From 35e2abd1fa34c2b7f8e91281886c5cf1fd452c82 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Wed, 29 May 2013 13:39:29 +0200 Subject: [PATCH] sAb: updated a picture --- thesis/3_SimplicialAbelianGroups.tex | 8 +- thesis/images/delta_cat.pdf | Bin 6152 -> 5510 bytes thesis/images/delta_cat.svg | 20 +-- thesis/images/simplicial_abelian_group.pdf | Bin 7219 -> 0 bytes thesis/images/simplicial_set.pdf | Bin 0 -> 7657 bytes ...l_abelian_group.svg => simplicial_set.svg} | 162 ++++++++++-------- 6 files changed, 101 insertions(+), 89 deletions(-) delete mode 100644 thesis/images/simplicial_abelian_group.pdf create mode 100644 thesis/images/simplicial_set.pdf rename thesis/images/{simplicial_abelian_group.svg => simplicial_set.svg} (89%) diff --git a/thesis/3_SimplicialAbelianGroups.tex b/thesis/3_SimplicialAbelianGroups.tex index c092aad..24bc90e 100644 --- a/thesis/3_SimplicialAbelianGroups.tex +++ b/thesis/3_SimplicialAbelianGroups.tex @@ -39,12 +39,12 @@ This category $\DELTA$ will act as a protoype for these kind of geometric struct (Or equivalently a covariant functor $X: \DELTA^{op} \to \Set.$) \end{definition} -So the category of all simplicial sets, $\sSet$, is the functor category $\Set^{\DELTA^{op}}$, where morphisms are natural transformations. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $X: \DELTA^{op} \to \Set$. And hence we can picture a simplicial set as done in figure~\ref{fig:simplicial_abelian_group}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $X$ is a contravariant functor. +So the category of all simplicial sets, $\sSet$, is the functor category $\Set^{\DELTA^{op}}$, where morphisms are natural transformations. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $X: \DELTA^{op} \to \Set$. And hence we can picture a simplicial set as done in figure~\ref{fig:simplicial_set}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $X$ is a contravariant functor. \begin{figure} - \includegraphics{simplicial_abelian_group} - \caption{A simplicial abelian group.} - \label{fig:simplicial_abelian_group} + \includegraphics{simplicial_set} + \caption{A simplicial set.} + \label{fig:simplicial_set} \end{figure} diff --git a/thesis/images/delta_cat.pdf b/thesis/images/delta_cat.pdf index e3635856740feae3527d2c1d029974b90dbadd57..41f07f9ffd4e571e081542a5938740f08d53afe9 100644 GIT binary patch delta 4786 zcmZXRbyO1o*T$6=DUt3_1j!9XNhw{@?LZhKMmLBsLTQkW(bC-^jly&^2!l}rL=Z;9 zP}0};`}2G6Ilpu7dG2|hbI-XkVo`$0x^M)zUPz$+Jr+VI6VWrMNb!TJIAiAEjWph~ zds`GN8jNY=)qXsOiRU7NC=6s@_v1p>`~O~F;T8XO+5Ejd%)mn1%u?a7H#axrGRQYK zL1h6Vrvg4fdR5oa84b}B`23%RiVauthiB>6GllaulX`z&orY|iSsu<-JDFUB!k=7h z1~y1q%k+MiSCI%x#s)lX_)ys`u|R~uI{UWq#}DqWID{I!wZ_Yv`B-2Bw4I=+rJpNk zl~bBRMwkZgNn5OAmE?j!{C&h9F99?PH~bc~D!}*vyr*%KA3UvB(PLfJ8}C~Jb5vQj 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