From f050a2e9771737047964ed6c4c3caf68c76b7326 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Sun, 28 Apr 2013 18:28:04 +0200 Subject: [PATCH] Thesis: added example D[0] and D[1] --- thesis/3_SimplicialAbelianGroups.tex | 60 ++++++++++++++++++++++++++-- thesis/preamble.tex | 6 +++ 2 files changed, 62 insertions(+), 4 deletions(-) diff --git a/thesis/3_SimplicialAbelianGroups.tex b/thesis/3_SimplicialAbelianGroups.tex index 8e8c560..dc9ee89 100644 --- a/thesis/3_SimplicialAbelianGroups.tex +++ b/thesis/3_SimplicialAbelianGroups.tex @@ -91,8 +91,60 @@ Of course the abstract definition of simplicial abelian group can easilty be gen $$\Delta[n] = \Hom{\DELTA}{-}{[n]} : \DELTA^{op} \to \Set.$$ \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} +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). Still it is useful to write out some cases. + +\begin{example} + We will compute how $\Delta[0]$ look like. Note that $[0]$ is an one-element set, so for any set $X$, there is only one function $\ast : X \to [0]$. Hence $\Delta[0]_n = \{\ast\}$ for all $n$. The face and degeneracy maps are now functions from $\{\ast\}$ to $\{\ast\}$. Again there is only one, namely $\id : \{\ast\} \to \{\ast\}$. This gives: + \todo{sAb: insert picture} +\end{example} + +\begin{example} + $\Delta[1]$ is a bit more interesting, but still not too hard. We will compute the first three abelian groups $\Delta[1]_0$, $\Delta[1]_1$ and $\Delta[1]_2$. We can use the fact that any monotone increasing map $f: [n] \to [m]$ is a composition of first applying degeneracy maps, and then face maps, ie.: $f: [n] \tot{\sigma^{i_0} \cdots \sigma^{i_M}} [k] \tot{\delta^{j_0} \cdots \delta^{j_N}} [m]$, where $k \leq m, n$. + + For $\Delta[1]_0$ we have to consider maps from $[0]$ to $[1]$, we cannot first apply degeneracy maps (there is no object $[-1]$). So this leaves us with the face maps: $\Delta[1]_0 = \{\delta_0, \delta_1\}$. For $\Delta[1]_1$ we of course have the identity function and two functions $\delta_0\sigma_0, \delta_1\sigma_0$. Now $\Delta[1]_2$ are the maps from $[2]$ to $[1]$. + + We will compute the two face maps $\delta^0$ and $\delta^1$ from $\Delta[1]_1$ to $\Delta[1]_0$. Recall that the $\mathbf{Hom}$-functor in the first argument (the contravariant argument) works with precomposition. So this gives: + \begin{align*} + \delta^0(id) &= \id \delta_0 = \delta_0 \\ + \delta^0(\delta_0\sigma_0) &= \delta_0 \sigma_0 \delta_0 = \delta_0 \\ + \delta^0(\delta_1\sigma_0) &= \delta_0 \sigma_0 \delta_0 = \delta_1. + \end{align*} + Where we in the first calculation used the identity law. In the second and third line we used the third simplicial equation, asserting that $\sigma_0 \delta_0 = \id$. Similarly we can calculate the face map $\delta^1$: + \begin{align*} + \delta^1(id) &= \id \delta_1 = \delta_1 \\ + \delta^1(\delta_0\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_0 \\ + \delta^1(\delta_1\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_1. + \end{align*} +\end{example} + +As we are interested in simplicial abelian group, it would be nice to make these $n$-simplices into simplicial abelian groups. We have seen how to make an abelian group out of any set using the free abelian group. We can use this functor $\Z[-] : \Set \to \Ab$ to induce a functor $\Z^\ast[-] : \sSet \to \sAb$ as shown in the diagram~\ref{fig:diagram_Z}. +\begin{figure} + \begin{tikzpicture} + \matrix (m) [matrix of math nodes]{ + \DELTA^{op} & \Set \\ + & \Ab \\ + }; + \path[->] + (m-1-1) edge node[auto] {$ X $} (m-1-2) + (m-1-2) edge node[auto] {$ \Z[-] $} (m-2-2) + (m-1-1) edge node[auto] {$ X' $} (m-2-2); + \end{tikzpicture} + \caption{The simplicial set $X$ can be made into a simplicial abelian group $X'$ by postcomposing with $\Z[-]$} + \label{fig:diagram_Z} +\end{figure} +\begin{example} + We can apply this to the standard $n$-simplex $\Delta[1]$. This gives $\Delta[1]_0 \iso \Z^2$, since $\Delta[1]_0$ had two elements, and $\Delta[1]_1 \iso \Z^3$, where the isomorphisms are taken such that: + \begin{align*} + \delta_0 &\mapstot{\iso} (1, 0) \\ + \delta_1 &\mapstot{\iso} (0, 1) \\ + \delta_0\sigma_0 &\mapstot{\iso} (1, 0, 0) \\ + \id &\mapstot{\iso} (0, 1, 0) \\ + \delta_1\sigma_0 &\mapstot{\iso} (0, 0, 1) + \end{align*} + The face maps from $\Delta[1]_1$ to $\Delta[1]_0$ under these isomorphisms are then given by: + \begin{align*} + \delta^0(x, y, z) &= (x+y, z) \\ + \delta^1(x, y, z) &= (x, y+z) + \end{align*} +\end{example} diff --git a/thesis/preamble.tex b/thesis/preamble.tex index eb3d74b..407ffa0 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -3,6 +3,11 @@ \usepackage{amssymb} \usepackage{color} \usepackage{listings} +\usepackage{mathtools} + +\usepackage{tikz} % http://pdp7.org/blog/?p=133 +\usetikzlibrary{matrix,arrows} +\tikzset{node distance=3em, row sep=3em, column sep=3em, auto} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} @@ -20,6 +25,7 @@ \newcommand{\iso}{\cong} \newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} +\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} \newcommand{\eps}{\varepsilon} \newcommand{\I}{\,\mid\,} \newcommand{\then}{\Rightarrow}