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1 %!TEX root = ../blob1.tex |
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2 |
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3 \section{The blob complex for $A_\infty$ $n$-categories} |
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4 \label{sec:ainfblob} |
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5 |
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6 Given an $A_\infty$ $n$-category $\cC$ and an $n$-manifold $M$, we define the blob |
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7 complex $\bc_*(M)$ to the be the colimit $\cC(M)$ of Section \ref{sec:ncats}. |
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8 \nn{say something about this being anticlimatically tautological?} |
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9 We will show below |
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10 \nn{give ref} |
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11 that this agrees (up to homotopy) with our original definition of the blob complex |
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12 in the case of plain $n$-categories. |
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13 When we need to distinguish between the new and old definitions, we will refer to the |
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14 new-fangled and old-fashioned blob complex. |
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15 |
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16 \medskip |
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17 |
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18 Let $M^n = Y^k\times F^{n-k}$. |
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19 Let $C$ be a plain $n$-category. |
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20 Let $\cF$ be the $A_\infty$ $k$-category which assigns to a $k$-ball |
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21 $X$ the old-fashioned blob complex $\bc_*(X\times F)$. |
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22 |
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23 \begin{thm} |
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24 The old-fashioned blob complex $\bc_*^C(Y\times F)$ is homotopy equivalent to the |
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25 new-fangled blob complex $\bc_*^\cF(Y)$. |
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26 \end{thm} |
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27 |
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28 \begin{proof} |
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29 We will use the concrete description of the colimit from Subsection \ref{ss:ncat_fields}. |
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30 |
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31 First we define a map from $\bc_*^\cF(Y)$ to $\bc_*^C(Y\times F)$. |
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32 In filtration degree 0 we just glue together the various blob diagrams on $X\times F$ |
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33 (where $X$ is a component of a permissible decomposition of $Y$) to get a blob diagram on |
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34 $Y\times F$. |
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35 In filtration degrees 1 and higher we define the map to be zero. |
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36 It is easy to check that this is a chain map. |
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37 |
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38 Next we define a map from $\bc_*^C(Y\times F)$ to $\bc_*^\cF(Y)$. |
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39 Actually, we will define it on the homotopy equivalent subcomplex |
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40 $\cS_* \sub \bc_*^C(Y\times F)$ generated by blob diagrams which are small with respect to some open cover |
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41 of $Y\times F$. |
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42 \nn{need reference to small blob lemma} |
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43 We will have to show eventually that this is independent (up to homotopy) of the choice of cover. |
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44 Also, for a fixed choice of cover we will only be able to define the map for blob degree less than |
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45 some bound, but this bound goes to infinity as the cover become finer. |
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46 |
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47 \nn{....} |
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48 \end{proof} |
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49 |
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50 \nn{need to say something about dim $< n$ above} |
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51 |
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52 |
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53 |
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54 \medskip |
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55 \hrule |
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56 \medskip |
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57 |
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58 \nn{to be continued...} |
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59 \medskip |
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60 |