3 \subsection{Basic properties}

     3 \subsection{Basic properties}

     4 \label{sec:basic-properties}

     4 \label{sec:basic-properties}

     5 

     5 

     6 In this section we complete the proofs of Properties \ref{property:disjoint-union}--\ref{property:contractibility}.

     6 In this section we complete the proofs of Properties \ref{property:disjoint-union}--\ref{property:contractibility}.

     7 Throughout the paper, where possible, we prove results using Properties \ref{property:functoriality}--\ref{property:contractibility}, 

     7 Throughout the paper, where possible, we prove results using Properties \ref{property:functoriality}--\ref{property:contractibility}, 

     9 This allows the possibility of future improvements on or alternatives to our definition.

     9 This allows the possibility of future improvements on or alternatives to our definition.

    10 In fact, we hope that there may be a characterization of the blob complex in 

    10 In fact, we hope that there may be a characterization of the blob complex in 

    11 terms of Properties \ref{property:functoriality}--\ref{property:contractibility}, but at this point we are unaware of one.

    11 terms of Properties \ref{property:functoriality}--\ref{property:contractibility}, but at this point we are unaware of one.

    12 

    12 

    13 Recall Property \ref{property:disjoint-union}, 

    13 Recall Property \ref{property:disjoint-union}, 

   110 \eq{

   110 \eq{

   111     \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl).

   111     \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl).

   112 }

   112 }

   113 The sum is over all fields $a$ on $Y$ compatible at their

   113 The sum is over all fields $a$ on $Y$ compatible at their

   114 ($n{-}2$-dimensional) boundaries with $c$.

   114 ($n{-}2$-dimensional) boundaries with $c$.

   116 In degree zero the map agrees with the gluing map coming from the underlying system of fields.

   116 In degree zero the map agrees with the gluing map coming from the underlying system of fields.

   117 \end{prop}

   117 \end{prop}

   118 

   118 

   119 This map is very far from being an isomorphism, even on homology.

   119 This map is very far from being an isomorphism, even on homology.