diff --git a/thesis/Definitions.tex b/thesis/1_Algebra.tex similarity index 70% rename from thesis/Definitions.tex rename to thesis/1_Algebra.tex index 5334c6f..b3dedba 100644 --- a/thesis/Definitions.tex +++ b/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 -$$ M \tensor N \iso \bigoplus_{n \in Z} \bigoplus_{m \in Z} M_m \tensor N_n \iso \bigoplus_{n \in Z} \bigoplus_{i + j = n} M_i \tensor N_j. $$ -This defines a natural grading on the tensor product. +Recall that the tensor product of modules distributes over direct sums. This defines a natural grading on the ordinary 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}} -\newcommand{\Fib}{\mathfrak{Fib}} -\newcommand{\Cof}{\mathfrak{Cof}} +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. \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 $p \in \Fib$, 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} + Given a differential graded modules $(M, d)$ we define the \emph{homology} of $M$ as: $H(M, d) = \ker(d) / \im(d)$. + It is naturally graded as follows: + $$ H(M, d)_i = H_i(M, d) = \ker(\restr{d}{M_i}) / d(M_{i+1}). $$ + If $d$ has degree $+1$ we define the \emph{cohomology} as: + $$ H(M, d)^i = H^i(M, d) = \ker(\restr{d}{M^i}) / d(M^{i-1}). $$ \end{definition} -\todo{define notation $\cof$ $\fib$} -\todo{define (co)fibrant objects} -\todo{maybe some basic propositions} +For differential graded algebras we can consider the (co)homology by forgetting the multiplicative structure. However this multiplication will actually pass to (co)homology: + +\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} + diff --git a/thesis/2_Model_Cats.tex b/thesis/2_Model_Cats.tex new file mode 100644 index 0000000..ff3dfb3 --- /dev/null +++ b/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} \ No newline at end of file diff --git a/thesis/CDGA_Model.tex b/thesis/CDGA_Model.tex index 09b7bad..210f2d6 100644 --- a/thesis/CDGA_Model.tex +++ b/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. diff --git a/thesis/preamble.tex b/thesis/preamble.tex index 6976610..b5f7b9d 100644 --- a/thesis/preamble.tex +++ b/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 diff --git a/thesis/thesis.tex b/thesis/thesis.tex index 0687a9a..4b17612 100644 --- a/thesis/thesis.tex +++ b/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{*}