293 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
293 of Stasheff polyhedra and $A_\infty$ categories), it becomes undesirably complex for higher categories. |
294 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
294 In a Moore loop space, we have a separate space $\Omega_r$ for each interval $[0,r]$, and a |
295 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
295 {\it strictly associative} composition $\Omega_r\times \Omega_s\to \Omega_{r+s}$. |
296 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
296 Thus we can have the simplicity of strict associativity in exchange for more morphisms. |
297 We wish to imitate this strategy in higher categories. |
297 We wish to imitate this strategy in higher categories. |
298 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
298 Because we are mainly interested in the case of pivotal $n$-categories, we replace the intervals $[0,r]$ not with |
299 a product of $k$ intervals (c.f. \cite{ulrike-tillmann-2008,0909.2212}) but rather with any $k$-ball, that is, |
299 a product of $k$ intervals (c.f. \cite{ulrike-tillmann-2008,0909.2212}) but rather with any $k$-ball, that is, |
300 any $k$-manifold which is homeomorphic |
300 any $k$-manifold which is homeomorphic |
301 to the standard $k$-ball $B^k$. |
301 to the standard $k$-ball $B^k$. |
302 |
302 |
303 By default our balls are unoriented, |
303 By default our balls are unoriented, |
314 homeomorphisms to the category of sets and bijections. |
314 homeomorphisms to the category of sets and bijections. |
315 \end{axiom} |
315 \end{axiom} |
316 |
316 |
317 Note that the functoriality in the above axiom allows us to operate via |
317 Note that the functoriality in the above axiom allows us to operate via |
318 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
318 homeomorphisms which are not the identity on the boundary of the $k$-ball. |
319 The action of these homeomorphisms gives the ``strong duality" structure. |
319 The action of these homeomorphisms gives the pivotal structure. |
320 For this reason we don't subdivide the boundary of a morphism |
320 For this reason we don't subdivide the boundary of a morphism |
321 into domain and range in the next axiom --- the duality operations can convert between domain and range. |
321 into domain and range in the next axiom --- the duality operations can convert between domain and range. |
322 |
322 |
323 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
323 Later we inductively define an extension of the functors $\cC_k$ to functors $\cl{\cC}_k$ |
324 defined on arbitrary manifolds. |
324 defined on arbitrary manifolds. |
502 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
502 For $X$ an $n$-ball define $\pi_{\le n}^\infty(T)(X)$ to be the space of all maps from $X$ to $T$ |
503 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
503 (if we are enriching over spaces) or the singular chains on that space (if we are enriching over chain complexes). |
504 |
504 |
505 |
505 |
506 \subsection{Example (string diagrams)} \mbox{} |
506 \subsection{Example (string diagrams)} \mbox{} |
507 Fix a ``traditional" $n$-category $C$ with strong duality (e.g.\ a pivotal 2-category). |
507 Fix a ``traditional" pivotal $n$-category $C$ (e.g.\ a pivotal 2-category). |
508 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
508 Let $X$ be a $k$-ball and define $\cS_C(X)$ to be the set of $C$ string diagrams drawn on $X$; |
509 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
509 that is, certain cell complexes embedded in $X$, with the codimension-$j$ cells labeled by $j$-morphisms of $C$. |
510 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
510 If $X$ is an $n$-ball, identify two such string diagrams if they evaluate to the same $n$-morphism of $C$. |
511 Boundary restrictions and gluing are again straightforward to define. |
511 Boundary restrictions and gluing are again straightforward to define. |
512 Define product morphisms via product cell decompositions. |
512 Define product morphisms via product cell decompositions. |