equal
deleted
inserted
replaced
214 \] |
214 \] |
215 and |
215 and |
216 \[ |
216 \[ |
217 (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
217 (a\times D')\bullet(a\times D'') = a\times (D'\bullet D'') . |
218 \] |
218 \] |
|
219 \nn{problem: if pinched boundary, then only one factor} |
219 Product morphisms are associative: |
220 Product morphisms are associative: |
220 \[ |
221 \[ |
221 (a\times D)\times D' = a\times (D\times D') . |
222 (a\times D)\times D' = a\times (D\times D') . |
222 \] |
223 \] |
223 (Here we are implicitly using functoriality and the obvious homeomorphism |
224 (Here we are implicitly using functoriality and the obvious homeomorphism |
446 |
447 |
447 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
448 Finally, define $\cC(W)$ to be the colimit of $\psi_\cC$. |
448 In other words, for each decomposition $x$ there is a map |
449 In other words, for each decomposition $x$ there is a map |
449 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
450 $\psi_\cC(x)\to \cC(W)$, these maps are compatible with the refinement maps |
450 above, and $\cC(W)$ is universal with respect to these properties. |
451 above, and $\cC(W)$ is universal with respect to these properties. |
|
452 \nn{in A-inf case, need to say more} |
|
453 |
|
454 \nn{should give more concrete description (two cases)} |
451 |
455 |
452 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
456 $\cC(W)$ is functorial with respect to homeomorphisms of $k$-manifolds. |
453 |
457 |
454 It is easy to see that |
458 It is easy to see that |
455 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
459 there are well-defined maps $\cC(W)\to\cC(\bd W)$, and that these maps |
632 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
636 M\times D \ar[r]^{\tilde{f}} \ar[d]_{\pi} & M'\times D' \ar[d]^{\pi} \\ |
633 M \ar[r]^{f} & M' |
637 M \ar[r]^{f} & M' |
634 } \] |
638 } \] |
635 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
639 commutes, then we have $\tilde{f}(a\times D) = f(a)\times D'$.} |
636 |
640 |
637 \nn{Need to say something about compatibility with gluing (of both $M$ and $D$) above.} |
641 \nn{Need to add compatibility with various things, as in the n-cat version of this axiom above.} |
638 |
642 |
639 \nn{** marker --- resume revising here **} |
643 \nn{** marker --- resume revising here **} |
640 |
644 |
641 There are two alternatives for the next axiom, according whether we are defining |
645 There are two alternatives for the next axiom, according whether we are defining |
642 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
646 modules for plain $n$-categories or $A_\infty$ $n$-categories. |
717 $\cN$ determines |
721 $\cN$ determines |
718 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
722 a functor $\psi_\cN$ from $\cJ(W)$ to the category of sets |
719 (possibly with additional structure if $k=n$). |
723 (possibly with additional structure if $k=n$). |
720 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
724 For a decomposition $x = (X_a, M_{ib})$ in $\cJ(W)$, define $\psi_\cN(x)$ to be the subset |
721 \[ |
725 \[ |
722 \psi_\cN(x) \sub (\prod_a \cC(X_a)) \prod (\prod_{ib} \cN_i(M_{ib})) |
726 \psi_\cN(x) \sub (\prod_a \cC(X_a)) \times (\prod_{ib} \cN_i(M_{ib})) |
723 \] |
727 \] |
724 such that the restrictions to the various pieces of shared boundaries amongst the |
728 such that the restrictions to the various pieces of shared boundaries amongst the |
725 $X_a$ and $M_{ib}$ all agree. |
729 $X_a$ and $M_{ib}$ all agree. |
726 (Think fibered product.) |
730 (Think fibered product.) |
727 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
731 If $x$ is a refinement of $y$, define a map $\psi_\cN(x)\to\psi_\cN(y)$ |
783 $\cC$, $\cM$ and $\cM'$ determine |
787 $\cC$, $\cM$ and $\cM'$ determine |
784 a functor $\psi$ from $\cJ(D)$ to the category of sets |
788 a functor $\psi$ from $\cJ(D)$ to the category of sets |
785 (possibly with additional structure if $k=n$). |
789 (possibly with additional structure if $k=n$). |
786 For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to be the subset |
790 For a decomposition $x = (X_a, M_b, M'_c)$ in $\cJ(D)$, define $\psi(x)$ to be the subset |
787 \[ |
791 \[ |
788 \psi(x) \sub (\prod_a \cC(X_a)) \prod (\prod_b \cM(M_b)) \prod (\prod_c \cM'(M'_c)) |
792 \psi(x) \sub (\prod_a \cC(X_a)) \times (\prod_b \cM(M_b)) \times (\prod_c \cM'(M'_c)) |
789 \] |
793 \] |
790 such that the restrictions to the various pieces of shared boundaries amongst the |
794 such that the restrictions to the various pieces of shared boundaries amongst the |
791 $X_a$, $M_b$ and $M'_c$ all agree. |
795 $X_a$, $M_b$ and $M'_c$ all agree. |
792 (Think fibered product.) |
796 (Think fibered product.) |
793 If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$ |
797 If $x$ is a refinement of $y$, define a map $\psi(x)\to\psi(y)$ |