908 

   908 \nn{Some salvaged paragraphs that we might want to work back in:}

   909 \hrule

   910 

   911 Appendix \ref{sec:comparing-A-infty} explains the translation between this definition of an $A_\infty$ $1$-category and the usual one expressed in terms of `associativity up to higher homotopy', as in \cite{MR1854636}. (In this version of the paper, that appendix is incomplete, however.)

   912 

   913 The motivating example is `chains of maps to $M$' for some fixed target space $M$. This is a topological $A_\infty$ category $\Xi_M$ with $\Xi_M(J) = C_*(\Maps(J \to M))$. The gluing maps $\Xi_M(J) \tensor \Xi_M(J') \to \Xi_M(J \cup J')$  takes the product of singular chains, then glues maps to $M$ together; the associativity condition is automatically satisfied. The evaluation map $\ev_{J,J'} : \CD{J \to J'} \tensor \Xi_M(J) \to \Xi_M(J')$ is the composition

   914 \begin{align*}

   915 \CD{J \to J'} \tensor C_*(\Maps(J \to M)) & \to C_*(\Diff(J \to J') \times \Maps(J \to M)) \\ & \to C_*(\Maps(J' \to M)),

   916 \end{align*}

   917 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism.

   918 

   919 We now give two motivating examples, as theorems constructing other homological systems of fields,

   920 

   921 

   922 \begin{thm}

   923 For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as

   924 \begin{equation*}

   925 \Xi(M) = \CM{M}{X}.

   926 \end{equation*}

   927 \end{thm}

   928 

   929 \begin{thm}

   930 Given an $n$-dimensional system of fields $\cF$, and a $k$-manifold $F$, there is an $n-k$ dimensional homological system of fields $\cF^{\times F}$ defined by

   931 \begin{equation*}

   932 \cF^{\times F}(M) = \cB_*(M \times F, \cF).

   933 \end{equation*}

   934 \end{thm}

   935 We might suggestively write $\cF^{\times F}$ as  $\cB_*(F \times [0,1]^b, \cF)$, interpreting this as an (undefined!) $A_\infty$ $b$-category, and then as the resulting homological system of fields, following a recipe analogous to that given above for $A_\infty$ $1$-categories.

   936 

   937 

   938 In later sections, we'll prove the following two unsurprising theorems, about the (as-yet-undefined) blob homology of these homological systems of fields.

   939 

   940 

   941 \begin{thm}

   942 \begin{equation*}

   943 \cB_*(M, \Xi) \iso \Xi(M)

   944 \end{equation*}

   945 \end{thm}

   946 

   947 \begin{thm}[Product formula]

   948 Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields,

   949 there is a quasi-isomorphism

   950 \begin{align*}

   951 \cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F})

   952 \end{align*}

   953 \end{thm}

   954 

   955 \begin{question}

   956 Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber?

   957 \end{question}

   958 

   959 \hrule