457 to the identity on the boundary. |
456 |
458 |
457 |
459 |
458 \begin{axiom}[\textup{\textbf{[linear version]}} Extended isotopy invariance in dimension $n$.] |
460 \begin{axiom}[\textup{\textbf{[for isotopy $n$-categories]}} Extended isotopy invariance in dimension $n$.] |
459 \label{axiom:extended-isotopies} |
461 \label{axiom:extended-isotopies} |
460 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
462 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
461 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
463 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
462 Then $f$ acts trivially on $\cC(X)$. |
464 Then $f$ acts trivially on $\cC(X)$. |
463 In addition, collar maps act trivially on $\cC(X)$. |
465 In addition, collar maps act trivially on $\cC(X)$. |
470 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
472 For the moment, assume that our $n$-morphisms are enriched over chain complexes. |
471 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
473 Let $\Homeo_\bd(X)$ denote homeomorphisms of $X$ which fix $\bd X$ and |
472 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
474 $C_*(\Homeo_\bd(X))$ denote the singular chains on this space. |
473 |
475 |
474 |
476 |
475 \begin{axiom}[\textup{\textbf{[$A_\infty$ version]}} Families of homeomorphisms act in dimension $n$.] |
477 \begin{axiom}[\textup{\textbf{[for $A_\infty$ $n$-categories]}} Families of homeomorphisms act in dimension $n$.] |
476 \label{axiom:families} |
478 \label{axiom:families} |
477 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
479 For each $n$-ball $X$ and each $c\in \cl{\cC}(\bd X)$ we have a map of chain complexes |
478 \[ |
480 \[ |
479 C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
481 C_*(\Homeo_\bd(X))\tensor \cC(X; c) \to \cC(X; c) . |
480 \] |
482 \] |
568 We will use the term `field on $W$' to refer to a point of this functor, |
570 We will use the term `field on $W$' to refer to a point of this functor, |
569 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
571 that is, a permissible decomposition $x$ of $W$ together with an element of $\psi_{\cC;W}(x)$. |
570 |
572 |
571 |
573 |
572 \subsubsection{Colimits} |
574 \subsubsection{Colimits} |
573 Our definition of an $n$-category is essentially a collection of functors defined on $k$-balls (and homeomorphisms) |
575 Recall that our definition of an $n$-category is essentially a collection of functors |
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576 defined on the categories of homeomorphisms $k$-balls |
574 for $k \leq n$ satisfying certain axioms. |
577 for $k \leq n$ satisfying certain axioms. |
575 It is natural to consider extending such functors to the |
578 It is natural to hope to extend such functors to the |
576 larger categories of all $k$-manifolds (again, with homeomorphisms). |
579 larger categories of all $k$-manifolds (again, with homeomorphisms). |
577 In fact, the axioms stated above explicitly require such an extension to $k$-spheres for $k<n$. |
580 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
578 |
581 |
579 The natural construction achieving this is the colimit. For a linear $n$-category $\cC$, |
582 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
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583 For an isotopy $n$-category $\cC$, |
580 we denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
584 we denote the extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
581 this is defined to be the colimit of the function $\psi_{\cC;W}$. |
585 this is defined to be the colimit of the functor $\psi_{\cC;W}$. |
582 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
586 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
583 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
587 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
584 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
588 Recall that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
585 the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy X)$, |
589 the set $\cC(X;c)$ is a vector space. Using this, we note that for $c \in \cl{\cC}(\bdy X)$, |
586 for $X$ an arbitrary $n$-manifold, the set $\cl{\cC}(X;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
590 for $X$ an arbitrary $n$-manifold, the set $\cl{\cC}(X;c) = \bdy^{-1} (c)$ inherits the structure of a vector space. |
717 |
721 |
718 The blob complex has several important special cases. |
722 The blob complex has several important special cases. |
719 |
723 |
720 \begin{thm}[Skein modules] |
724 \begin{thm}[Skein modules] |
721 \label{thm:skein-modules} |
725 \label{thm:skein-modules} |
722 Suppose $\cC$ is a linear $n$-category |
726 Suppose $\cC$ is an isotopy $n$-category |
723 The $0$-th blob homology of $X$ is the usual |
727 The $0$-th blob homology of $X$ is the usual |
724 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
728 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
725 by $\cC$. |
729 by $\cC$. |
726 \begin{equation*} |
730 \begin{equation*} |
727 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
731 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
861 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
865 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
862 |
866 |
863 \begin{thm}[Product formula] |
867 \begin{thm}[Product formula] |
864 \label{thm:product} |
868 \label{thm:product} |
865 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
869 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
866 Let $\cC$ be a linear $n$-category. |
870 Let $\cC$ be an isotopy $n$-category. |
867 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
871 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
868 Then |
872 Then |
869 \[ |
873 \[ |
870 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
874 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
871 \] |
875 \] |