# HG changeset patch # User Kevin Walker # Date 1279837018 21600 # Node ID 4d0ca2fc4f2b48f31977858cecebe5bb5ddce761 # Parent 07c18e2abd8fafdd393606a08c1135ba0414bb4a dual module (non-)definition; other minor stuff diff -r 07c18e2abd8f -r 4d0ca2fc4f2b text/a_inf_blob.tex --- a/text/a_inf_blob.tex Thu Jul 22 15:35:26 2010 -0600 +++ b/text/a_inf_blob.tex Thu Jul 22 16:16:58 2010 -0600 @@ -69,7 +69,8 @@ Let $G_*\sub \bc_*(Y\times F;C)$ be the subcomplex generated by blob diagrams $a$ such that there exists a decomposition $K$ of $Y$ such that $a$ splits along $K\times F$. -It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ is homotopic to a subcomplex of $G_*$. +It follows from Proposition \ref{thm:small-blobs} that $\bc_*(Y\times F; C)$ +is homotopic to a subcomplex of $G_*$. (If the blobs of $a$ are small with respect to a sufficiently fine cover then their projections to $Y$ are contained in some disjoint union of balls.) Note that the image of $\psi$ is equal to $G_*$. @@ -95,7 +96,8 @@ \end{lemma} \begin{proof} -We will prove acyclicity in the first couple of degrees, and \nn{in this draft, at least} +We will prove acyclicity in the first couple of degrees, and +%\nn{in this draft, at least} leave the general case to the reader. Let $K$ and $K'$ be two decompositions (0-simplices) of $Y$ compatible with $a$. diff -r 07c18e2abd8f -r 4d0ca2fc4f2b text/ncat.tex --- a/text/ncat.tex Thu Jul 22 15:35:26 2010 -0600 +++ b/text/ncat.tex Thu Jul 22 16:16:58 2010 -0600 @@ -1538,7 +1538,7 @@ \[ (X_B\ot {_BY})^* \cong \hom_B(X_B \to (_BY)^*) . \] -We will establish the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ +We would like to have the analogous isomorphism for a topological $A_\infty$ 1-cat $\cC$ and modules $\cM_\cC$ and $_\cC\cN$, \[ (\cM_\cC\ot {_\cC\cN})^* \cong \hom_\cC(\cM_\cC \to (_\cC\cN)^*) . @@ -1547,7 +1547,8 @@ In the next few paragraphs we define the objects appearing in the above equation: $\cM_\cC\ot {_\cC\cN}$, $(\cM_\cC\ot {_\cC\cN})^*$, $(_\cC\cN)^*$ and finally $\hom_\cC$. - +(Actually, we give only an incomplete definition of $(_\cC\cN)^*$, but since we are only trying to motivate the +definition of $\hom_\cC$, this will suffice for our purposes.) \def\olD{{\overline D}} \def\cbar{{\bar c}} @@ -1597,7 +1598,7 @@ & \qquad + (-1)^{l + \deg m + \deg \cbar} f(\olD\ot m\ot\cbar\ot \bd n). \notag \end{align} -Next we define the dual module $(_\cC\cN)^*$. +Next we partially define the dual module $(_\cC\cN)^*$. This will depend on a choice of interval $J$, just as the tensor product did. Recall that $_\cC\cN$ is, among other things, a functor from right-marked intervals to chain complexes. @@ -1607,9 +1608,17 @@ \] where $({_\cC\cN}(J\setmin K))^*$ denotes the (linear) dual of the chain complex associated to the right-marked interval $J\setmin K$. -This extends to a functor from all left-marked intervals (not just those contained in $J$). -\nn{need to say more here; not obvious how homeomorphisms act} -It's easy to verify the remaining module axioms. +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 $\cM_\cC \to (_\cC\cN)^*$. @@ -1772,10 +1781,10 @@ \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.} +%\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.}