612 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
612 A cone-product polyhedra is obtained from a point by successively taking the cone or taking the |
613 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
613 product with another cone-product polyhedron. Just as simplices correspond to linear directed graphs, |
614 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
614 cone-product polyheda correspond to directed trees: taking cone adds a new root before the existing root, |
615 and taking product identifies the roots of several trees. |
615 and taking product identifies the roots of several trees. |
616 The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
616 The ``local homotopy colimit" is then defined according to the same formula as above, but with $\bar{x}$ a cone-product polyhedron in $\cell(W)$. |
617 We further require that any morphism in a directed tree is not expressible as a product. |
617 We further require that all (compositions of) morphisms in a directed tree are not expressible as a product. |
618 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
618 The differential acts on $(\bar{x},a)$ both on $a$ and on $\bar{x}$, applying the appropriate gluing map to $a$ when required. |
619 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
619 A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization. |
620 |
620 |
621 %When $\cC$ is a topological $n$-category, |
621 %When $\cC$ is a topological $n$-category, |
622 %the flexibility available in the construction of a homotopy colimit allows |
622 %the flexibility available in the construction of a homotopy colimit allows |
837 which can be thought of as a free resolution. |
837 which can be thought of as a free resolution. |
838 \end{rem} |
838 \end{rem} |
839 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
839 This result is described in more detail as Example 6.2.8 of \cite{1009.5025}. |
840 |
840 |
841 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
841 Fix a topological $n$-category $\cC$, which we'll now omit from notation. |
842 Recall (Theorem \ref{thm:blobs-ainfty}) that there is associated to |
842 From the above, associated to any $(n{-}1)$-manifold $Y$ is an $A_\infty$ category $\bc_*(Y)$. |
843 any $(n{-}1)$-manifold $Y$ an $A_\infty$ category $\bc_*(Y)$. |
|
844 |
843 |
845 \begin{thm}[Gluing formula] |
844 \begin{thm}[Gluing formula] |
846 \label{thm:gluing} |
845 \label{thm:gluing} |
847 \mbox{}\vspace{-0.2cm}% <-- gets the indenting right |
846 \mbox{}\vspace{-0.2cm}% <-- gets the indenting right |
848 \begin{itemize} |
847 \begin{itemize} |
903 A standard acyclic models argument now constructs the homotopy inverse. |
902 A standard acyclic models argument now constructs the homotopy inverse. |
904 \end{proof} |
903 \end{proof} |
905 |
904 |
906 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
905 %\nn{Theorem \ref{thm:product} is proved in \S \ref{ss:product-formula}, and Theorem \ref{thm:gluing} in \S \ref{sec:gluing}.} |
907 |
906 |
908 \section{Deligne conjecture for $n$-categories} |
907 \section{Deligne's conjecture for $n$-categories} |
909 \label{sec:applications} |
908 \label{sec:applications} |
910 |
909 |
911 Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
910 Let $M$ and $N$ be $n$-manifolds with common boundary $E$. |
912 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
911 Recall (Theorem \ref{thm:gluing}) that the $A_\infty$ category $A = \bc_*(E)$ |
913 acts on $\bc_*(M)$ and $\bc_*(N)$. |
912 acts on $\bc_*(M)$ and $\bc_*(N)$. |
914 Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
913 Let $\hom_A(\bc_*(M), \bc_*(N))$ denote the chain complex of $A_\infty$ module maps |
915 from $\bc_*(M)$ to $\bc_*(N)$. |
914 from $\bc_*(M)$ to $\bc_*(N)$. |
916 Let $R$ be another $n$-manifold with boundary $E^\text{op}$. |
915 Let $R$ be another $n$-manifold with boundary $E^\text{op}$. |
917 There is a chain map |
916 There is a chain map |
918 \[ |
917 \begin{equation*} |
919 \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
918 \hom_A(\bc_*(M), \bc_*(N)) \ot \bc_*(M) \ot_A \bc_*(R) \to \bc_*(N) \ot_A \bc_*(R) . |
920 \] |
919 \end{equation*} |
921 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
920 We think of this map as being associated to a surgery which cuts $M$ out of $M\cup_E R$ and |
922 replaces it with $N$, yielding $N\cup_E R$. |
921 replaces it with $N$, yielding $N\cup_E R$. |
923 (This is a more general notion of surgery that usual --- $M$ and $N$ can be any manifolds |
922 (This is a more general notion of surgery that usual: $M$ and $N$ can be any manifolds |
924 which share a common boundary.) |
923 which share a common boundary.) |
925 In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains". |
924 In analogy to Hochschild cochains, we will call elements of $\hom_A(\bc_*(M), \bc_*(N))$ ``blob cochains". |
926 |
925 |
927 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
926 Recall (Theorem \ref{thm:evaluation}) that chains on the space of mapping cylinders also act on the |
928 blob complex. |
927 blob complex. |
948 C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots \\ |
947 C_*(SC^n_{\overline{M}, \overline{N}})\otimes \hom(\bc_*(M_1), \bc_*(N_1))\otimes\cdots \\ |
949 \otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
948 \otimes \hom(\bc_*(M_{k}), \bc_*(N_{k})) \to \hom(\bc_*(M_0), \bc_*(N_0)) |
950 \end{multline*} |
949 \end{multline*} |
951 which satisfy the operad compatibility conditions. |
950 which satisfy the operad compatibility conditions. |
952 |
951 |
953 \begin{proof} |
952 \begin{proof} (Sketch.) |
954 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
953 We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, |
955 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
954 and the action of surgeries is just composition of maps of $A_\infty$-modules. |
956 We only need to check that the relations of the $n$-SC operad are satisfied. |
955 We only need to check that the relations of the surgery cylinded operad are satisfied. |
957 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
956 This follows from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity. |
958 \end{proof} |
957 \end{proof} |
959 |
958 |
960 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals. |
959 Consider the special case where $n=1$ and all of the $M_i$'s and $N_i$'s are intervals. |
961 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |
960 We have that $SC^1_{\overline{M}, \overline{N}}$ is homotopy equivalent to the little |