20 \input{text/smallblobs} |
20 \input{text/smallblobs} |
21 |
21 |
22 \subsection{A product formula} |
22 \subsection{A product formula} |
23 |
23 |
24 \begin{thm} \label{product_thm} |
24 \begin{thm} \label{product_thm} |
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 |
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 |
26 \begin{equation*} |
26 \begin{equation*} |
27 C^{\times F}(B) = \cB_*(B \times F, C). |
27 C^{\times F}(B) = \cB_*(B \times F, C). |
28 \end{equation*} |
28 \end{equation*} |
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 $C^{\times F}$: |
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 $C^{\times F}$: |
30 \begin{align*} |
30 \begin{align*} |
31 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
31 \cB_*(Y \times F, C) & \htpy \cB_*(Y, C^{\times F}) |
32 \end{align*} |
32 \end{align*} |
33 \end{thm} |
33 \end{thm} |
34 |
34 |
35 \begin{question} |
35 \nn{To do: remark on the case of a nontrivial fiber bundle. |
36 Is it possible to compute the blob homology of a non-trivial bundle in terms of the blob homology of its fiber? |
36 I can think of two approaches. |
37 \end{question} |
37 In the first (slick but maybe a little too tautological), we generalize the |
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38 notion of an $n$-category to an $n$-category {\it over a space $B$}. |
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39 (Should be able to find precedent for this in a paper of PT. |
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40 This idea came up in a conversation with him, so maybe should site him.) |
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41 In this generalization, we replace the categories of balls with the categories |
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42 of balls equipped with maps to $B$. |
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43 A fiber bundle $F\to E\to B$ gives an example of such an $n$-category: |
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44 assign to $p:D\to B$ the blob complex $\bc_*(p^*(E))$. |
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45 We can do the colimit thing over $B$ with coefficients in a n-cat-over-B. |
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46 The proof below works essentially unchanged in this case to show that the colimit is the blob complex of the total space $E$. |
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47 } |
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48 |
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49 \nn{The second approach: Choose a decomposition $B = \cup X_i$ |
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50 such that the restriction of $E$ to $X_i$ is a product $F\times X_i$. |
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51 Choose the product structure as well. |
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52 To each codim-1 face $D_i\cap D_j$ we have a bimodule ($S^0$-module). |
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53 And more generally to each codim-$j$ face we have an $S^{j-1}$-module. |
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54 Decorate the decomposition with these modules and do the colimit. |
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55 } |
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56 |
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57 \nn{There is a version of this last construction for arbitrary maps $E \to B$ |
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58 (not necessarily a fibration).} |
38 |
59 |
39 |
60 |
40 \begin{proof}[Proof of Theorem \ref{product_thm}] |
61 \begin{proof}[Proof of Theorem \ref{product_thm}] |
41 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
62 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
42 |
63 |