Fix errors in and otherwise improve bifunctors.tex

diff --git a/bifunctors.tex b/bifunctors.tex
index 8ebefd1cabca07ccb0fba63f669f3bee327f1a03..491bb4c4ae91f305ac7f94c901515deffcfb31cf 100644 (file)
--- a/bifunctors.tex
+++ b/bifunctors.tex
@@ -263,7 +263,7 @@ $R^\star_C id_F = id_{F C}$.
   \node at (m) {$\circ$} ;
 
   \matrix (n) [matrix of math nodes,column sep={80pt,between origins},
-          row sep={30pt,between origins},] at (6,0) {
+          row sep={30pt,between origins}] at (6,0) {
 %
     |[name=ntl]| F C & |[name=ntr]| G C \\
 %
@@ -294,8 +294,12 @@ includes the identity natural transformations of every functor in
 $\mathcal{E}^{\mathcal{C}}$.  We are free to pick the maximal such category,
 and (abusively) suppress the $R^\star$ notation, to claim that the morphisms
 of $\mathcal{E}^{\mathcal{C}}$ {\em are} the natural transformations between
-its functors.
+its functors.  That is, rather than writing $R^\star_C \alpha$ we will now
+write $\alpha C$; some texts use the notation $\alpha_C$ to reflect the
+alternate characterization of natural transformations as object-indexed
+collections of arrows.
 
+\pagebreak
 \subsection{Composition of Nat. Trans}
 
 This has always confused me, so here's an excellent opportunity to expando
@@ -309,19 +313,85 @@ $\textrm{---} \bigcirc \textrm{---}$.  It is an object of the (visually
 intimidating) category
 $\paren{\mathcal{E}^\mathcal{C}}^{\mathcal{E}^{\mathcal{D}} \times
 \mathcal{D}^{\mathcal{C}}}$.  Adopting and extending the paper's naming
-scheme, let $G,K,Q \in \mathcal{E}^{\mathcal{D}}$; $\alpha,\alpha' \in
-\Hom{\mathcal{E}^{\mathcal{D}}}{G}{K}$; $F,H,P \in
-\mathcal{D}^{\mathcal{C}}$; $\beta,\beta' \in
-\Hom{\mathcal{D}^{\mathcal{C}}}{F}{H}$; $C,C' \in \mathcal{C}$; $f \in
-\Hom{\mathcal{C}}{C}{C'}$; $D,D' \in \mathcal{D}$; and $g \in
-\Hom{\mathcal{D}}{D}{D'}$.  First note, as the paper does, that we have that
-$G \bigcirc F \in \mathcal{E}^{\mathcal{C}}$ and $\beta \bigcirc \alpha \in
-\Hom{\mathcal{E}^{\mathcal{C}}}{G \bigcirc F}{K \bigcirc H}$. (These hold
-under exchange of similarly-quantified names, of course.)
+scheme, let
+%
+\begin{itemize}
+%
+\item $F,H,P \in \mathcal{D}^{\mathcal{C}}$; $\alpha \in
+\Hom{\mathcal{D}^{\mathcal{C}}}{F}{H}$; $\alpha' \in
+\Hom{\mathcal{D}^{\mathcal{C}}}{H}{P}$;
+%
+\item $G,K,Q \in \mathcal{E}^{\mathcal{D}}$; $\beta \in
+\Hom{\mathcal{E}^{\mathcal{D}}}{G}{K}$; $\beta' \in
+\Hom{\mathcal{E}^{\mathcal{D}}}{K}{Q}$;
+%
+\item $C,C' \in \mathcal{C}$; $f \in \Hom{\mathcal{C}}{C}{C'}$; $D,D' \in
+\mathcal{D}$; and $g \in \Hom{\mathcal{D}}{D}{D'}$.
+%
+\end{itemize}
+%
+The behavior of $\bigcirc$ is given as follows:
+%
+\begin{itemize}
+%
+  \item $(G \bigcirc F) C = G \star (FC) = GFC \in \mathcal{E}^{\mathcal{C}}$ (i.e. functor composition)
+%
+  \item $(G \bigcirc \alpha) C = (L^\bigcirc_G \alpha) C = G (R^\star_C \alpha) = G (\alpha C) \in \mathcal{E}$
+%
+  \item $(\beta \bigcirc F) C = (R^\bigcirc_F \beta) C = R^\star_{L^\star_F C} \beta = \beta (L^\star_F C) = \beta (F C) \in \mathcal{E}$
+%
+\end{itemize}
+
+The paper asserts that ``the coherence conditions follow from naturality'', i.e. that $\forall_C$
+%
+\[\begin{tikzpicture}
+%
+  \matrix (m) [matrix of math nodes,column sep={80pt,between origins},
+          row sep={30pt,between origins}] {
+%
+    |[name=tl]| (G \bigcirc F)C & |[name=tr]| (K \bigcirc F)C \\
+%
+    |[name=bl]| (G \bigcirc H)C & |[name=br]| (K \bigcirc H)C \\
+%
+  } ;
+%
+  \draw [->] (tl) -- (tr) node [above,midway] {$R^\bigcirc_F \beta$} ;
+  \draw [->] (tl) -- (bl) node [left,midway]  {$L^\bigcirc_G \alpha$} ;
+  \draw [->] (bl) -- (br) node [below,midway] {$R^\bigcirc_H \beta$} ;
+  \draw [->] (tr) -- (br) node [right,midway] {$L^\bigcirc_K \alpha$} ;
+%
+
+  \node at (m) {$\circ$} ;
+
+  \matrix (n) [matrix of math nodes,column sep={80pt,between origins},
+          row sep={30pt,between origins}] at (6,0) {
+%
+    |[name=ntl]| GFC & |[name=ntr]| KFC \\
+%
+    |[name=nbl]| GHC & |[name=nbr]| KHC \\
+%
+  } ;
+
+  \node at (n) {$\circ$} ;
+
+  \draw [->] (ntl) -- (ntr) node [above,midway] {$\beta(FC)$} ;
+  \draw [->] (ntl) -- (nbl) node [left,midway]  {$G(\alpha C)$} ;
+  \draw [->] (nbl) -- (nbr) node [below,midway] {$\beta(HC)$} ;
+  \draw [->] (ntr) -- (nbr) node [right,midway] {$K(\alpha C)$} ;
+
+  \path (m) -- (n) node [midway] {$\equiv$} ;
+%
+\end{tikzpicture}\]
+%
+This indeed follows from the naturality of $\beta$ (not $\alpha$!).  So $\beta
+\bigcirc \alpha \in \Hom{\mathcal{E}^{\mathcal{C}}}{G \bigcirc F}{K \bigcirc
+H}$ is well-defined.
 
 If we just write down everything we know (a popular technique for earning
 sympathy on exams), we first get these two ``vertical composition''
-diagrams (in $\mathcal{D}$ and $\mathcal{E}$, respectively):
+diagrams (in $\mathcal{D}$ and $\mathcal{E}$, respectively; the use of
+$\circ$ takes place in $\mathcal{D}^\mathcal{C}$ on the left and
+$\mathcal{E}^\mathcal{D}$ on the right):
 \begin{equation}
 \begin{tikzpicture}
   \matrix[matrix of math nodes,column sep={60pt,between origins},
@@ -374,7 +444,8 @@ diagrams (in $\mathcal{D}$ and $\mathcal{E}$, respectively):
 %
 \end{equation}
 %
-We also get this ``horizontal composition'' diagram in $\mathcal{E}$:
+We also get this ``horizontal composition'' diagram in $\mathcal{E}$; on the
+left is the diagram using $\bigcirc$ and on the right is a version with all $\bigcirc$ evaluated.
 %
 \begin{equation}
 \begin{tikzpicture}
@@ -398,17 +469,46 @@ We also get this ``horizontal composition'' diagram in $\mathcal{E}$:
   \draw [->] (mbl) -- (mbm) node [below,midway] {$(\beta \bigcirc \alpha) C'$} ;
   \draw [->] (mbm) -- (mbr) node [below,midway] {$(\beta' \bigcirc \alpha') C'$} ;
 %
-  \draw [->] (mtl) -- (mbl) node [left,midway]  {$(G \bigcirc F) f$} ;
-  \draw [->] (mtm) -- (mbm) node [left,midway]  {$(K \bigcirc H) f$} ;
-  \draw [->] (mtr) -- (mbr) node [right,midway] {$(Q \bigcirc P) f$} ;
+  \draw [->] (mtl) -- (mbl) node [left,midway] {$(G \bigcirc F) f$} ;
+  \draw [->] (mtm) -- (mbm) node [left,midway] {$(K \bigcirc H) f$} ;
+  \draw [->] (mtr) -- (mbr) node [left,midway] {$(Q \bigcirc P) f$} ;
 %
   \draw [->] (mtl) to[bend left] node [above] {$((\beta' \circ \beta) \bigcirc (\alpha' \circ \alpha)) C$} (mtr) ;
 
   \draw [->] (mbl) to[bend right] node [below] {$((\beta' \circ \beta) \bigcirc (\alpha' \circ \alpha)) C'$} (mbr) ;
 
+
+  \matrix[matrix of math nodes,column sep={100pt,between origins},
+          row sep={40pt,between origins}] at (9,0) {
+%
+       |[name=mtl]| GFC 
+  & |[name=mtm]| KHC 
+  & |[name=mtr]| QPC \\
+%
+    |[name=mbl]| GFC'
+  & |[name=mbm]| KHC'
+  & |[name=mbr]| QPC' \\
+%
+  } ;
+
+  \draw [->] (mtl) -- (mtm) node [above,midway] {$K (\alpha C) \circ \beta(FC)$} ;
+  \draw [->] (mtm) -- (mtr) node [above,midway] {$Q (\alpha C') \circ \beta (KC)$} ;
+%
+  \draw [->] (mbl) -- (mbm) node [below,midway] {$K (\alpha C') \circ \beta (FC')$} ;
+  \draw [->] (mbm) -- (mbr) node [below,midway] {$Q (\alpha' C') \circ \beta' (KC')$} ;
+%
+  \draw [->] (mtl) -- (mbl) node [left,midway] {$GFf$} ;
+  \draw [->] (mtm) -- (mbm) node [left,midway] {$KHf$} ;
+  \draw [->] (mtr) -- (mbr) node [left,midway] {$QPf$} ;
+
+  \draw [->] (mtl) to[bend left] node [above] {$Q((\alpha' \circ \alpha) C) \circ (\beta' \circ \beta)(F C)$} (mtr) ;
+
+  \draw [->] (mbl) to[bend right] node [below] {$Q((\alpha' \circ \alpha) C') \circ (\beta' \circ \beta)(F C')$} (mbr) ;
+
 \end{tikzpicture}
 %
 \end{equation}
+%
 The rectangles commute by definition of natural transformations while the
 upper and lower faces commute by bifunctorality of $\bigcirc$ (namely, that it
 preserves composition).