24 %Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.

    24 %Fix a top dimension $n$, and a symmetric monoidal category $\cS$ whose objects are sets. While reading the definition, you should just think about the case $\cS = \Set$ with cartesian product, until you reach the discussion of a \emph{linear system of fields} later in this section, where $\cS = \Vect$, and \S \ref{sec:homological-fields}, where $\cS = \Kom$.

    25 

    25 

    26 A $n$-dimensional {\it system of fields} in $\cS$

    26 A $n$-dimensional {\it system of fields} in $\cS$

    27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$

    27 is a collection of functors $\cC_k : \cM_k \to \Set$ for $0 \leq k \leq n$

    28 together with some additional data and satisfying some additional conditions, all specified below.

    28 together with some additional data and satisfying some additional conditions, all specified below.

    31 

    29 

    32 Before finishing the definition of fields, we give two motivating examples

    30 Before finishing the definition of fields, we give two motivating examples

    33 (actually, families of examples) of systems of fields.

    31 (actually, families of examples) of systems of fields.

    34 

    32 

    35 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps

    33 The first examples: Fix a target space $B$, and let $\cC(X)$ be the set of continuous maps

   388 \item A local relation field $u \in U(B; c)$

   386 \item A local relation field $u \in U(B; c)$

   389 (same $c$ as previous bullet).

   387 (same $c$ as previous bullet).

   390 \end{itemize}

   388 \end{itemize}

   391 (See Figure \ref{blob1diagram}.)

   389 (See Figure \ref{blob1diagram}.)

   392 \begin{figure}[!ht]\begin{equation*}

   390 \begin{figure}[!ht]\begin{equation*}

   394 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}

   392 \end{equation*}\caption{A 1-blob diagram.}\label{blob1diagram}\end{figure}

   395 In order to get the linear structure correct, we (officially) define

   393 In order to get the linear structure correct, we (officially) define

   396 \[

   394 \[

   397 	\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .

   395 	\bc_1(X) \deq \bigoplus_B \bigoplus_c U(B; c) \otimes \lf(X \setmin B; c) .

   398 \]

   396 \]

   421 A disjoint 2-blob diagram consists of

   419 A disjoint 2-blob diagram consists of

   422 \begin{itemize}

   420 \begin{itemize}

   423 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.

   421 \item A pair of closed balls (blobs) $B_0, B_1 \sub X$ with disjoint interiors.

   424 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$

   422 \item A field $r \in \cC(X \setmin (B_0 \cup B_1); c_0, c_1)$

   425 (where $c_i \in \cC(\bd B_i)$).

   423 (where $c_i \in \cC(\bd B_i)$).

   427 \end{itemize}

   425 \end{itemize}

   428 (See Figure \ref{blob2ddiagram}.)

   426 (See Figure \ref{blob2ddiagram}.)

   429 \begin{figure}[!ht]\begin{equation*}

   427 \begin{figure}[!ht]\begin{equation*}

   431 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}

   429 \end{equation*}\caption{A disjoint 2-blob diagram.}\label{blob2ddiagram}\end{figure}

   432 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;

   430 We also identify $(B_0, B_1, u_0, u_1, r)$ with $-(B_1, B_0, u_1, u_0, r)$;

   433 reversing the order of the blobs changes the sign.

   431 reversing the order of the blobs changes the sign.

   434 Define $\bd(B_0, B_1, u_0, u_1, r) = 

   432 Define $\bd(B_0, B_1, u_0, u_1, r) = 

   435 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.

   433 (B_1, u_1, u_0\bullet r) - (B_0, u_0, u_1\bullet r) \in \bc_1(X)$.

   445 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.

   443 (for some $c_0 \in \cC(\bd B_0)$), which is splittable along $\bd B_1$.

   446 \item A local relation field $u_0 \in U(B_0; c_0)$.

   444 \item A local relation field $u_0 \in U(B_0; c_0)$.

   447 \end{itemize}

   445 \end{itemize}

   448 (See Figure \ref{blob2ndiagram}.)

   446 (See Figure \ref{blob2ndiagram}.)

   449 \begin{figure}[!ht]\begin{equation*}

   447 \begin{figure}[!ht]\begin{equation*}

   451 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}

   449 \end{equation*}\caption{A nested 2-blob diagram.}\label{blob2ndiagram}\end{figure}

   452 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$

   450 Let $r = r_1 \bullet r'$, where $r_1 \in \cC(B_1 \setmin B_0; c_0, c_1)$

   453 (for some $c_1 \in \cC(B_1)$) and

   451 (for some $c_1 \in \cC(B_1)$) and

   454 $r' \in \cC(X \setmin B_1; c_1)$.

   452 $r' \in \cC(X \setmin B_1; c_1)$.

   455 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.

   453 Define $\bd(B_0, B_1, u_0, r) = (B_1, u_0\bullet r_1, r') - (B_0, u_0, r)$.