From e145e3c91efc5ffc3548379e838d5228be1e3332 Mon Sep 17 00:00:00 2001 From: Nathaniel Wesley Filardo Date: Tue, 1 May 2012 14:54:06 -0400 Subject: [PATCH] Tweak ctsheat-joy --- ctcheat-joy.tex | 19 ++++++++++++------- 1 file changed, 12 insertions(+), 7 deletions(-) diff --git a/ctcheat-joy.tex b/ctcheat-joy.tex index 32ed9b3..e9d66e7 100644 --- a/ctcheat-joy.tex +++ b/ctcheat-joy.tex @@ -245,6 +245,10 @@ Entries within each section are roughly sorted by definition, alphabetically. $A \in \mathbf{A}$ to $\tau_A : FA \to GA$ s.t. $\forall_{f : A \to A'} . G f \circ \tau_A = \tau_{A'} \circ F f$. (\S6.1) + There is special notation for functors applied to natural transformations + and vice-versa (\S6.2): $(F\tau)_A = F(\tau_A)$ and $(\tau F)_A = + \tau_{FA}$. + All functors \defn{preserve} (in $\mathbf{A}$ implies in $\mathbf{B}$)$\dots$ \begin{itemize} \item isomorphisms. (\S3.21) @@ -307,14 +311,15 @@ Entries within each section are roughly sorted by definition, alphabetically. $(\eta,\epsilon) : F \dashv G : \mathbf{A} \to \mathbf{B}$ is a \defn{adjoint situation} if the above relationships hold. (\S19.7) - A \defn{monad} on $\mathbf{X}$ is $(T : \mathbf{X} \to \mathbf{X}, - \eta : id_{\mathbf{X}} \natto, \mu : T^2 \natto T)$ s.t. - \[\xymatrix@R=10pt{ - T^3 \ar[r]^{T\mu} \ar[d]^{\mu T} & T^2 \ar[d]^\mu \\ - T^2 \ar[r]^{\mu} & T + A \defn{monad} (\S20.1) on $\mathbf{X}$ is $(T : \mathbf{X} \to \mathbf{X}, + \eta : id_{\mathbf{X}} \natto T, \mu : T^2 \natto T)$ s.t. + \[\forall_X \quad + \xymatrix@R=10pt{ + T^3X \ar[r]^{T(\mu_X)} \ar[d]^{\mu_{TX}} & T^2X \ar[d]^{\mu_{X}} \\ + T^2X \ar[r]^{\mu_X} & TX } \quad \xymatrix@R=10pt{ - T \ar[r]^{T\eta} \ar[dr]_{id} & T^2 \ar[d]^\mu & T \ar[l]_{\eta T} \ar[dl]^{id} \\ - & T & + TX \ar[r]^{T(\eta_X)} \ar[dr]_{id_{TX}} & T^2X \ar[d]^{\mu_X} & TX \ar[l]_{\eta_{TX}} \ar[dl]^{id_{TX}} \\ + & TX & }\] \end{document} -- 2.50.1