282 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
282 Because we are mainly interested in the case of strong duality, we replace the intervals $[0,r]$ not with |
283 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
283 a product of $k$ intervals (c.f. \cite{0909.2212}) but rather with any $k$-ball, that is, any $k$-manifold which is homeomorphic |
284 to the standard $k$-ball $B^k$. |
284 to the standard $k$-ball $B^k$. |
285 \nn{maybe add that in addition we want functoriality} |
285 \nn{maybe add that in addition we want functoriality} |
286 |
286 |
287 By default our balls are oriented, |
287 By default our balls are unoriented, |
288 but it is useful at times to vary this, |
288 but it is useful at times to vary this, |
289 for example by considering unoriented or Spin balls. |
289 for example by considering oriented or Spin balls. |
290 We can also consider more exotic structures, such as balls with a map to some target space, |
290 We can also consider more exotic structures, such as balls with a map to some target space, |
291 or equipped with $m$ independent vector fields. |
291 or equipped with $m$ independent vector fields. |
292 (The latter structure would model $n$-categories with less duality than we usually assume.) |
292 (The latter structure would model $n$-categories with less duality than we usually assume.) |
293 |
293 |
294 \begin{axiom}[Morphisms] |
294 \begin{axiom}[Morphisms] |