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sAb: updated a picture

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Joshua Moerman 12 years ago
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  1. 8
      thesis/3_SimplicialAbelianGroups.tex
  2. BIN
      thesis/images/delta_cat.pdf
  3. 20
      thesis/images/delta_cat.svg
  4. BIN
      thesis/images/simplicial_abelian_group.pdf
  5. BIN
      thesis/images/simplicial_set.pdf
  6. 146
      thesis/images/simplicial_set.svg

8
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 $X: \DELTA^{op} \to \Set.$) (Or equivalently a covariant functor $X: \DELTA^{op} \to \Set.$)
\end{definition} \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} \begin{figure}
\includegraphics{simplicial_abelian_group} \includegraphics{simplicial_set}
\caption{A simplicial abelian group.} \caption{A simplicial set.}
\label{fig:simplicial_abelian_group} \label{fig:simplicial_set}
\end{figure} \end{figure}

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