Added part about homology. Split tex files.

thesis/Definitions.tex → thesis/1_Algebra.tex

@@ -1,11 +1,11 @@
-% -*- root: thesis.tex -*-
-\section{Definitions}
-\label{sec:definitions}
 
-\subsection{Graded algebra}
+\section{Differential Graded Algebra}
+\label{sec:algebra}
 
 In this section $\k$ will be any commutative ring. We will recap some of the basic definitions of commutative algebra in a graded setting. By \emph{linear}, \emph{module}, \emph{tensor product}, etc \dots we always mean $\k$-linear, $\k$-module, tensor product over $\k$, etc \dots.
 
+\subsection{Graded algebra}
+
 \begin{definition}
 	A \emph{graded module} $M$ is a family of modules $\{M_n\}_{n\in\Z}$. An element $x \in M_n$ is called a \emph{homogeneous element} and said to be of \emph{degree $\deg{x} = n$}. We will often identify $M = \bigoplus_{n \in \Z} M_n$.
 \end{definition}
@@ -13,7 +13,8 @@ In this section $\k$ will be any commutative ring. We will recap some of the bas
 For an ordinary module $M$ we can consider the graded module $M[0]$ \emph{concentrated in degree $0$} defined by setting $M[0]_0 = M$ and $M[0]_n = 0$ for $i \neq 0$. If clear from the context we will denote this graded module by $M$. In particular $\k$ is a graded module concentrated in degree $0$.
 
 \begin{definition}
-	A linear map $f: M \to N$ between graded modules is \emph{graded of degree $p$} if it respects the grading, i.e. $\restr{f}{M_n} : M_n \to N_{n+p}$.
+	A linear map $f: M \to N$ between graded modules is \emph{graded of degree $p$} if it respects the grading and raises the degree by $p$, i.e.
+	$$ \restr{f}{M_n} : M_n \to N_{n+p}. $$
 \end{definition}
 
 \begin{definition}
@@ -23,9 +24,7 @@ For an ordinary module $M$ we can consider the graded module $M[0]$ \emph{concen
 
 Note that not all linear maps can be decomposed into a sum of graded maps, so that $\Hom_{gr}(M, N) \subset \Hom(M, N)$ may be proper for some $M$ and $N$.
 
-Recall that the tensor product of modules distributes over direct sums. So if $M = \bigoplus_{n \in \Z} M_n$ and $N = \bigoplus_{n \in \Z} N_n$, then
+Recall that the tensor product of modules distributes over direct sums. This defines a natural grading on the ordinary tensor product.
-$$ M \tensor N \iso \bigoplus_{n \in Z} \bigoplus_{m \in Z} M_m \tensor N_n \iso \bigoplus_{n \in Z} \bigoplus_{i + j = n} M_i \tensor N_j. $$
-This defines a natural grading on the tensor product.
 
 \begin{definition}
 	The graded tensor product is defined as:
@@ -42,7 +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 form a category, denoted as $\grAlg{\k}$.
+Again these objects and maps form a category, denoted as $\grAlg{\k}$.
 
 \begin{definition}
 	A graded algebra $A$ is \emph{commutative} if for all $x, y \in A$
@@ -74,38 +73,26 @@ 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}.
 
 
-\subsection{Model categories}
+\subsection{Homology}
 
-\newcommand{\W}{\mathfrak{W}}
+Whenever we have a differential graded module we have $d \circ d = 0$, or put in other words: the image of $d$ is a submodule of the kernel of $d$. The quotient of the two graded modules will be of interest.
-\newcommand{\Fib}{\mathfrak{Fib}}
-\newcommand{\Cof}{\mathfrak{Cof}}
 
 \begin{definition}
-	A \emph{model category} is a category $\cat{C}$ together with three subcategories:
+	Given a differential graded modules $(M, d)$ we define the \emph{homology} of $M$ as: $H(M, d) = \ker(d) / \im(d)$.
-	\begin{itemize}
+	It is naturally graded as follows:
-		\item the class of weak equivalences $\W$,
+	$$ H(M, d)_i = H_i(M, d) = \ker(\restr{d}{M_i}) / d(M_{i+1}). $$
-		\item the class of fibrations $\Fib$ and
+	If $d$ has degree $+1$ we define the \emph{cohomology} as:
-		\item the class of cofibrations $\Cof$,
+	$$ H(M, d)^i = H^i(M, d) = \ker(\restr{d}{M^i}) / d(M^{i-1}). $$
-	\end{itemize}
-	such that the following five axioms hold:
-	\begin{itemize}
-		\item[MC1] All finite limits and colimits exist in $\cat{C}$.
-		\item[MC2] If $f$, $g$ and $fg$ are maps such that two of them are weak equivalences, then so it the third. This is called the \emph{2-out-of-3} property.
-		\item[MC3] All three classes of maps are closed under retracts\todo{Either draw the diagram or define a retract earlier}.
-		\item[MC4] In any commuting square as follows where $i \in \Cof$ and $p \in \Fib$, there exist a lift if either 
-		\begin{itemize}
-			\item[a)] $i \in \W$ or
-			\item[b)] $p \in \W$.
-		\end{itemize}
-		\todo{insert diagram}
-		\item[MC5] Any map $f : A \to B$ can be factored as $f = pi$, where either
-		\begin{itemize}
-			\item[a)] $i \in \Cof \cap \W$ and $p \in \Fib$ or
-			\item[b)] $i \in \Cof$ and $p \in \Fib \cap \W$.
-		\end{itemize}
-	\end{itemize}
 \end{definition}
 
-\todo{define notation $\cof$ $\fib$}
+For differential graded algebras we can consider the (co)homology by forgetting the multiplicative structure. However this multiplication will actually pass to (co)homology:
-\todo{define (co)fibrant objects}
+
-\todo{maybe some basic propositions}
+\begin{lemma}
+	Let $(A, d)$ be a differential graded algebra. The kernel $\ker(d)$ is a subalgebra of $A$ and the image $d(A)$ is an ideal, so that the quotient
+	$$ H(A) = \ker(d) / \im(d) $$
+	is a graded algebra.
+\end{lemma}
+\begin{proof}
+	\todo{}
+\end{proof}
+

thesis/2_Model_Cats.tex

@@ -0,0 +1,89 @@
+
+\section{Model categories}
+\label{sec:model_cats}
+
+\newcommand{\W}{\mathfrak{W}}
+\newcommand{\Fib}{\mathfrak{Fib}}
+\newcommand{\Cof}{\mathfrak{Cof}}
+
+\begin{definition}
+	A \emph{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
+		\item the class of cofibrations $\Cof$,
+	\end{itemize}
+	such that the following five axioms hold:
+	\begin{itemize}
+		\item[MC1] All finite limits and colimits exist in $\cat{C}$.
+		\item[MC2] If $f$, $g$ and $fg$ are maps such that two of them are weak equivalences, then so it the third. This is called the \emph{2-out-of-3} property.
+		\item[MC3] All three classes of maps are closed under retracts\todo{Either draw the diagram or define a retract earlier}.
+		\item[MC4] In any commuting square as follows where $i \in \Cof$ and $p \in \Fib$,
+		\begin{center}
+		\begin{tikzpicture}
+		\matrix (m) [matrix of math nodes]{
+			A & X \\
+			B & Y \\
+		};
+
+		\path[->] (m-1-1) edge (m-1-2);
+		\path[->] (m-2-1) edge (m-2-2);
+		\path[->] (m-1-1) edge node[auto] {$i$} (m-2-1);
+		\path[->] (m-1-2) edge node[auto] {$p$} (m-2-2);
+
+		\end{tikzpicture}
+		\end{center}
+
+		 there exist a lift $h: B \to Y$ if either 
+		\begin{itemize}
+			\item[a)] $i \in \W$ or
+			\item[b)] $p \in \W$.
+		\end{itemize}
+		\item[MC5] Any map $f : A \to B$ can be factored in two ways:
+		\begin{itemize}
+			\item[a)] as $f = pi$, where $i \in \Cof \cap \W$ and $p \in \Fib$ and
+			\item[b)] as $f = pi$, where $i \in \Cof$ and $p \in \Fib \cap \W$.
+		\end{itemize}
+	\end{itemize}
+\end{definition}
+
+\begin{notation} For brevity
+	\begin{itemize}
+		\item we write $f: A \fib B$ when $f$ is a fibration,
+		\item we write $f: A \cof B$ when $f$ is a cofibration and
+		\item we write $f: A \we B$ when $f$ is a weak equivalence.
+	\end{itemize}
+\end{notation}
+
+\begin{definition}
+	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:
+
+\begin{center}
+\begin{tikzpicture}
+\matrix (m) [matrix of math nodes]{
+	\cat{0} &         & X \\
+	        & X^{cof} &   \\
+};
+
+\path[->] (m-1-1) edge (m-1-3);
+\path[right hook->] (m-1-1) edge (m-2-2);
+\path[->>] (m-2-2) edge node[auto] {$ \simeq $} (m-1-3);
+
+\end{tikzpicture}\quad
+\begin{tikzpicture}
+\matrix (m) [matrix of math nodes]{
+	X &         & \cat{1} \\
+	  & X^{fib} &         \\
+};
+
+\path[->] (m-1-1) edge (m-1-3);
+\path[right hook->] (m-1-1) edge node[auto] {$ \simeq $} (m-2-2);
+\path[->>] (m-2-2) edge (m-1-3);
+
+\end{tikzpicture}
+\end{center}
+
+\todo{maybe some basic propositions}

thesis/CDGA_Model.tex

@@ -1,4 +1,4 @@
-% -*- root: thesis.tex -*-
+
 \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.

thesis/preamble.tex

@@ -1,4 +1,4 @@
-% -*- root: thesis.tex -*-
+
 % clickable tocs
 \usepackage{hyperref}
 
@@ -62,6 +62,7 @@
 \newcommand{\eq}{\sim}					% homotopic
 \newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}}	% arrow with name
 \newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}}	% mapsto with name
+\DeclareMathOperator*{\im}{im}
 \DeclareMathOperator*{\colim}{colim}
 \DeclareMathOperator*{\tensor}{\otimes}
 \DeclareMathOperator*{\bigtensor}{\bigotimes}
@@ -89,6 +90,7 @@
 
 \theoremstyle{definition}
 \newtheorem{definition}[theorem]{Definition}
+\newtheorem{notation}[theorem]{Notation}
 \newtheorem{example}[theorem]{Example}
 
 % headings for a table

thesis/thesis.tex

@@ -11,7 +11,15 @@
 \maketitle
 \tableofcontents
 
-\input{Definitions} \newpage
+Some general notation: \todo{leave this out, or define somewhere else?}
+\begin{itemize}
+	\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
 
 \nocite{*}