1536 \end{eqnarray*} |
1536 \end{eqnarray*} |
1537 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
1537 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
1538 \[ |
1538 \[ |
1539 (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
1539 (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
1540 \] |
1540 \] |
1541 We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
1541 We would like to have the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
1542 and modules $\cM_\cC$ and $_\cC\cN$, |
1542 and modules $\cM_\cC$ and $_\cC\cN$, |
1543 \[ |
1543 \[ |
1544 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1544 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1545 \] |
1545 \] |
1546 |
1546 |
1547 In the next few paragraphs we define the objects appearing in the above equation: |
1547 In the next few paragraphs we define the objects appearing in the above equation: |
1548 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
1548 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
1549 $\hom_\cC$. |
1549 $\hom_\cC$. |
1550 |
1550 (Actually, we give only an incomplete definition of $(_\cC\cN)^*$, but since we are only trying to motivate the |
|
1551 definition of $\hom_\cC$, this will suffice for our purposes.) |
1551 |
1552 |
1552 \def\olD{{\overline D}} |
1553 \def\olD{{\overline D}} |
1553 \def\cbar{{\bar c}} |
1554 \def\cbar{{\bar c}} |
1554 In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
1555 In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
1555 for general $n$. |
1556 for general $n$. |
1595 (-1)^{\deg f +1} (\bd f)(\olD\ot m\ot\cbar\ot n) & = f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) + f((\bd_+ \olD)\ot m\ot\cbar\ot n) + \\ |
1596 (-1)^{\deg f +1} (\bd f)(\olD\ot m\ot\cbar\ot n) & = f((\bd_0 \olD)\ot \rho(m\ot\cbar\ot n)) + f((\bd_+ \olD)\ot m\ot\cbar\ot n) + \\ |
1596 & \qquad + (-1)^{l} f(\olD\ot\bd m\ot\cbar \ot n) + (-1)^{l + \deg m} f(\olD\ot m\ot\bd \cbar \ot n) + \notag \\ |
1597 & \qquad + (-1)^{l} f(\olD\ot\bd m\ot\cbar \ot n) + (-1)^{l + \deg m} f(\olD\ot m\ot\bd \cbar \ot n) + \notag \\ |
1597 & \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
1598 & \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
1598 \end{align} |
1599 \end{align} |
1599 |
1600 |
1600 Next we define the dual module $(_\cC\cN)^*$. |
1601 Next we partially define the dual module $(_\cC\cN)^*$. |
1601 This will depend on a choice of interval $J$, just as the tensor product did. |
1602 This will depend on a choice of interval $J$, just as the tensor product did. |
1602 Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
1603 Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
1603 to chain complexes. |
1604 to chain complexes. |
1604 Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
1605 Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
1605 \[ |
1606 \[ |
1606 (_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
1607 (_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
1607 \] |
1608 \] |
1608 where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
1609 where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
1609 to the right-marked interval $J\setmin K$. |
1610 to the right-marked interval $J\setmin K$. |
1610 This extends to a functor from all left-marked intervals (not just those contained in $J$). |
1611 We define the action map |
1611 \nn{need to say more here; not obvious how homeomorphisms act} |
1612 \[ |
1612 It's easy to verify the remaining module axioms. |
1613 (_\cC\cN)^*(K) \ot \cC(I) \to (_\cC\cN)^*(K\cup I) |
|
1614 \] |
|
1615 to be the (partial) adjoint of the map |
|
1616 \[ |
|
1617 \cC(I)\ot {_\cC\cN}(J\setmin (K\cup I)) \to {_\cC\cN}(J\setmin K) . |
|
1618 \] |
|
1619 This falls short of fully defining the module $(_\cC\cN)^*$ (in particular, |
|
1620 we have no action of homeomorphisms of left-marked intervals), but it will be enough to motivate |
|
1621 the definition of $\hom_\cC$ below. |
1613 |
1622 |
1614 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1623 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
1615 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1624 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
1616 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1625 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
1617 Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
1626 Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
1770 |
1779 |
1771 |
1780 |
1772 \medskip |
1781 \medskip |
1773 |
1782 |
1774 |
1783 |
1775 \nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
1784 %\nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
1776 of the $\cC$ functors which commute with gluing only up to higher morphisms? |
1785 %of the $\cC$ functors which commute with gluing only up to higher morphisms? |
1777 perhaps worth having both definitions available. |
1786 %perhaps worth having both definitions available. |
1778 certainly the simple kind (strictly commute with gluing) arise in nature.} |
1787 %certainly the simple kind (strictly commute with gluing) arise in nature.} |
1779 |
1788 |
1780 |
1789 |
1781 |
1790 |
1782 |
1791 |
1783 \subsection{The $n{+}1$-category of sphere modules} |
1792 \subsection{The $n{+}1$-category of sphere modules} |