equal
deleted
inserted
replaced
216 \] |
216 \] |
217 and |
217 and |
218 \[ |
218 \[ |
219 (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
219 (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
220 \] |
220 \] |
221 \nn{problem: if pinched boundary, then only one factor} |
221 \nn{if pinched boundary, then remove first case above} |
222 Product morphisms are associative: |
222 Product morphisms are associative: |
223 \[ |
223 \[ |
224 (a\times D)\times D' = a\times (D\times D') . |
224 (a\times D)\times D' = a\times (D\times D') . |
225 \] |
225 \] |
226 (Here we are implicitly using functoriality and the obvious homeomorphism |
226 (Here we are implicitly using functoriality and the obvious homeomorphism |
231 \] |
231 \] |
232 for $E\sub \bd D$ and $a\in \cC(X)$. |
232 for $E\sub \bd D$ and $a\in \cC(X)$. |
233 } |
233 } |
234 |
234 |
235 \nn{need even more subaxioms for product morphisms?} |
235 \nn{need even more subaxioms for product morphisms?} |
|
236 |
|
237 \nn{Almost certainly we need a little more than the above axiom. |
|
238 More specifically, in order to bootstrap our way from the top dimension |
|
239 properties of identity morphisms to low dimensions, we need regular products, |
|
240 pinched products and even half-pinched products. |
|
241 I'm not sure what the best way to cleanly axiomatize the properties of these various is. |
|
242 For the moment, I'll assume that all flavors of the product are at |
|
243 our disposal, and I'll plan on revising the axioms later.} |
236 |
244 |
237 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
245 All of the axioms listed above hold for both ordinary $n$-categories and $A_\infty$ $n$-categories. |
238 The last axiom (below), concerning actions of |
246 The last axiom (below), concerning actions of |
239 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
247 homeomorphisms in the top dimension $n$, distinguishes the two cases. |
240 |
248 |
258 |
266 |
259 Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity. |
267 Second, we require that product (a.k.a.\ identity) $n$-morphisms act as the identity. |
260 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
268 Let $X$ be an $n$-ball and $Y\sub\bd X$ be an $n{-}1$-ball. |
261 Let $J$ be a 1-ball (interval). |
269 Let $J$ be a 1-ball (interval). |
262 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
270 We have a collaring homeomorphism $s_{Y,J}: X\cup_Y (Y\times J) \to X$. |
|
271 (Here we use the ``pinched" version of $Y\times J$. |
|
272 \nn{need notation for this}) |
263 We define a map |
273 We define a map |
264 \begin{eqnarray*} |
274 \begin{eqnarray*} |
265 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
275 \psi_{Y,J}: \cC(X) &\to& \cC(X) \\ |
266 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
276 a & \mapsto & s_{Y,J}(a \cup ((a|_Y)\times J)) . |
267 \end{eqnarray*} |
277 \end{eqnarray*} |
871 \item traditional $A_\infty$ 1-cat def implies our def |
881 \item traditional $A_\infty$ 1-cat def implies our def |
872 \item ... and vice-versa (already done in appendix) |
882 \item ... and vice-versa (already done in appendix) |
873 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
883 \item say something about unoriented vs oriented vs spin vs pin for $n=1$ (and $n=2$?) |
874 \item spell out what difference (if any) Top vs PL vs Smooth makes |
884 \item spell out what difference (if any) Top vs PL vs Smooth makes |
875 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
885 \item explain relation between old-fashioned blob homology and new-fangled blob homology |
876 (follows as special case of product formula (product with a point). |
886 (follows as special case of product formula (product with a point)). |
877 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
887 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
878 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
888 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
879 \end{itemize} |
889 \end{itemize} |
880 |
890 |
881 |
891 |