equal
deleted
inserted
replaced
1397 This of course depends (functorially) |
1397 This of course depends (functorially) |
1398 on the choice of 1-ball $J$. |
1398 on the choice of 1-ball $J$. |
1399 |
1399 |
1400 We will define a more general self tensor product (categorified coend) below. |
1400 We will define a more general self tensor product (categorified coend) below. |
1401 |
1401 |
1402 %\nn{what about self tensor products /coends ?} |
|
1403 |
|
1404 \nn{maybe ``tensor product" is not the best name?} |
|
1405 |
|
1406 %\nn{start with (less general) tensor products; maybe change this later} |
|
1407 |
|
1408 |
|
1409 |
|
1410 |
1402 |
1411 \subsection{Morphisms of $A_\infty$ $1$-category modules} |
1403 \subsection{Morphisms of $A_\infty$ $1$-category modules} |
1412 \label{ss:module-morphisms} |
1404 \label{ss:module-morphisms} |
1413 |
1405 |
1414 In order to state and prove our version of the higher dimensional Deligne conjecture |
1406 In order to state and prove our version of the higher dimensional Deligne conjecture |
1606 \] |
1598 \] |
1607 constitutes a null homotopy of |
1599 constitutes a null homotopy of |
1608 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
1600 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
1609 should be interpreted as above). |
1601 should be interpreted as above). |
1610 |
1602 |
1611 Define a {\it naive morphism} |
1603 Define a {\it strong morphism} |
1612 \nn{should consider other names for this} |
|
1613 of modules to be a collection of {\it chain} maps |
1604 of modules to be a collection of {\it chain} maps |
1614 \[ |
1605 \[ |
1615 h_K : \cX(K)\to \cY(K) |
1606 h_K : \cX(K)\to \cY(K) |
1616 \] |
1607 \] |
1617 for each left-marked interval $K$. |
1608 for each left-marked interval $K$. |
1621 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
1612 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_0}\ot \id} |
1622 \ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
1613 \ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
1623 \ar[d]^{\gl} \\ |
1614 \ar[d]^{\gl} \\ |
1624 \cX(K) \ar[r]^{h_{K}} & \cY(K) |
1615 \cX(K) \ar[r]^{h_{K}} & \cY(K) |
1625 } \] |
1616 } \] |
1626 Given such an $h$ we can construct a non-naive morphism $g$, with $\bd g = 0$, as follows. |
1617 Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows. |
1627 Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
1618 Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
1628 If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
1619 If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
1629 \[ |
1620 \[ |
1630 g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
1621 g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
1631 \] |
1622 \] |