903 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
903 \item define $n{+}1$-cat of $n$-cats (a.k.a.\ $n{+}1$-category of generalized bimodules |
904 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
904 a.k.a.\ $n{+}1$-category of sphere modules); discuss Morita equivalence |
905 \item morphisms of modules; show that it's adjoint to tensor product |
905 \item morphisms of modules; show that it's adjoint to tensor product |
906 \end{itemize} |
906 \end{itemize} |
907 |
907 |
908 |
908 \nn{Some salvaged paragraphs that we might want to work back in:} |
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909 \hrule |
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910 |
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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.) |
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912 |
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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 |
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914 \begin{align*} |
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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)), |
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916 \end{align*} |
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917 where the first map is the product of singular chains, and the second is precomposition by the inverse of a diffeomorphism. |
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918 |
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919 We now give two motivating examples, as theorems constructing other homological systems of fields, |
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920 |
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921 |
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922 \begin{thm} |
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923 For a fixed target space $X$, `chains of maps to $X$' is a homological system of fields $\Xi$, defined as |
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924 \begin{equation*} |
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925 \Xi(M) = \CM{M}{X}. |
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926 \end{equation*} |
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927 \end{thm} |
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928 |
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929 \begin{thm} |
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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 |
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931 \begin{equation*} |
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932 \cF^{\times F}(M) = \cB_*(M \times F, \cF). |
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933 \end{equation*} |
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934 \end{thm} |
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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. |
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936 |
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937 |
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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. |
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939 |
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940 |
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941 \begin{thm} |
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942 \begin{equation*} |
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943 \cB_*(M, \Xi) \iso \Xi(M) |
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944 \end{equation*} |
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945 \end{thm} |
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946 |
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947 \begin{thm}[Product formula] |
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948 Given a $b$-manifold $B$, an $f$-manifold $F$ and a $b+f$ dimensional system of fields, |
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949 there is a quasi-isomorphism |
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950 \begin{align*} |
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951 \cB_*(B \times F, \cF) & \quismto \cB_*(B, \cF^{\times F}) |
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952 \end{align*} |
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953 \end{thm} |
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954 |
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955 \begin{question} |
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956 Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber? |
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957 \end{question} |
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958 |
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959 \hrule |