From 762787ec2883bde89d8dcb10ebb563138217e2b1 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Fri, 19 Apr 2013 15:52:18 +0200 Subject: [PATCH] Thesis: Some minor things + standard n simplex --- thesis/3_SimplicialAbelianGroups.tex | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/thesis/3_SimplicialAbelianGroups.tex b/thesis/3_SimplicialAbelianGroups.tex index 6675772..569005b 100644 --- a/thesis/3_SimplicialAbelianGroups.tex +++ b/thesis/3_SimplicialAbelianGroups.tex @@ -21,8 +21,9 @@ for each $n \in \N$. The nice things about these maps is that every map in $\DEL \end{figure} \begin{definition} - An simplicial abelian group $A$ is a contravariant functor: + An simplicial abelian group $A$ is a covariant functor: $$A: \DELTA^{op} \to \Ab.$$ + (Or equivalently a contravariant functor $A: \DELTA \to \Ab.$) \end{definition} So the category of all simplicial abelian groups, $\sAb$, is the functor category $\Ab^{\DELTA^{op}}$, 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 $A: \DELTA^{op} \to Ab$. And hence we can picture a simplicial abelian group 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 $A$ is a contravariant functor. @@ -75,9 +76,12 @@ When using a simplicial abelian group to construct another object, it is often h Of course the abstract definition of simplicial abelian group can easilty be generalized to other categories. For example $\Set^{\DELTA^{op}} = \sSet$ is the category of simplicial sets. There are very important simplicial sets: \begin{definition} - $\Delta[n]$ + The standard $n$-simplex is given by: + $$\Delta[n] = \Hom{\DELTA}{-}{[n]} : \DELTA^{op} \to \Set.$$ \end{definition} +Note that indeed $\Hom{\DELTA}{X}{[n]} \in \Set$, because the collection of morphisms in a category is per definition a set. We do not need to specify the face or degeneracy maps, as we already know that $\mathbf{Hom}$ is a functor (in both arguments). + \todo{sAb: as example do $\Delta[n]$} \todo{sAb: as example do the free abelian group pointwise}