Adds differential on tensor. Koszul sign conv. Adds todos.

thesis/1_Algebra.tex

@@ -29,9 +29,14 @@ Recall that the tensor product of modules distributes over direct sums. This def
 \begin{definition}
 	The graded tensor product is defined as:
 	$$ (M \tensor N)_n = \bigoplus_{i + j = n} M_i \tensor N_j. $$
+	The tensor product extends to graded maps. Let $f: A \to B$ and $g:X \to Y$ be two graded maps, then their tensor product $f \tensor g: A \tensor B \to X \tensor Y$ is defined as:
+	$$ (f \tensor g)(a \tensor x) = (-1)^{\deg{a}\deg{g}} \cdot f(a) \tensor g(x). $$
 \end{definition}
 
-The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. From now on we will simply refer to maps instead of graded maps. Together with the tensor product and the ground ring, $(\grMod{\k}, \tensor, \k)$ is a monoidal category. This now dictates the definition of a graded algebra.
+The sign is due to \emph{Koszuls sign convention}: whenever two elements next to each other are swapped (in this case $g$ and $a$) a minus sign appears if both elements are of odd degree. More formally we can define a swap map
+$$ \tau : A \tensor B \to B \tensor A : a \tensor b \mapsto (-1)^{\deg{a}\deg{b}} b \tensor a. $$
+
+The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. From now on we will simply refer to maps instead of graded maps. Together with the tensor product and the ground ring, $(\grMod{\k}, \tensor, \k)$ is a symmetric monoidal category (with the symmetry given by $\tau$). This now dictates the definition of a graded algebra.
 
 \begin{definition}
 	A \emph{graded algebra} consists of a graded module $A$ together with two maps of degree $0$:
@@ -48,8 +53,6 @@ Again these objects and maps form a category, denoted as $\grAlg{\k}$. We will d
 	$$ xy = (-1)^{\deg{x}\deg{y}} yx. $$
 \end{definition}
 
-\todo{Add a remark about the signs somewhere}
-
 
 \subsection{Differential graded algebra}
 
@@ -59,15 +62,22 @@ Again these objects and maps form a category, denoted as $\grAlg{\k}$. We will d
 
 A differential graded module $(M, d)$ with $M_i = 0$ for all $i < 0$ is a \emph{chain complex}. A differential graded module $(M, d)$ with $M_i = 0$ for all $i > 0$ is a \emph{cochain complex}. It will be convenient to define $M^i = M_{-i}$ in the latter case, so that $M = \bigoplus_{n \in \N} M^i$ and $d$ is a map of \emph{upper degree} $+1$.
 
-The tensor product of two differential graded modules is again a differential graded module if we define the differential as follows. \todo{Define this}
+\begin{definition}
+	Let $(M, d_M)$ and $(N, d_N)$ be two differential graded modules, their tensor product $M \tensor N$ is a differential graded module with the differential given by:
+	$$ d_{M \tensor N} = d_M \tensor \id_N + \id_M \tensor d_N. $$
+\end{definition}
+
+\todo{Prove that this is in fact a differential?}
 
 Finally we come to the definition of a differential graded algebra. This will be a graded algebra with a differential. Of course we want this to be compatible with the algebra structure, or stated differently: we want $\mu$ and $\eta$ to be chain maps.
 
 \begin{definition}
-	A \emph{differential graded algebra (DGA)} is a graded algebra $A$ together with an differential $d$ such that in addition:
+	A \emph{differential graded algebra (DGA)} is a graded algebra $A$ together with an differential $d$ such that in addition the \emph{Leibniz rule} holds:
 	$$ d(xy) = d(x) y + (-1)^{\deg{x}} x d(y) \quad\text{ for all } x, y \in A. $$
 \end{definition}
 
+\todo{Define the notion of derivation?}
+
 It is not hard to see that this definition precisely defines the monoidal objects in the category of differential graded modules. The category of DGAs will be denoted by $\DGA_\k$, the category of commutative DGAs (CDGAs) will be denoted by $\CDGA_\k$. If no confusion can arise, the ground ring $\k$ will be suppressed in this notation.
 
 Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i = 0$ for $i < 0$. Similarly if $M^i = 0$ for all $i < 0$, then $M$ is a \emph{cochain algebra}.
@@ -94,6 +104,13 @@ For differential graded algebras we can consider the (co)homology by forgetting
 	is a graded algebra.
 \end{lemma}
 \begin{proof}
-	\todo{}
+	\todo{Maybe just state this?}
 \end{proof}
 
+\TODO{Discuss:
+\titem The Künneth theorem (especially in the case of fields)
+\titem The tensor algebra $T : Ch^\ast(\Q) \to \DGA_\Q$ and free cdga $\Lambda : Ch^\ast(\Q) \to \CDGA_\Q$
+\titem Coalgebras and Hopfalgebras?
+\titem Define reduced/connected differential graded things
+\titem Singular (co)homology as a quick example?
+}

thesis/2_Model_Cats.tex

@@ -86,18 +86,16 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival
 \end{tikzpicture}
 \end{center}
 
-\todo{Maybe some basic propositions:
-> Over/under category (or simply pointed objects)   \\
-> If a map has LLP/RLP wrt fib/cof, it is a cof/fib \\
-> Fibs are preserved under pullbacks/limits         \\
+\TODO{Maybe some basic propositions (refer to Dwyer \& Spalinski):
+\titem Over/under category (or simply pointed objects)
+\titem If a map has LLP/RLP wrt fib/cof, it is a cof/fib
+\titem Fibs are preserved under pullbacks/limits
+\titem Cofibrantly generated mod. cats.
+\titem Small object argument
 }
 
 \todo{Define homotopy category}
 
-\todo{Cofibrantly generated mod cats?}
-
-\todo{Small obj. argument?}
-
 \subsection{Quillen pairs}
-In order to relate model categories and their associated homotopy categories we need a notion of maps between them.
+In order to relate model categories and their associated homotopy categories we need a notion of maps between them. We want the maps such that they induce maps on the homotopy categories.
 \todo{Definition etc}

thesis/CDGA_Model.tex

@@ -1,7 +1,9 @@
 
 \section{Model structure on $\CDGA_\k$}
 
-In this section we will define a model structure on CDGAs over a field $\k$ \todo{Can $\k$ be a c. ring here?}, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations.
+\TODO{First discuss the model structure on (co)chain complexes. Then discuss that we want the adjunction $(\Lambda, U)$ to be a Quillen pair. Then state that (co)chain complexes are cofib. generated, so we can cofib. generate CDGAs.}
+
+In this section we will define a model structure on CDGAs over a field $\k$ of characteristic zero\todo{Can $\k$ be a c. ring here?}, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations.
 
 \begin{proposition}
 	There is a model structure on $\CDGA_\k$ where $f: A \to B$ is
@@ -51,7 +53,7 @@ Next we will prove the factorization property [MC5]. We will do this by Quillen'
 	The maps $i_n$ are trivial cofibrations and the maps $j_n$ are cofibrations.
 \end{lemma}
 \begin{proof}
-	Since $H(T(n)) = \k$ we see that indeed $H(i_n)$ is an isomorphism. For the lifting property of $i_n$ and $j_n$ simply use surjectivity of the fibrations. \todo{give a bit more detail}
+	Since $H(T(n)) = \k$ \todo{Note that this only hold when characteristic = 0} we see that indeed $H(i_n)$ is an isomorphism. For the lifting property of $i_n$ and $j_n$ simply use surjectivity of the fibrations. \todo{give a bit more detail}
 \end{proof}
 
 \begin{lemma}

thesis/preamble.tex

@@ -34,6 +34,7 @@
 \newcommand{\Np}{{\mathbb{N}^{>0}}}		% positive numbers
 \newcommand{\Z}{\mathbb{Z}}				% integers
 \newcommand{\R}{\mathbb{R}}				% reals
+\newcommand{\Q}{\mathbb{Q}}				% rationals
 \renewcommand{\k}{\mathbbm{k}}			% default ground ring
 
 % Basic category stuff
@@ -76,11 +77,25 @@
   \right|_{#2} % this is the delimiter
   }}
 
-% todos
+% Todos in the margin
 \newcommand{\todo}[1]{
 	\addcontentsline{tdo}{todo}{\protect{#1}}
 	$\ast$ \marginpar{\tiny $\ast$ #1}
 }
+% Big todos in text
+\newcommand{\TODO}[1]{
+	\addcontentsline{tdo}{todo}{\protect{#1}}
+	{\tiny $\ast$ #1}
+}
+% TODO item, as itemize does not work
+\newcommand{\titem}[0]{\\-\qquad}
+% List of todos
+\makeatletter
+	\newcommand \listoftodos{\section*{Todo list} \@starttoc{tdo}}
+	\newcommand\l@todo[2]{
+		\par\noindent \textit{#2}, \parbox{10cm}{#1}\par
+	}
+\makeatother
 
 \theoremstyle{plain}
 \newtheorem{theorem}{Theorem}[section]

thesis/thesis.tex

@@ -16,11 +16,14 @@ Some general notation: \todo{leave this out, or define somewhere else?}
 	\item $\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category.
 	\item $\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$.
 \end{itemize}
-\newpage
 
-\input{1_Algebra} \newpage
-\input{2_Model_Cats} \newpage
-\input{CDGA_Model} \newpage
+\vspace{1cm}
+
+\input{1_Algebra}    \vspace{2cm}
+\input{2_Model_Cats} \vspace{2cm}
+\input{CDGA_Model}   \vspace{2cm}
+
+% \listoftodos
 
 \nocite{*}
 \bibliographystyle{alpha}