1412 \bigskip |
1412 \bigskip |
1413 |
1413 |
1414 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
1414 \nn{need to define/recall def of (self) tensor product over an $A_\infty$ category} |
1415 |
1415 |
1416 |
1416 |
|
1417 \section{Commutative algebras as $n$-categories} |
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1418 |
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1419 \nn{this should probably not be a section by itself. i'm just trying to write down the outline |
|
1420 while it's still fresh in my mind.} |
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1421 |
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1422 If $C$ is a commutative algebra it |
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1423 can (and will) also be thought of as an $n$-category with trivial $j$-morphisms for |
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1424 $j<n$ and $n$-morphisms are $C$. |
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1425 The goal of this \nn{subsection?} is to compute |
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1426 $\bc_*(M^n, C)$ for various commutative algebras $C$. |
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1427 |
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1428 Let $k[t]$ denote the ring of polynomials in $t$ with coefficients in $k$. |
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1429 |
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1430 Let $\Sigma^i(M)$ denote the $i$-th symmetric power of $M$, the configuration space of $i$ |
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1431 unlabeled points in $M$. |
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1432 Note that $\Sigma^i(M)$ is a point. |
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1433 Let $\Sigma^\infty(M) = \coprod_{i=0}^\infty \Sigma^i(M)$. |
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1434 |
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1435 Let $C_*(X)$ denote the singular chain complex of the space $X$. |
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1436 |
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1437 \begin{prop} |
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1438 $\bc_*(M^n, k[t])$ is homotopy equivalent to $C_*(\Sigma^\infty(M))$. |
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1439 \end{prop} |
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1440 |
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1441 \begin{proof} |
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1442 To define the chain maps between the two complexes we will use the following lemma: |
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1443 |
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1444 \begin{lemma} |
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1445 Let $A_*$ and $B_*$ be chain complexes, and assume $A_*$ is equipped with |
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1446 a basis (e.g.\ blob diagrams or singular simplices). |
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1447 For each basis element $c \in A_*$ assume given a contractible subcomplex $R(c)_* \sub B_*$ |
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1448 such that $R(c') \sub R(c)$ whenever $c'$ is a basis element which is part of $\bd c$. |
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1449 Then the complex of chain maps (and (iterated) homotopies) $f:A_*\to B_*$ such that |
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1450 $f(c) \in R(c)_*$ for all $c$ is contractible (and in particular non-empty). |
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1451 \end{lemma} |
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1452 |
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1453 \begin{proof} |
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1454 \nn{easy, but should probably write the details eventually} |
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1455 \end{proof} |
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1456 |
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1457 \nn{...} |
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1458 |
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1459 \end{proof} |
|
1460 |
|
1461 |
|
1462 |
|
1463 |
1417 |
1464 |
1418 |
1465 |
1419 \appendix |
1466 \appendix |
1420 |
1467 |
1421 \section{Families of Diffeomorphisms} \label{sec:localising} |
1468 \section{Families of Diffeomorphisms} \label{sec:localising} |
1604 |
1651 |
1605 \nn{this completes proof} |
1652 \nn{this completes proof} |
1606 |
1653 |
1607 \input{text/explicit.tex} |
1654 \input{text/explicit.tex} |
1608 |
1655 |
|
1656 |
1609 % ---------------------------------------------------------------- |
1657 % ---------------------------------------------------------------- |
1610 %\newcommand{\urlprefix}{} |
1658 %\newcommand{\urlprefix}{} |
1611 \bibliographystyle{plain} |
1659 \bibliographystyle{plain} |
1612 %Included for winedt: |
1660 %Included for winedt: |
1613 %input "bibliography/bibliography.bib" |
1661 %input "bibliography/bibliography.bib" |
1614 \bibliography{bibliography/bibliography} |
1662 \bibliography{bibliography/bibliography} |
1615 % ---------------------------------------------------------------- |
1663 % ---------------------------------------------------------------- |
1616 |
1664 |
1617 This paper is available online at \arxiv{?????}, and at |
1665 This paper is available online at \arxiv{?????}, and at |
1618 \url{http://tqft.net/blobs}. |
1666 \url{http://tqft.net/blobs}, |
|
1667 and at \url{http://canyon23.net/math/}. |
1619 |
1668 |
1620 % A GTART necessity: |
1669 % A GTART necessity: |
1621 % \Addresses |
1670 % \Addresses |
1622 % ---------------------------------------------------------------- |
1671 % ---------------------------------------------------------------- |
1623 \end{document} |
1672 \end{document} |