182 A linear 0-category is a vector space, and a representation |
182 A linear 0-category is a vector space, and a representation |
183 of a vector space is an element of the dual space. |
183 of a vector space is an element of the dual space. |
184 Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
184 Thus a TQFT assigns to each closed $n$-manifold $Y$ a vector space $A(Y)$, |
185 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
185 and to each $(n{+}1)$-manifold $W$ an element of $A(\bd W)^*$. |
186 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
186 For the remainder of this paper we will in fact be interested in so-called $(n{+}\epsilon)$-dimensional |
187 TQFTs, which are slightly weaker structures and do not assign anything to general $(n{+}1)$-manifolds, |
187 TQFTs, which are slightly weaker structures in that they assign |
188 but only to mapping cylinders. |
188 invariants to mapping cylinders of homeomorphisms between $n$-manifolds, but not to general $(n{+}1)$-manifolds. |
189 |
189 |
190 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
190 When $k=n-1$ we have a linear 1-category $A(S)$ for each $(n{-}1)$-manifold $S$, |
191 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
191 and a representation of $A(\bd Y)$ for each $n$-manifold $Y$. |
192 The TQFT gluing rule in dimension $n$ states that |
192 The TQFT gluing rule in dimension $n$ states that |
193 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
193 $A(Y_1\cup_S Y_2) \cong A(Y_1) \ot_{A(S)} A(Y_2)$, |
194 where $Y_1$ and $Y_2$ and $n$-manifolds with common boundary $S$. |
194 where $Y_1$ and $Y_2$ are $n$-manifolds with common boundary $S$. |
195 |
195 |
196 When $k=0$ we have an $n$-category $A(pt)$. |
196 When $k=0$ we have an $n$-category $A(pt)$. |
197 This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$ |
197 This can be thought of as the local part of the TQFT, and the full TQFT can be reconstructed from $A(pt)$ |
198 via colimits (see below). |
198 via colimits (see below). |
199 |
199 |
205 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak |
205 (The WRT invariants need to be reinterpreted as $3{+}1$-dimensional theories with only a weak |
206 dependence on interiors in order to be |
206 dependence on interiors in order to be |
207 extended all the way down to dimension 0.) |
207 extended all the way down to dimension 0.) |
208 |
208 |
209 For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
209 For other non-semisimple TQFT-like invariants, however, the above framework seems to be inadequate. |
210 For example, the gluing rule for 3-manifolds in Ozsv\'{a}th-Szab\'{o}/Seiberg-Witten theory |
210 For example, the gluing rule for 3-manifolds in Ozsv\'ath-Szab\'o/Seiberg-Witten theory |
211 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
211 involves a tensor product over an $A_\infty$ 1-category associated to 2-manifolds \cite{1003.0598,1005.1248}. |
212 Long exact sequences are important computational tools in these theories, |
212 Long exact sequences are important computational tools in these theories, |
213 and also in Khovanov homology, but the colimit construction breaks exactness. |
213 and also in Khovanov homology, but the colimit construction breaks exactness. |
214 For these reasons and others, it is desirable to |
214 For these reasons and others, it is desirable to |
215 extend to above framework to incorporate ideas from derived categories. |
215 extend to above framework to incorporate ideas from derived categories. |
239 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
239 Of course, there are currently many interesting alternative notions of $n$-category and of TQFT. |
240 We note that our $n$-categories are both more and less general |
240 We note that our $n$-categories are both more and less general |
241 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
241 than the ``fully dualizable" ones which play a prominent role in \cite{0905.0465}. |
242 They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
242 They are more general in that we make no duality assumptions in the top dimension $n{+}1$. |
243 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
243 They are less general in that we impose stronger duality requirements in dimensions 0 through $n$. |
244 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional unoriented or oriented TQFTs, while |
244 Thus our $n$-categories correspond to $(n{+}\epsilon)$-dimensional {\it unoriented} or {\it oriented} TQFTs, while |
245 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional framed TQFTs. |
245 Lurie's (fully dualizable) $n$-categories correspond to $(n{+}1)$-dimensional {\it framed} TQFTs. |
246 |
246 |
247 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
247 At several points we only sketch an argument briefly; full details can be found in \cite{1009.5025}. |
248 In this paper we attempt to give a clear view of the big picture without getting |
248 In this paper we attempt to give a clear view of the big picture without getting |
249 bogged down in technical details. |
249 bogged down in technical details. |
250 |
250 |
316 \end{axiom} |
316 \end{axiom} |
317 |
317 |
318 Note that the functoriality in the above axiom allows us to operate via |
318 Note that the functoriality in the above axiom allows us to operate via |
319 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
319 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
320 The action of these homeomorphisms gives the ``strong duality" structure. |
320 The action of these homeomorphisms gives the ``strong duality" structure. |
321 As such, we don't subdivide the boundary of a morphism |
321 For this reason we don't subdivide the boundary of a morphism |
322 into domain and range --- the duality operations can convert between domain and range. |
322 into domain and range in the next axiom --- the duality operations can convert between domain and range. |
323 |
323 |
324 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
324 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
325 from arbitrary manifolds to sets. We need these functors for $k$-spheres, |
325 defined on arbitrary manifolds. |
326 for $k<n$, for the next axiom. |
326 We need these functors for $k$-spheres, for $k<n$, for the next axiom. |
327 |
327 |
328 \begin{axiom}[Boundaries]\label{nca-boundary} |
328 \begin{axiom}[Boundaries]\label{nca-boundary} |
329 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
329 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$. |
330 These maps, for various $X$, comprise a natural transformation of functors. |
330 These maps, for various $X$, comprise a natural transformation of functors. |
331 \end{axiom} |
331 \end{axiom} |
372 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
372 \gl_Y : \cC(B_1)_E \times_{\cC(Y)} \cC(B_2)_E \to \cC(B)_E |
373 \] |
373 \] |
374 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 |
375 to the intersection of the boundaries of $B$ and $B_i$. |
375 to the intersection of the boundaries of $B$ and $B_i$. |
376 If $k < n$, |
376 If $k < n$, |
377 or if $k=n$ and we are in the $A_\infty$ case \nn{Kevin: remind me why we ask this?}, |
377 or if $k=n$ and we are in the $A_\infty$ case, |
378 we require that $\gl_Y$ is injective. |
378 we require that $\gl_Y$ is injective. |
379 (For $k=n$ in the isotopy $n$-category case, see below. \nn{where?}) |
379 (For $k=n$ in the isotopy $n$-category case, see Axiom \ref{axiom:extended-isotopies}.) |
380 \end{axiom} |
380 \end{axiom} |
381 |
381 |
382 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
382 \begin{axiom}[Strict associativity] \label{nca-assoc}\label{axiom:associativity} |
383 The gluing maps above are strictly associative. |
383 The gluing maps above are strictly associative. |
384 Given any decomposition of a ball $B$ into smaller balls |
384 Given any decomposition of a ball $B$ into smaller balls |
385 $$\bigsqcup B_i \to B,$$ |
385 $$\bigsqcup B_i \to B,$$ |
386 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
386 any sequence of gluings (where all the intermediate steps are also disjoint unions of balls) yields the same result. |
387 \end{axiom} |
387 \end{axiom} |
388 This axiom is only reasonable because the definition assigns a set to every ball; |
388 %This axiom is only reasonable because the definition assigns a set to every ball; |
389 any identifications would limit the extent to which we can demand associativity. |
389 %any identifications would limit the extent to which we can demand associativity. |
|
390 %%%% KW: It took me quite a while figure out what you [or I??] meant by the above, so I'm attempting a rewrite. |
|
391 Note that even though our $n$-categories are ``weak" in the traditional sense, we can require |
|
392 strict associativity because we have more morphisms (cf.\ discussion of Moore loops above). |
|
393 |
390 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
394 For the next axiom, a \emph{pinched product} is a map locally modeled on a degeneracy map between simplices. |
391 \begin{axiom}[Product (identity) morphisms] |
395 \begin{axiom}[Product (identity) morphisms] |
392 \label{axiom:product} |
396 \label{axiom:product} |
393 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
397 For each pinched product $\pi:E\to X$, with $X$ a $k$-ball and $E$ a $k{+}m$-ball ($m\ge 1$), |
394 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
398 there is a map $\pi^*:\cC(X)\to \cC(E)$. |
484 These action maps are required to restrict to the usual action of homeomorphisms on $C_0$, be associative up to homotopy, |
488 These action maps are required to restrict to the usual action of homeomorphisms on $C_0$, be associative up to homotopy, |
485 and also be compatible with composition (gluing) in the sense that |
489 and also be compatible with composition (gluing) in the sense that |
486 a diagram like the one in Theorem \ref{thm:CH} commutes. |
490 a diagram like the one in Theorem \ref{thm:CH} commutes. |
487 \end{axiom} |
491 \end{axiom} |
488 |
492 |
489 \subsection{Example (the fundamental $n$-groupoid)} |
493 \subsection{Example (the fundamental $n$-groupoid)} \mbox{} |
490 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
494 We will define $\pi_{\le n}(T)$, the fundamental $n$-groupoid of a topological space $T$. |
491 When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
495 When $X$ is a $k$-ball with $k<n$, define $\pi_{\le n}(T)(X)$ |
492 to be the set of continuous maps from $X$ to $T$. |
496 to be the set of continuous maps from $X$ to $T$. |
493 When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
497 When $X$ is an $n$-ball, define $\pi_{\le n}(T)(X)$ to be homotopy classes (rel boundary) of such maps. |
494 Define boundary restrictions and gluing in the obvious way. |
498 Define boundary restrictions and gluing in the obvious way. |
498 We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
502 We can also define an $A_\infty$ version $\pi_{\le n}^\infty(T)$ of the fundamental $n$-groupoid. |
499 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
503 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
500 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
504 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
501 |
505 |
502 |
506 |
503 \subsection{Example (string diagrams)} |
507 \subsection{Example (string diagrams)} \mbox{} |
504 Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
508 Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
505 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
509 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
506 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
510 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
507 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
511 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
508 Boundary restrictions and gluing are again straightforward to define. |
512 Boundary restrictions and gluing are again straightforward to define. |
509 Define product morphisms via product cell decompositions. |
513 Define product morphisms via product cell decompositions. |
510 |
514 |
511 \subsection{Example (bordism)} |
515 \subsection{Example (bordism)} \mbox{} |
512 When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
516 When $X$ is a $k$-ball with $k<n$, $\Bord^n(X)$ is the set of all $k$-dimensional |
513 submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
517 submanifolds $W$ in $X\times \bbR^\infty$ which project to $X$ transversely |
514 to $\bd X$. |
518 to $\bd X$. |
515 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
519 For an $n$-ball $X$ define $\Bord^n(X)$ to be homeomorphism classes rel boundary of such $n$-dimensional submanifolds. |
516 |
520 |