From: Nathaniel Wesley Filardo Date: Fri, 1 Apr 2016 18:04:34 +0000 (-0400) Subject: Fix errors in and otherwise improve bifunctors.tex X-Git-Url: https://hydra-www.ietfng.org/gitweb/?a=commitdiff_plain;p=ctcheat Fix errors in and otherwise improve bifunctors.tex --- diff --git a/bifunctors.tex b/bifunctors.tex index 8ebefd1..491bb4c 100644 --- 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).