text/a_inf_blob.tex
changeset 420 257066702f60
parent 418 a96f3d2ef852
child 426 8aca80203f9d
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418:a96f3d2ef852 420:257066702f60
    39 
    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*}