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39 |
40 \begin{thm} \label{thm:product} |
40 \begin{thm} \label{thm:product} |
41 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
41 Given a topological $n$-category $C$ and a $n{-}k$-manifold $F$, recall from |
42 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by |
42 Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $\bc_*(F; C)$ defined by |
43 \begin{equation*} |
43 \begin{equation*} |
44 \bc_*(F; C) = \cB_*(B \times F, C). |
44 \bc_*(F; C)(B) = \cB_*(F \times B; C). |
45 \end{equation*} |
45 \end{equation*} |
46 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' |
46 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the ``old-fashioned'' |
47 blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled" |
47 blob complex for $Y \times F$ with coefficients in $C$ and the ``new-fangled" |
48 (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: |
48 (i.e.\ homotopy colimit) blob complex for $Y$ with coefficients in $\bc_*(F; C)$: |
49 \begin{align*} |
49 \begin{align*} |
50 \cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y) |
50 \cB_*(Y \times F; C) & \htpy \cl{\bc_*(F; C)}(Y) |
51 \end{align*} |
51 \end{align*} |