178 \begin{figure}[!ht]

   178 \begin{figure}[!ht] \centering

   179 $$

   235 If $k < n$ we require that $\gl_Y$ is injective.

   233 If $k < n$,

   236 (For $k=n$, see below.)

   234 or if $k=n$ and we are in the $A_\infty$ case, 

   235 we require that $\gl_Y$ is injective.

   236 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)

   239 \begin{figure}[!ht]

   239 \begin{figure}[!ht] \centering

   240 $$

  1171 \begin{module-axiom}[Module morphisms]

  1169 \begin{module-axiom}[Module morphisms] \label{module-axiom-funct}

  1279 If $k < n$ we require that $\gl_Y$ is injective.

  1277 If $k < n$,

  1280 (For $k=n$, see below.)}

  1278 or if $k=n$ and we are in the $A_\infty$ case, 

  1279 we require that $\gl_Y$ is injective.

  1280 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)}

  1301 If $k < n$ we require that $\gl_Y$ is injective.

  1301 If $k < n$,

  1302 (For $k=n$, see below.)}

  1302 or if $k=n$ and we are in the $A_\infty$ case, 

  1303 we require that $\gl_Y$ is injective.

  1304 (For $k=n$ in the plain (non-$A_\infty$) case, see below.)}

  1586 \subsection{Morphisms of \texorpdfstring{$A_\infty$}{A-infinity} 1-category modules}

  1592 

  1593 

  1594 \subsection{Morphisms of modules}

  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:

  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)

  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.}