268 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
268 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
269 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while |
269 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while |
270 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs. |
270 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs. |
271 |
271 |
272 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
272 We will define two variations simultaneously, as all but one of the axioms are identical in the two cases. |
273 These variations are ``plain $n$-categories", where homeomorphisms fixing the boundary |
273 These variations are ``ordinary $n$-categories", where homeomorphisms fixing the boundary |
274 act trivially on the sets associated to $n$-balls |
274 act trivially on the sets associated to $n$-balls |
275 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
275 (and these sets are usually vector spaces or more generally modules over a commutative ring) |
276 and ``$A_\infty$ $n$-categories", where there is a homotopy action of |
276 and ``$A_\infty$ $n$-categories", where there is a homotopy action of |
277 $k$-parameter families of homeomorphisms on these sets |
277 $k$-parameter families of homeomorphisms on these sets |
278 (which are usually chain complexes or topological spaces). |
278 (which are usually chain complexes or topological spaces). |
373 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
373 which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions |
374 to the intersection of the boundaries of $B$ and $B_i$. |
374 to the intersection of the boundaries of $B$ and $B_i$. |
375 If $k < n$, |
375 If $k < n$, |
376 or if $k=n$ and we are in the $A_\infty$ case, |
376 or if $k=n$ and we are in the $A_\infty$ case, |
377 we require that $\gl_Y$ is injective. |
377 we require that $\gl_Y$ is injective. |
378 (For $k=n$ in the plain $n$-category case, see Axiom \ref{axiom:extended-isotopies}.) |
378 (For $k=n$ in the ordinary $n$-category case, see Axiom \ref{axiom:extended-isotopies}.) |
379 \end{axiom} |
379 \end{axiom} |
380 |
380 |
381 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
381 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
382 The gluing maps above are strictly associative. |
382 The gluing maps above are strictly associative. |
383 Given any decomposition of a ball $B$ into smaller balls |
383 Given any decomposition of a ball $B$ into smaller balls |
459 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
459 where the first arrow is gluing with a product morphism on $Y\times_p J$ and |
460 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
460 the second is induced by a homeomorphism from $X\cup_Y (Y\times_p J)$ to $X$ which restricts |
461 to the identity on the boundary. |
461 to the identity on the boundary. |
462 |
462 |
463 |
463 |
464 \begin{axiom}[\textup{\textbf{[for plain $n$-categories]}} Extended isotopy invariance in dimension $n$.] |
464 \begin{axiom}[\textup{\textbf{[for ordinary $n$-categories]}} Extended isotopy invariance in dimension $n$.] |
465 \label{axiom:extended-isotopies} |
465 \label{axiom:extended-isotopies} |
466 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
466 Let $X$ be an $n$-ball and $f: X\to X$ be a homeomorphism which restricts |
467 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
467 to the identity on $\bd X$ and isotopic (rel boundary) to the identity. |
468 Then $f$ acts trivially on $\cC(X)$. |
468 Then $f$ acts trivially on $\cC(X)$. |
469 In addition, collar maps act trivially on $\cC(X)$. |
469 In addition, collar maps act trivially on $\cC(X)$. |
583 It is natural to hope to extend such functors to the |
583 It is natural to hope to extend such functors to the |
584 larger categories of all $k$-manifolds (again, with homeomorphisms). |
584 larger categories of all $k$-manifolds (again, with homeomorphisms). |
585 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
585 In fact, the axioms stated above already require such an extension to $k$-spheres for $k<n$. |
586 |
586 |
587 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
587 The natural construction achieving this is a colimit along the poset of permissible decompositions. |
588 Given a plain $n$-category $\cC$, |
588 Given an ordinary $n$-category $\cC$, |
589 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
589 we will denote its extension to all manifolds by $\cl{\cC}$. On a $k$-manifold $W$, with $k \leq n$, |
590 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
590 this is defined to be the colimit along $\cell(W)$ of the functor $\psi_{\cC;W}$. |
591 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
591 Note that Axioms \ref{axiom:composition} and \ref{axiom:associativity} |
592 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
592 imply that $\cl{\cC}(X) \iso \cC(X)$ when $X$ is a $k$-ball with $k<n$. |
593 Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
593 Suppose that $\cC$ is enriched in vector spaces: this means that given boundary conditions $c \in \cl{\cC}(\bdy X)$, for $X$ an $n$-ball, |
620 |
620 |
621 %When $\cC$ is a topological $n$-category, |
621 %When $\cC$ is a topological $n$-category, |
622 %the flexibility available in the construction of a homotopy colimit allows |
622 %the flexibility available in the construction of a homotopy colimit allows |
623 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
623 %us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$. |
624 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
624 %\todo{either need to explain why this is the same, or significantly rewrite this section} |
625 When $\cC$ is the plain $n$-category based on string diagrams for a traditional |
625 When $\cC$ is the ordinary $n$-category based on string diagrams for a traditional |
626 $n$-category $C$, |
626 $n$-category $C$, |
627 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
627 one can show \cite{1009.5025} that the above two constructions of the homotopy colimit |
628 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
628 are equivalent to the more concrete construction which we describe next, and which we denote $\bc_*(W; \cC)$. |
629 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
629 Roughly speaking, the generators of $\bc_k(W; \cC)$ are string diagrams on $W$ together with |
630 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
630 a configuration of $k$ balls (or ``blobs") in $W$ whose interiors are pairwise disjoint or nested. |
744 |
744 |
745 The blob complex has several important special cases. |
745 The blob complex has several important special cases. |
746 |
746 |
747 \begin{thm}[Skein modules] |
747 \begin{thm}[Skein modules] |
748 \label{thm:skein-modules} |
748 \label{thm:skein-modules} |
749 Suppose $\cC$ is a plain $n$-category. |
749 Suppose $\cC$ is an ordinary $n$-category. |
750 The $0$-th blob homology of $X$ is the usual |
750 The $0$-th blob homology of $X$ is the usual |
751 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
751 (dual) TQFT Hilbert space (a.k.a.\ skein module) associated to $X$ |
752 by $\cC$. |
752 by $\cC$. |
753 \begin{equation*} |
753 \begin{equation*} |
754 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
754 H_0(\bc_*(X;\cC)) \iso \cl{\cC}(X) |
888 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
888 blob complexes for the $A_\infty$ $n$-categories constructed as above. |
889 |
889 |
890 \begin{thm}[Product formula] |
890 \begin{thm}[Product formula] |
891 \label{thm:product} |
891 \label{thm:product} |
892 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
892 Let $W$ be a $k$-manifold and $Y$ be an $n{-}k$ manifold. |
893 Let $\cC$ be a plain $n$-category. |
893 Let $\cC$ be an ordinary $n$-category. |
894 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
894 Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above. |
895 Then |
895 Then |
896 \[ |
896 \[ |
897 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
897 \bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W). |
898 \] |
898 \] |