From 2f72999923d3586cd150026135691673dfd84a73 Mon Sep 17 00:00:00 2001 From: Nathaniel Wesley Filardo Date: Tue, 28 Feb 2012 18:36:22 -0500 Subject: [PATCH] Er, post-composition! --- yoneda.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/yoneda.tex b/yoneda.tex index 273d70a..58fd5c1 100644 --- a/yoneda.tex +++ b/yoneda.tex @@ -83,7 +83,7 @@ note the similarities---the {\em representation} of the structure reachable from & \txt{$\{f', f' \circ a,$\\ $b \circ f', b \circ f' \circ a\}$} \\ } \] The two arrows from $\mbox{hom}(A,A) = \set{id_A, a}$ to $\mbox{hom}(A,B) -= \set{f, f \circ a}$ are obtained by pre-composition: += \set{f, f \circ a}$ are obtained by post-composition: \begin{align*} \mbox{hom}(A,f) &= \set{ id_A \mapsto f, a \mapsto f \circ a } \\ \mbox{hom}(A,f \circ a) &= \set{ id_a \mapsto f \circ a, a \mapsto f } @@ -91,14 +91,14 @@ The two arrows from $\mbox{hom}(A,A) = \set{id_A, a}$ to $\mbox{hom}(A,B) (the last entry holds because $(f \circ a) \circ a = f \circ (a \circ a) = f \circ id_A = f$). The four arrows from $\mbox{hom}(A,A)$ to $\mbox{hom}(A,B') = \set{f', f' \circ a, b \circ f', b \circ f' \circ a}$ are -again obtained by pre-composition: +again obtained by post-composition: \begin{align*} \mbox{hom}(A,f') &= \set{id_A \mapsto f', a \mapsto f' \circ a} \\ \mbox{hom}(A,f' \circ a) &= \set{id_a \mapsto f' \circ a, a \mapsto f'} \\ \mbox{hom}(A,b \circ f') &= \set{id_a \mapsto b \circ f', a \mapsto b \circ f' \circ a} \\ \mbox{hom}(A,b \circ f' \circ a) &= \set{id_a \mapsto b \circ f' \circ a, a \mapsto b \circ f'} \end{align*} -The two vertical arrows are (again by precomposition, and recall that $f' = g \circ f$): +The two vertical arrows are (again by post-composition, and recall that $f' = g \circ f$): \begin{align*} \mbox{hom}(A,g) &= \set{f \mapsto f', f \circ a \mapsto f' \circ a} \\ \mbox{hom}(A,b \circ g) &= \set{f \mapsto b \circ f', f \circ a \mapsto b \circ f' \circ a} @@ -149,7 +149,7 @@ define $\tau_A$ at inputs other than $id_A$ just as well!) that &= F(f)(a_0) & \text{requirement} \end{align*} So $\tau$ is fully determined by naturality and the requirement given, -precisely because $\mbox{hom}(A,-)$ on arrows captures pre-composition. +precisely because $\mbox{hom}(A,-)$ on arrows captures post-composition. So: given a choice of $a_0 \in FA$, we can fully specify a natural transformation $\tau$. -- 2.50.1