More about model cats

thesis/1_Algebra.tex

@@ -41,7 +41,7 @@ The graded modules together with graded maps of degree $0$ form the category $\g
 	A map between two graded algebra will be called a \emph{graded algebra map} if the map is compatible with the multiplication and unit. Such a map is necessarily of degree $0$.
 \end{definition}
 
-Again these objects and maps form a category, denoted as $\grAlg{\k}$.
+Again these objects and maps form a category, denoted as $\grAlg{\k}$. We will denote multiplication by a dot or juxtaposition, instead of explicitly mentioning $\mu$.
 
 \begin{definition}
 	A graded algebra $A$ is \emph{commutative} if for all $x, y \in A$
@@ -57,7 +57,7 @@ Again these objects and maps form a category, denoted as $\grAlg{\k}$.
 	A \emph{differential graded module} $(M, d)$ is a graded module $M$ together with a map $d: M \to M$ of degree $-1$, called a \emph{differential}, such that $dd = 0$. A map $f: M \to N$ is a \emph{chain map} if it is compatible with the differential, i.e. $d_N f = f d_M$.
 \end{definition}
 
-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.
+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}
 
@@ -72,6 +72,7 @@ It is not hard to see that this definition precisely defines the monoidal object
 
 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}.
 
+\todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary CDGAs.}
 
 \subsection{Homology}
 

thesis/2_Model_Cats.tex

@@ -7,7 +7,7 @@
 \newcommand{\Cof}{\mathfrak{Cof}}
 
 \begin{definition}
-	A \emph{model category} is a category $\cat{C}$ together with three subcategories:
+	A \emph{(closed) model category} is a category $\cat{C}$ together with three subcategories:
 	\begin{itemize}
 		\item the class of weak equivalences $\W$,
 		\item the class of fibrations $\Fib$ and
@@ -59,7 +59,7 @@
 	An object $A$ in a model category $\cat{C}$ will be called \emph{fibrant} if $A \to \cat{1}$ is a fibration and \emph{cofibrant} if $\cat{0} \to A$ is a cofibration.
 \end{definition}
 
-Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equivalent fibrant object $X^{fib}$ and a weakly equivalent cofibrant object $X^{cof}$, as seen in the following diagram:
+Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equivalent fibrant object $X^{fib}$ and by [MC5b] by a weakly equivalent cofibrant object $X^{cof}$, as seen in the following diagram:
 
 \begin{center}
 \begin{tikzpicture}
@@ -86,4 +86,18 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival
 \end{tikzpicture}
 \end{center}
 
-\todo{maybe some basic propositions}
+\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{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.
+\todo{Definition etc}