1130 	J = I_1\cup \cdots\cup I_m

  1130 	J = I_1\cup \cdots\cup I_p

  1187 Let $\olD$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$, and let

  1187 Let $\olD = (D_0\cdots D_l)$ be a chain of subdivisions with $D_0 = [J = I_1\cup\cdots\cup I_m]$.

  1188 $m\ot \cbar \in \cM(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{m-1})$.

  1188 Recall that $(_\cC\cN)^*(I_1\cup\cdots\cup I_{p-1}) = (_\cC\cN(I_p))^*$.

  1189 

  1189 Then for each such $\olD$ we have a degree $l$ map

  1190 

  1190 \begin{eqnarray*}

  1191 

  1191 	\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}) \\

  1192 

  1192 	m\ot \cbar &\mapsto& [n\mapsto f(\olD\ot m\ot \cbar\ot n)]

  1193 

  1193 \end{eqnarray*}

  1194 

  1195 We are almost ready to give the definition of morphisms between arbitrary modules

  1196 $\cX_\cC$ and $\cY_\cC$.

  1197 Note that the rightmost interval $I_m$ does not appear above, except implicitly in $\olD$.

  1198 To fix this, we define subdivisions are antirefinements of left-marked intervals.

  1199 Subdivisions are just the obvious thing, but antirefinements are defined to mimic

  1200 the above antirefinements of the fixed interval $J$, but with the rightmost subinterval $I_m$ always

  1201 omitted.

  1202 More specifically, $D\to D'$ is an antirefinement if $D'$ is obtained from $D$ by 

  1203 gluing subintervals together and/or omitting some of the rightmost subintervals.

  1204 (See Figure xxxx.)

  1205 

  1206 Now we define the chain complex $\hom_\cC(\cX_\cC \to \cY_\cC)$.

  1207 The underlying vector space is 

  1208 \[

  1209 	\prod_l \prod_{\olD} \hom[l]\left(

  1210 				\cX(I_1)\ot\cC(I_2)\ot\cdots\ot\cC(I_{p-1}) \to 

  1211 							\cY(I_1\cup\cdots\cup I_{p-1}) \rule{0pt}{1.1em}\right) ,

  1212 \]

  1213 where, as usual $\olD = (D_0\cdots D_l)$ is a chain of antirefinements

  1214 (but now of left-marked intervals) and $D_0$ is the subdivision $I_1\cup\cdots\cup I_{p-1}$.

  1215 $\hom[l](- \to -)$ means graded linear maps of degree $l$.

  1216 

  1217 \nn{small issue (pun intended): 

  1218 the above is a vector space only if the class of subdivisions is a set, e.g. only if

  1219 all of our left-marked intervals are contained in some universal interval (like $J$ above).

  1220 perhaps we should give another version of the definition in terms of natural transformations of functors.}

  1221 

  1222 Abusing notation slightly, we will denote elements of the above space by $g$, with

  1223 \[

  1224 	\olD\ot x \ot \cbar \mapsto g(\olD\ot x \ot \cbar) \in \cY(I_1\cup\cdots\cup I_{p-1}) .

  1225 \]

  1226 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

  1227 which are dropped off the right side.

  1228 (Either $\cbar'$ or $\cbar''$ might be empty.)

  1229 Translating from the boundary map for $(\cM_\cC\ot {_\cC\cN})^*$ \nn{give ref?},

  1230 we have

  1231 \begin{eqnarray*}

  1232 	(\bd g)(\olD\ot x \ot \cbar) &=& \bd(g(\olD\ot x \ot \cbar)) + g(\olD\ot\bd(x\ot\cbar)) + \\

  1233 	& & \;\; g((\bd_+\olD)\ot x\ot\cbar) + \gl(g((\bd_0\olD)\ot x\ot\cbar')\ot\cbar'') .

  1234 \end{eqnarray*}

  1235 Here $\gl$ denotes the module action in $\cY_\cC$.

  1236 This completes the definition of $\hom_\cC(\cX_\cC \to \cY_\cC)$.

  1237 

  1238 Note that if $\bd g = 0$, then each 

  1239 \[

  1240 	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})

  1241 \]

  1242 constitutes a null homotopy of

  1243 $g((\bd \olD)\ot -)$ (where the $g((\bd_0 \olD)\ot -)$ part of $g((\bd \olD)\ot -)$

  1244 should be interpreted as above).

  1245 

  1246 \nn{do we need to say anything about composing morphisms of modules?}

  1247 

  1248 \nn{should we define functors between $n$-cats in a similar way?}