1943 An inner product is {\it non-degenerate} if 

  1944 An inner product induces a linear map

  1945 \begin{eqnarray*}

  1946 	\varphi: \cS(Y) &\to& \cS(Y)^* \\

  1947 	a &\mapsto& \langle a, \cdot \rangle

  1948 \end{eqnarray*}

  1949 which satisfies, for all morphisms $e$ of $\cS(\bd Y)$,

  1950 \[

  1951 	\varphi(ae)(b) = \langle ae, b \rangle = z_Y(a\bullet e\bullet b) = 

  1952 			\langle a, eb \rangle = \varphi(a)(eb) .

  1953 \]

  1954 In other words, $\varphi$ is a map of $\cS(\bd Y)$ modules.

  1955 An inner product is {\it non-degenerate} if $\varphi$ is an isomorphism.

  1956 This implies that $\cS(Y; c)$ is finite dimensional for all boundary conditions $c$.

  1957 (One can think of these inner products as giving some duality in dimension $n{+}1$;

  1958 heretofore we have only assumed duality in dimensions 0 through $n$.)

  1959 

  1960 Next we define compatibility.

  1961 Let $Y = Y_1\cup Y_2$ with $D = Y_1\cap Y_2$.

  1962 Let $X_1$ and $X_2$ be the two components of $Y\times I$ (pinched) cut along

  1963 $D\times I$.

  1964 (Here we are overloading notation and letting $D$ denote both a decorated and an undecorated

  1965 manifold.)

  1966 We have $\bd X_i = Y_i \cup \ol{Y}_i \cup (D\times I)$

  1967 (see Figure xxxx).

  1968 Given $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$ and $v\in\cS(D\times I)$

  1969 which agree on their boundaries, we can evaluate

  1970 \[

  1971 	z_{Y_i}(a_i\bullet b_i\bullet v) \in \c .

  1972 \]

  1973 (This requires a choice of homeomorphism $Y_i \cup \ol{Y}_i \cup (D\times I) \cong

  1974 Y_i \cup \ol{Y}_i$, but the value of $z_{Y_i}$ is independent of this choice.)

  1975 We can think of $z_{Y_i}$ as giving a function

  1976 \[

  1977 	\psi_i : \cS(Y_i) \ot \cS(\ol{Y}_i) \to \cS(D\times I)^* 

  1978 					\stackrel{\varphi\inv}{\longrightarrow} \cS(D\times I) .

  1979 \]

  1980 We can now finally define a family of inner products to be {\it compatible} if

  1981 for all decompositions $Y = Y_1\cup Y_2$ as above and all $a_i\in \cS(Y_i)$, $b_i\in \cS(\ol{Y}_i)$

  1982 we have

  1983 \[

  1984 	z_Y(a_1\bullet a_2\bullet b_1\bullet b_2) = 

  1985 				z_{D\times I}(\psi_1(a_1\ot b_1)\bullet \psi_2(a_2\ot b_2)) .

  1986 \]

  1987 In other words, the inner product on $Y$ is determined by the inner products on

  1988 $Y_1$, $Y_2$ and $D\times I$.