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20 \input{text/smallblobs} |
20 \input{text/smallblobs} |
21 |
21 |
22 \subsection{A product formula} |
22 \subsection{A product formula} |
23 |
23 |
24 Let $M^n = Y^k\times F^{n-k}$. |
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25 Let $C$ be a plain $n$-category. |
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26 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball |
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27 $X$ the old-fashioned blob complex $\bc_*(X\times F)$. |
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28 |
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29 \begin{thm} \label{product_thm} |
24 \begin{thm} \label{product_thm} |
30 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
25 Given a topological $n$-category $C$ and a $n-k$-manifold $F$, recall from Example \ref{ex:blob-complexes-of-balls} that there is an $A_\infty$ $k$-category $C^{\times F}$ defined by |
31 new-fangled blob complex $\bc_*^\cF(Y)$. |
26 \begin{equation*} |
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27 C^{\times F}(B) = \cB_*(B \times F, C). |
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28 \end{equation*} |
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29 Now, given a $k$-manifold $Y$, there is a homotopy equivalence between the `old-fashioned' blob complex for $Y \times F$ with coefficients in $C$ and the `new-fangled' (i.e. homotopy colimit) blob complex for $Y$ with coefficients in $C^{\times F}$: |
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30 \begin{align*} |
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31 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
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32 \end{align*} |
32 \end{thm} |
33 \end{thm} |
33 |
34 |
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35 \begin{question} |
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36 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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37 \end{question} |
34 |
38 |
35 |
39 |
36 \begin{proof}[Proof of Theorem \ref{product_thm}] |
40 \begin{proof}[Proof of Theorem \ref{product_thm}] |
37 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
41 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
38 |
42 |
340 |
344 |
341 |
345 |
342 \end{proof} |
346 \end{proof} |
343 |
347 |
344 \nn{maybe should also mention version where we enrich over |
348 \nn{maybe should also mention version where we enrich over |
345 spaces rather than chain complexes; should comment on Lurie's (and others') similar result |
349 spaces rather than chain complexes; should comment on Lurie's \cite{0911.0018} (and others') similar result |
346 for the $E_\infty$ case, and mention that our version does not require |
350 for the $E_\infty$ case, and mention that our version does not require |
347 any connectivity assumptions} |
351 any connectivity assumptions} |
348 |
352 |
349 \medskip |
353 \medskip |
350 \hrule |
354 \hrule |