# HG changeset patch # User Kevin Walker # Date 1285003457 25200 # Node ID 689ef4edbdd7d6cdd673ce385e743d4da6b1c814 # Parent 8f33a46597c4462aa35b80896ac24793822e0535 new def of mophisms between modules diff -r 8f33a46597c4 -r 689ef4edbdd7 text/ncat.tex --- a/text/ncat.tex Mon Sep 20 06:39:25 2010 -0700 +++ b/text/ncat.tex Mon Sep 20 10:24:17 2010 -0700 @@ -175,8 +175,7 @@ becomes a normal product.) \end{lem} -\begin{figure}[!ht] -$$ +\begin{figure}[!ht] \centering \begin{tikzpicture}[%every label/.style={green} ] \node[fill=black, circle, label=below:$E$, inner sep=1.5pt](S) at (0,0) {}; @@ -186,7 +185,6 @@ \node[left] at (-1,1) {$B_1$}; \node[right] at (1,1) {$B_2$}; \end{tikzpicture} -$$ \caption{Combining two balls to get a full boundary.}\label{blah3}\end{figure} Note that we insist on injectivity above. @@ -232,12 +230,13 @@ \] which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions to the intersection of the boundaries of $B$ and $B_i$. -If $k < n$ we require that $\gl_Y$ is injective. -(For $k=n$, see below.) +If $k < n$, +or if $k=n$ and we are in the $A_\infty$ case, +we require that $\gl_Y$ is injective. +(For $k=n$ in the plain (non-$A_\infty$) case, see below.) \end{axiom} -\begin{figure}[!ht] -$$ +\begin{figure}[!ht] \centering \begin{tikzpicture}[%every label/.style={green}, x=1.5cm,y=1.5cm] \node[fill=black, circle, label=below:$E$, inner sep=2pt](S) at (0,0) {}; @@ -249,7 +248,6 @@ \node[right] at (1/4,1) {$B_2$}; \node at (1/6,3/2) {$Y$}; \end{tikzpicture} -$$ \caption{From two balls to one ball.}\label{blah5}\end{figure} \begin{axiom}[Strict associativity] \label{nca-assoc} @@ -1168,7 +1166,7 @@ A homeomorphism between marked $k$-balls is a homeomorphism of balls which restricts to a homeomorphism of markings. -\begin{module-axiom}[Module morphisms] +\begin{module-axiom}[Module morphisms] \label{module-axiom-funct} {For each $0 \le k \le n$, we have a functor $\cM_k$ from the category of marked $k$-balls and homeomorphisms to the category of sets and bijections.} @@ -1276,8 +1274,10 @@ \] which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions to the intersection of the boundaries of $M$ and $M_i$. -If $k < n$ we require that $\gl_Y$ is injective. -(For $k=n$, see below.)} +If $k < n$, +or if $k=n$ and we are in the $A_\infty$ case, +we require that $\gl_Y$ is injective. +(For $k=n$ in the plain (non-$A_\infty$) case, see below.)} \end{module-axiom} @@ -1298,8 +1298,10 @@ \] which is natural with respect to the actions of homeomorphisms, and also compatible with restrictions to the intersection of the boundaries of $X$ and $M'$. -If $k < n$ we require that $\gl_Y$ is injective. -(For $k=n$, see below.)} +If $k < n$, +or if $k=n$ and we are in the $A_\infty$ case, +we require that $\gl_Y$ is injective. +(For $k=n$ in the plain (non-$A_\infty$) case, see below.)} \end{module-axiom} \begin{module-axiom}[Strict associativity] @@ -1505,6 +1507,10 @@ \ref{ex:linearized-maps-to-a-space} are also possible, and there is an $A_\infty$ version analogous to Example \ref{ex:chains-of-maps-to-a-space} given by taking singular chains. + + + + \subsection{Modules as boundary labels (colimits for decorated manifolds)} \label{moddecss} @@ -1583,271 +1589,79 @@ We will define a more general self tensor product (categorified coend) below. -\subsection{Morphisms of \texorpdfstring{$A_\infty$}{A-infinity} 1-category modules} + + +\subsection{Morphisms of modules} \label{ss:module-morphisms} -In order to state and prove our version of the higher dimensional Deligne conjecture -(\S\ref{sec:deligne}), -we need to define morphisms of $A_\infty$ $1$-category modules and establish -some of their elementary properties. - -To motivate the definitions which follow, consider algebras $A$ and $B$, -right modules $X_B$ and $Z_A$ and a bimodule $\leftidx{_B}{Y}{_A}$, and the familiar adjunction -\begin{eqnarray*} - \hom_A(X_B\ot {_BY_A} \to Z_A) &\cong& \hom_B(X_B \to \hom_A( {_BY_A} \to Z_A)) \\ - f &\mapsto& [x \mapsto f(x\ot -)] \\ - {}[x\ot y \mapsto g(x)(y)] & \mapsfrom & g . -\end{eqnarray*} -If $A$ and $Z_A$ are both the ground field $\k$, this simplifies to -\[ - (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . -\] -We would like to have the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ -and modules $\cM_\cC$ and $_\cC\cN$, -\[ - (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . -\] - -In the next few paragraphs we define the objects appearing in the above equation: -$\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally -$\hom_\cC$. -(Actually, we give only an incomplete definition of $(_\cC\cN)^*$, but since we are only trying to motivate the -definition of $\hom_\cC$, this will suffice for our purposes.) - -\def\olD{{\overline D}} -\def\cbar{{\bar c}} -In the previous subsection we defined a tensor product of $A_\infty$ $n$-category modules -for general $n$. -For $n=1$ this definition is a homotopy colimit indexed by subdivisions of a fixed interval $J$ -and their gluings (antirefinements). -(This tensor product depends functorially on the choice of $J$.) -To a subdivision $D$ -\[ - J = I_1\cup \cdots\cup I_p -\] -we associate the chain complex -\[ - \psi(D) = \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})\ot\cN(I_m) . -\] -To each antirefinement we associate a chain map using the composition law of $\cC$ and the -module actions of $\cC$ on $\cM$ and $\cN$. -The underlying graded vector space of the homotopy colimit is -\[ - \bigoplus_l \bigoplus_{\olD} \psi(D_0)[l] , -\] -where $l$ runs through the natural numbers, $\olD = (D_0\to D_1\to\cdots\to D_l)$ -runs through chains of antirefinements of length $l+1$, and $[l]$ denotes a grading shift. -We will denote an element of the summand indexed by $\olD$ by -$\olD\ot m\ot\cbar\ot n$, where $m\ot\cbar\ot n \in \psi(D_0)$. -The boundary map is given by -\begin{align*} - \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 \; + \\ - & \qquad + (-1)^l \olD\ot\bd m\ot\cbar\ot n + (-1)^{l+\deg m} \olD\ot m\ot\bd \cbar\ot n + \\ - & \qquad + (-1)^{l+\deg m + \deg \cbar} \olD\ot m\ot \cbar\ot \bd n -\end{align*} -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 -boundary which retain $D_0$), $\bd_0 \olD = (D_1 \to \cdots \to D_l)$, -and $\rho$ is the gluing map associated to the antirefinement $D_0\to D_1$. +Modules are collections of functors together with some additional data, so we define morphisms +of modules to be collections of natural transformations which are compatible with this +additional data. -$(\cM_\cC\ot {_\cC\cN})^*$ is just the dual chain complex to $\cM_\cC\ot {_\cC\cN}$: -\[ - \prod_l \prod_{\olD} (\psi(D_0)[l])^* , -\] -where $(\psi(D_0)[l])^*$ denotes the linear dual. -The boundary is given by -\begin{align} -\label{eq:tensor-product-boundary} - (-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) + \\ - & \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 \\ - & \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag -\end{align} +More specifically, let $\cX$ and $\cY$ be $\cC$ modules, i.e.\ collections of functors +$\{\cX_k\}$ and $\{\cY_k\}$, for $0\le k\le n$, from marked $k$-balls to sets +as in Module Axiom \ref{module-axiom-funct}. +A morphism $g:\cX\to\cY$ is a collection of natural transformations $g_k:\cX_k\to\cY_k$ +satisfying: +\begin{itemize} +\item Each $g_k$ commutes with $\bd$. +\item Each $g_k$ commutes with gluing (module composition and $\cC$ action). +\item Each $g_k$ commutes with taking products. +\item In the top dimension $k=n$, $g_n$ preserves whatever additional structure we are enriching over (e.g.\ vector +spaces). +In the $A_\infty$ case (e.g.\ enriching over chain complexes) $g_n$ should live in +an appropriate derived hom space, as described below. +\end{itemize} -Next we partially define the dual module $(_\cC\cN)^*$. -This will depend on a choice of interval $J$, just as the tensor product did. -Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals -to chain complexes. -Given $J$, we define for each $K\sub J$ which contains the {\it left} endpoint of $J$ -\[ - (_\cC\cN)^*(K) \deq ({_\cC\cN}(J\setmin K))^* , -\] -where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated -to the right-marked interval $J\setmin K$. -We define the action map -\[ - (_\cC\cN)^*(K) \ot \cC(I) \to (_\cC\cN)^*(K\cup I) -\] -to be the (partial) adjoint of the map -\[ - \cC(I)\ot {_\cC\cN}(J\setmin (K\cup I)) \to {_\cC\cN}(J\setmin K) . -\] -This falls short of fully defining the module $(_\cC\cN)^*$ (in particular, -we have no action of homeomorphisms of left-marked intervals), but it will be enough to motivate -the definition of $\hom_\cC$ below. - -Now we reinterpret $(\cM_\cC\ot {_\cC\cN})^*$ -as some sort of morphism $\cM_\cC \to (_\cC\cN)^*$. -Let $f\in (\cM_\cC\ot {_\cC\cN})^*$. -Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$. -Recall that for any subdivision $J = I_1\cup\cdots\cup I_p$, $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$. -Then for each such $\olD$ we have a degree $l$ map -\begin{eqnarray*} - \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}) \\ - m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)] -\end{eqnarray*} - -We are almost ready to give the definition of morphisms between arbitrary modules -$\cX_\cC$ and $\cY_\cC$. -Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$. -To fix this, we define subdivisions as antirefinements of left-marked intervals. -Subdivisions are just the obvious thing, but antirefinements are defined to mimic -the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always -omitted. -More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by -gluing subintervals together and/or omitting some of the rightmost subintervals. -(See Figure \ref{fig:lmar}.) -\begin{figure}[t] \centering -\definecolor{arcolor}{rgb}{.75,.4,.1} -\begin{tikzpicture}[line width=1pt] -\fill (0,0) circle (.1); -\draw (0,0) -- (2,0); -\draw (1,0.1) -- (1,-0.1); - -\draw [->, arcolor] (1,0.25) -- (1,0.75); +We will be mainly interested in the case $n=1$ and enriched over chain complexes, +since this is the case that's relevant to the generalized Deligne conjecture of \S\ref{sec:deligne}. +So we treat this case in more detail. -\fill (0,1) circle (.1); -\draw (0,1) -- (2,1); -\end{tikzpicture} -\qquad -\begin{tikzpicture}[line width=1pt] -\fill (0,0) circle (.1); -\draw (0,0) -- (2,0); -\draw (1,0.1) -- (1,-0.1); - -\draw [->, arcolor] (1,0.25) -- (1,0.75); - -\fill (0,1) circle (.1); -\draw (0,1) -- (1,1); -\end{tikzpicture} -\qquad -\begin{tikzpicture}[line width=1pt] -\fill (0,0) circle (.1); -\draw (0,0) -- (3,0); -\foreach \x in {0.5, 1.0, 1.25, 1.5, 2.0, 2.5} { - \draw (\x,0.1) -- (\x,-0.1); -} - -\draw [->, arcolor] (1,0.25) -- (1,0.75); - -\fill (0,1) circle (.1); -\draw (0,1) -- (2,1); -\foreach \x in {1.0, 1.5} { - \draw (\x,1.1) -- (\x,0.9); -} - -\end{tikzpicture} -\caption{Antirefinements of left-marked intervals}\label{fig:lmar}\end{figure} - -Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$. -The underlying vector space is +First we explain the remark about derived hom above. +Let $L$ be a marked 1-ball and let $\cl{\cX}(L)$ denote the local homotopy colimit construction +associated to $L$ by $\cX$ and $\cC$. +(See \S \ref{ss:ncat_fields} and \S \ref{moddecss}.) +Define $\cl{\cY}(L)$ similarly. +For $K$ an unmarked 1-ball let $\cl{\cC(K)}$ denote the local homotopy colimit +construction associated to $K$ by $\cC$. +Then we have an injective gluing map \[ - \prod_l \prod_{\olD} \hom[l]\left( - \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to - \cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) , -\] -where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements -(but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$. -$\hom[l](- \to -)$ means graded linear maps of degree $l$. - -\nn{small issue (pun intended): -the above is a vector space only if the class of subdivisions is a set, e.g. only if -all of our left-marked intervals are contained in some universal interval (like $J$ above). -perhaps we should give another version of the definition in terms of natural transformations of functors.} - -Abusing notation slightly, we will denote elements of the above space by $g$, with -\[ - \olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) . + \gl: \cl{\cX}(L) \ot \cl{\cC}(K) \to \cl{\cX}(L\cup K) \] -For fixed $D_0$ and $D_1$, let $\cbar = \cbar'\ot\cbar''$, -where $\cbar'$ corresponds to the subintervals of $D_0$ which map to $D_1$ and -$\cbar''$ corresponds to the subintervals -which are dropped off the right side. -(If no such subintervals are dropped, then $\cbar''$ is empty.) -Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ appearing in Equation \eqref{eq:tensor-product-boundary}, -we have -\begin{eqnarray*} - (\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\ - & & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl''(g((\bd_0\olD)\ot \gl'(x\ot\cbar'))\ot\cbar'') . -\end{eqnarray*} -\nn{put in signs, rearrange terms to match order in previous formulas} -Here $\gl''$ denotes the module action in $\cY_\cC$ -and $\gl'$ denotes the module action in $\cX_\cC$. -This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$. - -Note that if $\bd g = 0$, then each -\[ - g(\olD\ot -) : \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to \cY(I_1\cup\cdots\cup I_{p-1}) -\] -constitutes a null homotopy of -$g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$ -should be interpreted as above). - -Define a {\it strong morphism} -of modules to be a collection of {\it chain} maps -\[ - h_K : \cX(K)\to \cY(K) -\] -for each left-marked interval $K$. -These are required to commute with gluing; -for each subdivision $K = I_1\cup\cdots\cup I_q$ the following diagram commutes: +which is also a chain map. +(For simplicity we are suppressing mention of boundary conditions on the unmarked +boundary components of the 1-balls.) +We define $\hom_\cC(\cX \to \cY)$ to be a collection of (graded linear) natural transformations +$g: \cl{\cX}(L)\to \cl{\cY}(L)$ such that the following diagram commutes for all $L$ and $K$: \[ \xymatrix{ - \cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) \ar[r]^{h_{I_1}\ot \id} - \ar[d]_{\gl} & \cY(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_q) - \ar[d]^{\gl} \\ - \cX(K) \ar[r]^{h_{K}} & \cY(K) + \cl{\cX}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} \ar[d]_{g\ot \id} & \cl{\cX}(L\cup K) \ar[d]^{g}\\ + \cl{\cY}(L) \ot \cl{\cC}(K) \ar[r]^{\gl} & \cl{\cY}(L\cup K) } \] -Given such an $h$ we can construct a morphism $g$, with $\bd g = 0$, as follows. -Define $g(\olD\ot - ) = 0$ if the length/degree of $\olD$ is greater than 0. -If $\olD$ consists of the single subdivision $K = I_0\cup\cdots\cup I_q$ then define -\[ - g(\olD\ot x\ot \cbar) \deq h_K(\gl(x\ot\cbar)) . -\] -Trivially, we have $(\bd g)(\olD\ot x \ot \cbar) = 0$ if $\deg(\olD) > 1$. -If $\deg(\olD) = 1$, $(\bd g) = 0$ is equivalent to the fact that $h$ commutes with gluing. -If $\deg(\olD) = 0$, $(\bd g) = 0$ is equivalent to the fact -that each $h_K$ is a chain map. -We can think of a general closed element $g\in \hom_\cC(\cX_\cC \to \cY_\cC)$ -as a collection of chain maps which commute with the module action (gluing) up to coherent homotopy. -\nn{ideally should give explicit examples of this in low degrees, -but skip that for now.} -\nn{should also say something about composition of morphisms; well-defined up to homotopy, or maybe -should make some arbitrary choice} -\medskip +The usual differential on graded linear maps between chain complexes induces a differential +on $\hom_\cC(\cX \to \cY)$, giving it the structure of a chain complex. -Given $_\cC\cZ$ and $g: \cX_\cC \to \cY_\cC$ with $\bd g = 0$ as above, we next define a chain map +Let $\cZ$ be another $\cC$ module. +We define a chain map \[ - g\ot\id : \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . -\] -\nn{...} -More generally, we have a chain map -\[ - \hom_\cC(\cX_\cC \to \cY_\cC) \ot \cX_\cC \ot {}_\cC\cZ \to \cY_\cC \ot {}_\cC\cZ . + a: \hom_\cC(\cX \to \cY) \ot (\cX \ot_\cC \cZ) \to \cY \ot_\cC \cZ \] - -\nn{not sure whether to do low degree examples or try to state the general case; ideally both, -but maybe just low degrees for now.} - - -\nn{...} - - -\medskip - - -%\nn{should we define functors between $n$-cats in a similar way? i.e.\ natural transformations -%of the $\cC$ functors which commute with gluing only up to higher morphisms? -%perhaps worth having both definitions available. -%certainly the simple kind (strictly commute with gluing) arise in nature.} +as follows. +Recall that the tensor product $\cX \ot_\cC \cZ$ depends on a choice of interval $J$, labeled +by $\cX$ on one boundary component and $\cZ$ on the other. +Because we are using the {\it local} homotopy colimit, any generator +$D\ot x\ot \bar{c}\ot z$ of $\cX \ot_\cC \cZ$ can be written (perhaps non-uniquely) as a gluing +$(D'\ot x \ot \bar{c}') \bullet (D''\ot \bar{c}''\ot z)$, for some decomposition $J = L'\cup L''$ +and with $D'\ot x \ot \bar{c}'$ a generator of $\cl{\cX}(L')$ and +$D''\ot \bar{c}''\ot z$ a generator of $\cl{\cZ}(L'')$. +(Such a splitting exists because the blob diagram $D$ can be split into left and right halves, +since no blob can include both the leftmost and rightmost intervals in the underlying decomposition. +This step would fail if we were using the usual hocolimit instead of the local hocolimit.) +We now define +\[ + a: g\ot (D\ot x\ot \bar{c}\ot z) \mapsto g(D'\ot x \ot \bar{c}')\bullet (D''\ot \bar{c}''\ot z) . +\] +This does not depend on the choice of splitting $D = D'\bullet D''$ because $g$ commutes with gluing.