From 9567805b5d8680940974dbfab12a35ab5ffa10d3 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Fri, 10 May 2013 21:18:23 +0200 Subject: [PATCH] Thesis: added some todos --- thesis/1_CategoryTheory.tex | 1 + thesis/2_ChainComplexes.tex | 2 +- thesis/5_Homotopy.tex | 2 ++ thesis/DoldKan.tex | 3 +++ 4 files changed, 7 insertions(+), 1 deletion(-) diff --git a/thesis/1_CategoryTheory.tex b/thesis/1_CategoryTheory.tex index b46467b..e58ba96 100644 --- a/thesis/1_CategoryTheory.tex +++ b/thesis/1_CategoryTheory.tex @@ -65,3 +65,4 @@ For example the cyclic group $\Z_4$ and the klein four-group $V_4$ are not isomo \todo{CT: Equivalence / natro} \todo{CT: Adjunction} +\todo{CT: Yoneda?} \ No newline at end of file diff --git a/thesis/2_ChainComplexes.tex b/thesis/2_ChainComplexes.tex index 06e1720..ea42663 100644 --- a/thesis/2_ChainComplexes.tex +++ b/thesis/2_ChainComplexes.tex @@ -7,7 +7,7 @@ In other words a chain complex is the following diagram. $$ \cdots \to C_4 \to C_3 \to C_2 \to C_1 \to C_0 $$ -Of course we can make this more general by taking for example $R$-modules instead of abelian groups. We will later see which kind of algebraic objects make sense to use in this definition. The boundary operators give rise to certain subgroups, because all groups are abelian, subgroups are normal subgroups. +Of course we can make this more general by taking for example $R$-modules instead of abelian groups. We will later see which kind of algebraic objects make sense to use in this definition \todo{Ch: Will I discuss ab. cat. ?}. The boundary operators give rise to certain subgroups, because all groups are abelian, subgroups are normal subgroups. \begin{definition} Given a chain complex $C$ we define the following subgroups: diff --git a/thesis/5_Homotopy.tex b/thesis/5_Homotopy.tex index 21fd403..0d5b397 100644 --- a/thesis/5_Homotopy.tex +++ b/thesis/5_Homotopy.tex @@ -5,6 +5,8 @@ We've already seen homology in chain complexes. We can of course now translate t When dealing with homotopy in a topological space $X$ we always need a base-point $\ast \in X$. This is also the case for homotopy in simplicial sets. We will notate the chosen base-point of a simplicial set $X$ with $\ast \in X_0$. Note that it is a $0$-simplex, but in fact the base-point is present in all sets $X_n$, because we can consider its degenerate simplices $s_0(\ldots(s_0(\ast))\ldots) \in X_n$, we will also denote these elements as $\ast$. Of course in our situation we are concerned about simplicial abelien groups, where there is an obvious choice for the base-point, namely $0$. +\todo{Htp: Do I want to define homotopy between maps?} + \begin{definition} Given a simplicial set $X$ with base-point $\ast$, we define $Z_n(X)$ to be the set of $n$-simplices with the base-point as boundary, i.e.: $$ Z_n(X) = \{ x \in X_n | d_i(x) = \ast \text{ for all } i < n \}. $$ diff --git a/thesis/DoldKan.tex b/thesis/DoldKan.tex index 0aeeffa..213b61f 100644 --- a/thesis/DoldKan.tex +++ b/thesis/DoldKan.tex @@ -52,6 +52,9 @@ In the first section some definitions from category theory are given, because we \newpage \input{../thesis/5_Homotopy} +\newpage +\todo{References: Lamotke, Friedman, Weibel} + \newpage \listoftodos % \nocite{*}