equal
deleted
inserted
replaced
204 %\nn{we might want a more official looking proof...} |
204 %\nn{we might want a more official looking proof...} |
205 |
205 |
206 We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples |
206 We do not insist on surjectivity of the gluing map, since this is not satisfied by all of the examples |
207 we are trying to axiomatize. |
207 we are trying to axiomatize. |
208 If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is |
208 If our $k$-morphisms $\cC(X)$ are labeled cell complexes embedded in $X$ (c.f. Example \ref{ex:traditional-n-categories} below), then a $k$-morphism is |
209 in the image of the gluing map precisely which the cell complex is in general position |
209 in the image of the gluing map precisely when the cell complex is in general position |
210 with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective |
210 with respect to $E$. On the other hand, in categories based on maps to a target space (c.f. Example \ref{ex:maps-to-a-space} below) the gluing map is always surjective. |
211 |
211 |
212 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
212 If $S$ is a 0-sphere (the case $k=1$ above), then $S$ can be identified with the {\it disjoint} union |
213 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
213 of two 0-balls $B_1$ and $B_2$ and the colimit construction $\cl{\cC}(S)$ can be identified |
214 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
214 with the (ordinary, not fibered) product $\cC(B_1) \times \cC(B_2)$. |
215 |
215 |
997 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}); |
997 \item if enriching in an auxiliary category, additional structure on $\cC_n(X; c)$ (Axiom \ref{axiom:enriched}); |
998 %\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
998 %\item in the $A_\infty$ case, an action of $C_*(\Homeo_\bd(X))$, and similarly for families of collar maps (Axiom \ref{axiom:families}). |
999 \item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions |
999 \item in the $A_\infty$ case, actions of the topological spaces of homeomorphisms preserving boundary conditions |
1000 and collar maps (Axiom \ref{axiom:families}). |
1000 and collar maps (Axiom \ref{axiom:families}). |
1001 \end{itemize} |
1001 \end{itemize} |
1002 The above data must satisfy the following conditions: |
1002 The above data must satisfy the following conditions. |
1003 \begin{itemize} |
1003 \begin{itemize} |
1004 \item The gluing maps are compatible with actions of homeomorphisms and boundary |
1004 \item The gluing maps are compatible with actions of homeomorphisms and boundary |
1005 restrictions (Axiom \ref{axiom:composition}). |
1005 restrictions (Axiom \ref{axiom:composition}). |
1006 \item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}). |
1006 \item For $k<n$ the gluing maps are injective (Axiom \ref{axiom:composition}). |
1007 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}). |
1007 \item The gluing maps are strictly associative (Axiom \ref{nca-assoc}). |
2408 non-degenerate inner products", then there is a coherent family of isomorphisms |
2408 non-degenerate inner products", then there is a coherent family of isomorphisms |
2409 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
2409 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
2410 This will allow us to define $\cS(X; c)$ independently of the choice of $E$. |
2410 This will allow us to define $\cS(X; c)$ independently of the choice of $E$. |
2411 |
2411 |
2412 First we must define ``inner product", ``non-degenerate" and ``compatible". |
2412 First we must define ``inner product", ``non-degenerate" and ``compatible". |
2413 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
2413 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ its mirror image. |
2414 (We assume we are working in the unoriented category.) |
2414 (We assume we are working in the unoriented category.) |
2415 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
2415 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
2416 along their common boundary. |
2416 along their common boundary. |
2417 An {\it inner product} on $\cS(Y)$ is a dual vector |
2417 An {\it inner product} on $\cS(Y)$ is a dual vector |
2418 \[ |
2418 \[ |