From c52defc4588a70f60e76de43b3ff6009d3760bcd Mon Sep 17 00:00:00 2001 From: Andreas <andschu@uzh.ch> Date: Tue, 24 Nov 2020 11:04:01 +0100 Subject: [PATCH] Andreas --- Exercises/Exercises.log | 4 ++-- Exercises/Exercises.pdf | Bin 340314 -> 340311 bytes Exercises/Exercises.synctex.gz | Bin 517615 -> 517546 bytes Exercises/Exercises.tex | 4 ++-- 4 files changed, 4 insertions(+), 4 deletions(-) diff --git a/Exercises/Exercises.log b/Exercises/Exercises.log index bab4a62..fbd120b 100644 --- a/Exercises/Exercises.log +++ b/Exercises/Exercises.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 20.11) (preloaded format=pdflatex 2020.11.9) 23 NOV 2020 15:35 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 20.11) (preloaded format=pdflatex 2020.11.9) 24 NOV 2020 11:03 entering extended mode **./Exercises.tex (Exercises.tex @@ -1643,7 +1643,7 @@ gram Files/MiKTeX/fonts/type1/public/txfonts/txsya.pfb><C:/Program Files/MiKTeX ic/txfonts/txsyc.pfb><C:/Program Files/MiKTeX/fonts/type1/urw/times/utmb8a.pfb> <C:/Program Files/MiKTeX/fonts/type1/urw/times/utmr8a.pfb><C:/Program Files/MiK TeX/fonts/type1/urw/times/utmri8a.pfb> -Output written on Exercises.pdf (46 pages, 340314 bytes). +Output written on Exercises.pdf (46 pages, 340311 bytes). 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100644 --- a/Exercises/Exercises.tex +++ b/Exercises/Exercises.tex @@ -2107,7 +2107,7 @@ U_X:P^2(X)&\longrightarrow P(X)\\ \end{itemize} Determine the Elienberg-Moore category of algebras for this monad. (\emph{Hint}: consider complete lattices) \hfill \break -\iam{Andreas} First let's check if $s_X,U_X$ are natural transformation, meaning the diagrams +\iam{Andreas} First let's check if $s_X,U_X$ are natural transformations, meaning the diagrams \begin{center} \begin{tikzcd} @@ -2141,7 +2141,7 @@ Indeed we have & \mathcal{P}(X) & \end{tikzcd} \end{center} -Then $$U_X(\mathcal{P}(U_X(\alpha))=U_X(\mathcal{P}(\bigcup \alpha)=\bigcup\mathcal{P}(\bigcup \alpha)=\bigcup\mathcal{P}(U_X(\alpha))=(U_X)_{\mathcal{P}}(U_X(\alpha))$$ +Then $$U_X(\mathcal{P}(U_X(\alpha))=U_X(\mathcal{P}(\bigcup \alpha)=\bigcup\mathcal{P}(U_X(\alpha))=U_X \circ (U_X)_{\mathcal{P}}(\alpha)$$ and $$U_X(\mathcal{P}(s_X(x))=U_X(\mathcal{P}(\{x\})=\bigcup\mathcal{P}(\{x\})=\bigcup s_X(\mathcal{P}(x))=U_X(s_X\mathcal{P}(x))$$ \subsection{} Show that $(D,\mu,\eta)$ is a monad over $\mathbf{Vec}_k$, the category of vector fields over the field $k$, where -- GitLab