Bachelor thesis about the Dold-Kan correspondence
https://github.com/Jaxan/Dold-Kan
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47 lines
2.4 KiB
47 lines
2.4 KiB
11 years ago
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\section{Simplicial Abelian Groups}
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\label{sec:Simplicial Abelian Groups}
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\begin{definition}
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We define a category $\DELTA$, where the objects are the finite ordinals $[n] = \{0, \dots, n\}$ and maps are monotone increasing functions.
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\end{definition}
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There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \emph{degeneracy} maps, defined as (resp.):
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$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
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$$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
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for each $n \in \N$. The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of these maps. So in a certain sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as follows.
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\begin{figure}[h!]
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\label{fig:delta_cat}
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\includegraphics{delta_cat}
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\caption{The category $\DELTA$ with the face and degeneracy maps.}
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\end{figure}
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\todo{sAb: Epi-mono factorization}
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Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. 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 $F: \DELTA^{op} \to Ab$. And hence we can picture a simplicial abelian group as follows.
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\begin{figure}
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\label{fig:simplicial_abelian_group}
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\includegraphics{simplicial_abelian_group}
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\caption{A simplicial abelian group.}
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\end{figure}
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Of course the maps $\delta_i$ and $\sigma_i$ satisfy certain equations, these are the so called \emph{simplicial equations}.
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\todo{sAb: Is \emph{simplicial equations} really a thing?}
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\begin{lemma}
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The face and degeneracy maps in $\DELTA$ satisfy the simplicial equations, ie.:
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\begin{align}
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\delta_j\delta_i &= \delta_i\delta_{j-1} \hspace{0.5cm} \text{ if } i < j,\\
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\sigma_j\delta_i &= \delta_i\sigma_{j-1} \hspace{0.5cm} \text{ if } i < j,\\
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\sigma_j\delta_j &= \sigma_j\delta_{j+1} = \text{id},\\
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\sigma_j\delta_i &= \delta_{i-1}\sigma_j \hspace{0.5cm} \text{ if } i > j+1,\\
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\sigma_j\sigma_i &= \sigma_i\sigma_{j+1} \hspace{0.5cm} \text{ if } i \leq j.
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\end{align}
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\end{lemma}
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\begin{proof}
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By writing out the definitions given above.
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\end{proof}
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\todo{sAb: Say a bit more (because Mueger will not like this)}
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