1581 on the choice of 1-ball $J$. |
1587 on the choice of 1-ball $J$. |
1582 |
1588 |
1583 We will define a more general self tensor product (categorified coend) below. |
1589 We will define a more general self tensor product (categorified coend) below. |
1584 |
1590 |
1585 |
1591 |
1586 \subsection{Morphisms of \texorpdfstring{$A_\infty$}{A-infinity} 1-category modules} |
1592 |
|
1593 |
|
1594 \subsection{Morphisms of modules} |
1587 \label{ss:module-morphisms} |
1595 \label{ss:module-morphisms} |
1588 |
1596 |
1589 In order to state and prove our version of the higher dimensional Deligne conjecture |
1597 Modules are collections of functors together with some additional data, so we define morphisms |
1590 (\S\ref{sec:deligne}), |
1598 of modules to be collections of natural transformations which are compatible with this |
1591 we need to define morphisms of $A_\infty$ $1$-category modules and establish |
1599 additional data. |
1592 some of their elementary properties. |
1600 |
1593 |
1601 More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors |
1594 To motivate the definitions which follow, consider algebras $A$ and $B$, |
1602 $\{\cX_k\}$ and $\{\cY_k\}$, for $0\le k\le n$, from marked $k$-balls to sets |
1595 right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction |
1603 as in Module Axiom \ref{module-axiom-funct}. |
1596 \begin{eqnarray*} |
1604 A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$ |
1597 \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ |
1605 satisfying: |
1598 f &\mapsto& [x \mapsto f(x\ot -)] \\ |
1606 \begin{itemize} |
1599 {}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . |
1607 \item Each $g_k$ commutes with $\bd$. |
1600 \end{eqnarray*} |
1608 \item Each $g_k$ commutes with gluing (module composition and $\cC$ action). |
1601 If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to |
1609 \item Each $g_k$ commutes with taking products. |
1602 \[ |
1610 \item In the top dimension $k=n$, $g_n$ preserves whatever additional structure we are enriching over (e.g.\ vector |
1603 (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . |
1611 spaces). |
1604 \] |
1612 In the $A_\infty$ case (e.g.\ enriching over chain complexes) $g_n$ should live in |
1605 We would like to have the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ |
1613 an appropriate derived hom space, as described below. |
1606 and modules $\cM_\cC$ and $_\cC\cN$, |
1614 \end{itemize} |
1607 \[ |
1615 |
1608 (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . |
1616 We will be mainly interested in the case $n=1$ and enriched over chain complexes, |
1609 \] |
1617 since this is the case that's relevant to the generalized Deligne conjecture of \S\ref{sec:deligne}. |
1610 |
1618 So we treat this case in more detail. |
1611 In the next few paragraphs we define the objects appearing in the above equation: |
1619 |
1612 $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally |
1620 First we explain the remark about derived hom above. |
1613 $\hom_\cC$. |
1621 Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction |
1614 (Actually, we give only an incomplete definition of $(_\cC\cN)^*$, but since we are only trying to motivate the |
1622 associated to $L$ by $\cX$ and $\cC$. |
1615 definition of $\hom_\cC$, this will suffice for our purposes.) |
1623 (See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) |
1616 |
1624 Define $\cl{\cY}(L)$ similarly. |
1617 \def\olD{{\overline D}} |
1625 For $K$ an unmarked 1-ball let $\cl{\cC(K)}$ denote the local homotopy colimit |
1618 \def\cbar{{\bar c}} |
1626 construction associated to $K$ by $\cC$. |
1619 In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules |
1627 Then we have an injective gluing map |
1620 for general $n$. |
1628 \[ |
1621 For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ |
1629 \gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K) |
1622 and their gluings (antirefinements). |
1630 \] |
1623 (This tensor product depends functorially on the choice of $J$.) |
1631 which is also a chain map. |
1624 To a subdivision $D$ |
1632 (For simplicity we are suppressing mention of boundary conditions on the unmarked |
1625 \[ |
1633 boundary components of the 1-balls.) |
1626 J = I_1\cup \cdots\cup I_p |
1634 We define $\hom_\cC(\cX \to \cY)$ to be a collection of (graded linear) natural transformations |
1627 \] |
1635 $g: \cl{\cX}(L)\to \cl{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$: |
1628 we associate the chain complex |
|
1629 \[ |
|
1630 \psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . |
|
1631 \] |
|
1632 To each antirefinement we associate a chain map using the composition law of $\cC$ and the |
|
1633 module actions of $\cC$ on $\cM$ and $\cN$. |
|
1634 The underlying graded vector space of the homotopy colimit is |
|
1635 \[ |
|
1636 \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , |
|
1637 \] |
|
1638 where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ |
|
1639 runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. |
|
1640 We will denote an element of the summand indexed by $\olD$ by |
|
1641 $\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. |
|
1642 The boundary map is given by |
|
1643 \begin{align*} |
|
1644 \bd(\olD\ot m\ot\cbar\ot n) &= (\bd_0 \olD)\ot \rho(m\ot\cbar\ot n) + (\bd_+ \olD)\ot m\ot\cbar\ot n \; + \\ |
|
1645 & \qquad + (-1)^l \olD\ot\bd m\ot\cbar\ot n + (-1)^{l+\deg m} \olD\ot m\ot\bd \cbar\ot n + \\ |
|
1646 & \qquad + (-1)^{l+\deg m + \deg \cbar} \olD\ot m\ot \cbar\ot \bd n |
|
1647 \end{align*} |
|
1648 where $\bd_+ \olD = \sum_{i>0} (-1)^i (D_0\to \cdots \to \widehat{D_i} \to \cdots \to D_l)$ (those parts of the simplicial |
|
1649 boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, |
|
1650 and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. |
|
1651 |
|
1652 $(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: |
|
1653 \[ |
|
1654 \prod_l \prod_{\olD} (\psi(D_0)[l])^* , |
|
1655 \] |
|
1656 where $(\psi(D_0)[l])^*$ denotes the linear dual. |
|
1657 The boundary is given by |
|
1658 \begin{align} |
|
1659 \label{eq:tensor-product-boundary} |
|
1660 (-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) + \\ |
|
1661 & \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 \\ |
|
1662 & \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag |
|
1663 \end{align} |
|
1664 |
|
1665 Next we partially define the dual module $(_\cC\cN)^*$. |
|
1666 This will depend on a choice of interval $J$, just as the tensor product did. |
|
1667 Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals |
|
1668 to chain complexes. |
|
1669 Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ |
|
1670 \[ |
|
1671 (_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , |
|
1672 \] |
|
1673 where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated |
|
1674 to the right-marked interval $J\setmin K$. |
|
1675 We define the action map |
|
1676 \[ |
|
1677 (_\cC\cN)^*(K) \ot \cC(I) \to (_\cC\cN)^*(K\cup I) |
|
1678 \] |
|
1679 to be the (partial) adjoint of the map |
|
1680 \[ |
|
1681 \cC(I)\ot {_\cC\cN}(J\setmin (K\cup I)) \to {_\cC\cN}(J\setmin K) . |
|
1682 \] |
|
1683 This falls short of fully defining the module $(_\cC\cN)^*$ (in particular, |
|
1684 we have no action of homeomorphisms of left-marked intervals), but it will be enough to motivate |
|
1685 the definition of $\hom_\cC$ below. |
|
1686 |
|
1687 Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ |
|
1688 as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. |
|
1689 Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. |
|
1690 Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. |
|
1691 Recall that for any subdivision $J = I_1\cup\cdots\cup I_p$, $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. |
|
1692 Then for each such $\olD$ we have a degree $l$ map |
|
1693 \begin{eqnarray*} |
|
1694 \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) &\to& (_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) \\ |
|
1695 m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] |
|
1696 \end{eqnarray*} |
|
1697 |
|
1698 We are almost ready to give the definition of morphisms between arbitrary modules |
|
1699 $\cX_\cC$ and $\cY_\cC$. |
|
1700 Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. |
|
1701 To fix this, we define subdivisions as antirefinements of left-marked intervals. |
|
1702 Subdivisions are just the obvious thing, but antirefinements are defined to mimic |
|
1703 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always |
|
1704 omitted. |
|
1705 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by |
|
1706 gluing subintervals together and/or omitting some of the rightmost subintervals. |
|
1707 (See Figure \ref{fig:lmar}.) |
|
1708 \begin{figure}[t] \centering |
|
1709 \definecolor{arcolor}{rgb}{.75,.4,.1} |
|
1710 \begin{tikzpicture}[line width=1pt] |
|
1711 \fill (0,0) circle (.1); |
|
1712 \draw (0,0) -- (2,0); |
|
1713 \draw (1,0.1) -- (1,-0.1); |
|
1714 |
|
1715 \draw [->, arcolor] (1,0.25) -- (1,0.75); |
|
1716 |
|
1717 \fill (0,1) circle (.1); |
|
1718 \draw (0,1) -- (2,1); |
|
1719 \end{tikzpicture} |
|
1720 \qquad |
|
1721 \begin{tikzpicture}[line width=1pt] |
|
1722 \fill (0,0) circle (.1); |
|
1723 \draw (0,0) -- (2,0); |
|
1724 \draw (1,0.1) -- (1,-0.1); |
|
1725 |
|
1726 \draw [->, arcolor] (1,0.25) -- (1,0.75); |
|
1727 |
|
1728 \fill (0,1) circle (.1); |
|
1729 \draw (0,1) -- (1,1); |
|
1730 \end{tikzpicture} |
|
1731 \qquad |
|
1732 \begin{tikzpicture}[line width=1pt] |
|
1733 \fill (0,0) circle (.1); |
|
1734 \draw (0,0) -- (3,0); |
|
1735 \foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { |
|
1736 \draw (\x,0.1) -- (\x,-0.1); |
|
1737 } |
|
1738 |
|
1739 \draw [->, arcolor] (1,0.25) -- (1,0.75); |
|
1740 |
|
1741 \fill (0,1) circle (.1); |
|
1742 \draw (0,1) -- (2,1); |
|
1743 \foreach \x in {1.0, 1.5} { |
|
1744 \draw (\x,1.1) -- (\x,0.9); |
|
1745 } |
|
1746 |
|
1747 \end{tikzpicture} |
|
1748 \caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} |
|
1749 |
|
1750 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
|
1751 The underlying vector space is |
|
1752 \[ |
|
1753 \prod_l \prod_{\olD} \hom[l]\left( |
|
1754 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to |
|
1755 \cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) , |
|
1756 \] |
|
1757 where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements |
|
1758 (but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$. |
|
1759 $\hom[l](- \to -)$ means graded linear maps of degree $l$. |
|
1760 |
|
1761 \nn{small issue (pun intended): |
|
1762 the above is a vector space only if the class of subdivisions is a set, e.g. only if |
|
1763 all of our left-marked intervals are contained in some universal interval (like $J$ above). |
|
1764 perhaps we should give another version of the definition in terms of natural transformations of functors.} |
|
1765 |
|
1766 Abusing notation slightly, we will denote elements of the above space by $g$, with |
|
1767 \[ |
|
1768 \olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . |
|
1769 \] |
|
1770 For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, |
|
1771 where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and |
|
1772 $\cbar''$ corresponds to the subintervals |
|
1773 which are dropped off the right side. |
|
1774 (If no such subintervals are dropped, then $\cbar''$ is empty.) |
|
1775 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, |
|
1776 we have |
|
1777 \begin{eqnarray*} |
|
1778 (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ |
|
1779 & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . |
|
1780 \end{eqnarray*} |
|
1781 \nn{put in signs, rearrange terms to match order in previous formulas} |
|
1782 Here $\gl''$ denotes the module action in $\cY_\cC$ |
|
1783 and $\gl'$ denotes the module action in $\cX_\cC$. |
|
1784 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. |
|
1785 |
|
1786 Note that if $\bd g = 0$, then each |
|
1787 \[ |
|
1788 g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) |
|
1789 \] |
|
1790 constitutes a null homotopy of |
|
1791 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ |
|
1792 should be interpreted as above). |
|
1793 |
|
1794 Define a {\it strong morphism} |
|
1795 of modules to be a collection of {\it chain} maps |
|
1796 \[ |
|
1797 h_K : \cX(K)\to \cY(K) |
|
1798 \] |
|
1799 for each left-marked interval $K$. |
|
1800 These are required to commute with gluing; |
|
1801 for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: |
|
1802 \[ \xymatrix{ |
1636 \[ \xymatrix{ |
1803 \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_1}\ot \id} |
1637 \cl{\cX}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \cl{\cX}(L\cup K) \ar[d]^{g}\\ |
1804 \ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) |
1638 \cl{\cY}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} & \cl{\cY}(L\cup K) |
1805 \ar[d]^{\gl} \\ |
|
1806 \cX(K) \ar[r]^{h_{K}} & \cY(K) |
|
1807 } \] |
1639 } \] |
1808 Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows. |
1640 |
1809 Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. |
1641 The usual differential on graded linear maps between chain complexes induces a differential |
1810 If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define |
1642 on $\hom_\cC(\cX \to \cY)$, giving it the structure of a chain complex. |
1811 \[ |
1643 |
1812 g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . |
1644 Let $\cZ$ be another $\cC$ module. |
1813 \] |
1645 We define a chain map |
1814 Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. |
1646 \[ |
1815 If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. |
1647 a: \hom_\cC(\cX \to \cY) \ot (\cX \ot_\cC \cZ) \to \cY \ot_\cC \cZ |
1816 If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact |
1648 \] |
1817 that each $h_K$ is a chain map. |
1649 as follows. |
1818 |
1650 Recall that the tensor product $\cX \ot_\cC \cZ$ depends on a choice of interval $J$, labeled |
1819 We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$ |
1651 by $\cX$ on one boundary component and $\cZ$ on the other. |
1820 as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy. |
1652 Because we are using the {\it local} homotopy colimit, any generator |
1821 \nn{ideally should give explicit examples of this in low degrees, |
1653 $D\ot x\ot \bar{c}\ot z$ of $\cX \ot_\cC \cZ$ can be written (perhaps non-uniquely) as a gluing |
1822 but skip that for now.} |
1654 $(D'\ot x \ot \bar{c}') \bullet (D''\ot \bar{c}''\ot z)$, for some decomposition $J = L'\cup L''$ |
1823 \nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe |
1655 and with $D'\ot x \ot \bar{c}'$ a generator of $\cl{\cX}(L')$ and |
1824 should make some arbitrary choice} |
1656 $D''\ot \bar{c}''\ot z$ a generator of $\cl{\cZ}(L'')$. |
1825 \medskip |
1657 (Such a splitting exists because the blob diagram $D$ can be split into left and right halves, |
1826 |
1658 since no blob can include both the leftmost and rightmost intervals in the underlying decomposition. |
1827 Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map |
1659 This step would fail if we were using the usual hocolimit instead of the local hocolimit.) |
1828 \[ |
1660 We now define |
1829 g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
1661 \[ |
1830 \] |
1662 a: g\ot (D\ot x\ot \bar{c}\ot z) \mapsto g(D'\ot x \ot \bar{c}')\bullet (D''\ot \bar{c}''\ot z) . |
1831 \nn{...} |
1663 \] |
1832 More generally, we have a chain map |
1664 This does not depend on the choice of splitting $D = D'\bullet D''$ because $g$ commutes with gluing. |
1833 \[ |
|
1834 \hom_\cC(\cX_\cC \to \cY_\cC) \ot \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . |
|
1835 \] |
|
1836 |
|
1837 \nn{not sure whether to do low degree examples or try to state the general case; ideally both, |
|
1838 but maybe just low degrees for now.} |
|
1839 |
|
1840 |
|
1841 \nn{...} |
|
1842 |
|
1843 |
|
1844 \medskip |
|
1845 |
|
1846 |
|
1847 %\nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations |
|
1848 %of the $\cC$ functors which commute with gluing only up to higher morphisms? |
|
1849 %perhaps worth having both definitions available. |
|
1850 %certainly the simple kind (strictly commute with gluing) arise in nature.} |
|
1851 |
1665 |
1852 |
1666 |
1853 |
1667 |
1854 |
1668 |
1855 \subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |
1669 \subsection{The \texorpdfstring{$n{+}1$}{n+1}-category of sphere modules} |