87 (This condition is of course trivial when $\cS = \Set$.) |
87 (This condition is of course trivial when $\cS = \Set$.) |
88 If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$ (chain complexes)), |
88 If the objects are sets with extra structure (e.g. $\cS = \Vect$ or $\Kom$ (chain complexes)), |
89 then this extra structure is considered part of the definition of $\cC_n$. |
89 then this extra structure is considered part of the definition of $\cC_n$. |
90 Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$. |
90 Any maps mentioned below between fields on $n$-manifolds must be morphisms in $\cS$. |
91 \item $\cC_k$ is compatible with the symmetric monoidal |
91 \item $\cC_k$ is compatible with the symmetric monoidal |
92 structures on $\cM_k$, $\Set$ and $\cS$: $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
92 structures on $\cM_k$, $\Set$ and $\cS$. |
|
93 For $k<n$ we have $\cC_k(X \du W) \cong \cC_k(X)\times \cC_k(W)$, |
93 compatibly with homeomorphisms and restriction to boundary. |
94 compatibly with homeomorphisms and restriction to boundary. |
94 We will call the projections $\cC(X_1 \du X_2) \to \cC(X_i)$ |
95 For $k=n$ we require $\cC_n(X \du W; c\du d) \cong \cC_k(X, c)\ot \cC_k(W, d)$. |
|
96 We will call the projections $\cC_k(X_1 \du X_2) \to \cC_k(X_i)$ |
95 restriction maps. |
97 restriction maps. |
96 \item Gluing without corners. |
98 \item Gluing without corners. |
97 Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
99 Let $\bd X = Y \du Y \du W$, where $Y$ and $W$ are closed $k{-}1$-manifolds. |
98 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
100 Let $X\sgl$ denote $X$ glued to itself along the two copies of $Y$. |
99 Using the boundary restriction and disjoint union |
101 Using the boundary restriction and disjoint union |