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Adds a stub form model structure on CDGAs and adds stuff about model cats

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Joshua Moerman 2014-05-01 12:33:03 +02:00
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108
thesis/CDGA_Model.tex Normal file
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% -*- 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.
\begin{proposition}
There is a model structure on $\CDGA_\k$ where $f: A \to B$ is
\begin{itemize}
\item a \emph{weak equivalence} if $H(f)$ is an isomorphism,
\item a \emph{fibration} if $f$ is an surjective and
\item a \emph{cofibration} if $f$ has the LLP w.r.t. trivial fibrations
\end{itemize}
\end{proposition}
We will prove the different axioms in the following lemmas. First observe that the classes as defined above are indeed closed under multiplication and contain all isomorphisms.
\begin{lemma}
[MC1] The category has all finite limits and colimits.
\end{lemma}
\begin{proof}
As discussed earlier \todo{really discuss this somewhere} products are given by direct sums and equalizers are kernels. Furthermore the coproducts are tensor products and coequalizers are quotients.
\end{proof}
\begin{lemma}
[MC2] The \emph{2-out-of-3} property for quasi isomorphisms.
\end{lemma}
\begin{proof}
Let $f$ and $g$ be two maps such that two out of $f$, $g$ and $fg$ are weak equivalences. This means that two out of $H(f)$, $H(g)$ and $H(f)H(g)$ are isomorphisms. The \emph{2-out-of-3} property holds for isomorphisms, proving the statement.
\end{proof}
\begin{lemma}
[MC3] All three classes are closed under retracts
\end{lemma}
\begin{proof}
\todo{Make some diagrams and write it out}
\end{proof}
Next we will prove the factorization property [MC5]. We will do this by Quillen's small object argument. When proved, we get an easy way to prove the missing lifting property of [MC4]. For the Quillen's small object argument we use classes of generating cofibrations.
\begin{definition}
Define the following objects and sets of maps:
\begin{itemize}
\item $S(n)$ is the CDGA generated by one element $a$ of degree $n$ such that $da = 0$.
\item $T(n)$ is the CDGA generated by two element $b$ and $c$ of degree $n$ and $n+1$ respectively, such that $db = c$ (and necessarily $dc = 0$).
\item $I = \{ i_n: \k \to T(n) \I n \in \N \}$ is the set of units of $T(n)$.
\item $J = \{ j_n: S(n+1) \to T(n) \I n \in \N \}$ is the set of inclusions $j_n$ defined by $j_n(a) = b$.
\end{itemize}
\end{definition}
\begin{lemma}
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}
\end{proof}
\begin{lemma}
The class of (trivial) cofibrations is saturated.
\end{lemma}
\begin{proof}
\todo{prove this}
\end{proof}
As a consequence of the above two lemmas, the class generated by $I$ is contained in the class of trivial cofibrations. Similarly the class generated by $J$ is contained in the class of cofibrations. We also have a similar lemma about (trivial) fibrations.
\begin{lemma}
If $p: X \to Y$ has the RLP w.r.t. $I$ then $p$ is a fibration.
\end{lemma}
\begin{proof}
Easy\todo{Define a lift}.
\end{proof}
\begin{lemma}
If $p: X \to Y$ has the RLP w.r.t. $J$ then $p$ is a trivial fibration.
\end{lemma}
\begin{proof}
As $p$ has the RLP w.r.t. $J$, it also has the RLP w.r.t. $I$. From the previous lemma it follows that $p$ is a fibration. To show that $p$ is a weak equivalence ... \todo{write out}
\end{proof}
We can use Quillen's small object argument with these sets. The argument directly proves the following lemma. Together with the above lemmas this translates to the required factorization.
\begin{lemma}
A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is in the class generated by $I$ and $p$ has the RLP w.r.t. $I$.
\end{lemma}
\begin{proof}
Quillen's small object argument. \todo{small = finitely generated?}
\end{proof}
\begin{corollary}
[MC5a] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a trivial cofibration and $p$ a fibration.
\end{corollary}
The previous factorization can also be described explicitly as seen in \cite{bous}. Let $f: A \to X$ be a map, define $E = A \tensor \bigtensor_{x \in X}T(\deg{x})$. Then $f$ factors as:
$$ A \tot{i} E \tot{p} X, $$
where $i$ is the obvious inclusion $i(a) = a \tensor 1$ and $p$ maps (products of) generators $a \tensor b_x$ with $b_x \in T(\deg{x})$ to $f(a) \cdot x \in X$.
\begin{lemma}
A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is in the class generated by $J$ and $p$ has the RLP w.r.t. $J$.
\end{lemma}
\begin{proof}
Quillen's small object argument.
\end{proof}
\begin{corollary}
[MC5b] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a cofibration and $p$ a trivial fibration.
\end{corollary}

75
thesis/Definitions.tex
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% -*- root: thesis.tex -*-
\section{Definitions}
\label{sec:definitions}
\subsection{Graded 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}, \dots we always mean $\k$-linear, $\k$-module, tensor product over $\k$, \dots.
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.
\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{homogenous element} and said to be of \emph{degree $\deg{x} = n$}. We will often identify $M = \bigoplus_{n \in \Z} M_n$.
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}
For an arbitrary 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 $0$.
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. $\restr{f}{M_n} : M_n \to N_{n+p}$.
@ -18,10 +18,10 @@ For an arbitrary module $M$ we can consider the graded module $M[0]$ \emph{conce
\begin{definition}
The graded maps $f: M \to N$ between graded modules can be arranged in a graded module by defining:
$$ \Hom{gr}{M}{N}_n = \{ f: M \to N \I f \text{ is graded of degree } n \}. $$
$$ \Hom_{gr}(M, N)_n = \{ f: M \to N \I f \text{ is graded of degree } n \}. $$
\end{definition}
Note that not all linear maps can be decomposed into a sum of graded maps. In other words $\Hom{gr}{M}{N} \subset \Hom{}{M}{N}$ might not be equal.
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
$$ M \tensor N \iso \bigoplus_{n \in Z} \bigoplus_{m \in Z} M_m \tensor N_n \iso \bigoplus_{n \in Z} \bigoplus_{i + j = n} M_i \tensor N_j. $$
@ -32,31 +32,80 @@ This defines a natural grading on the tensor product.
$$ (M \tensor N)_n = \bigoplus_{i + j = n} M_i \tensor N_j. $$
\end{definition}
The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. 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 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.
\begin{definition}
A \emph{graded algebra} consists of a graded module $A$ together with two graded maps of degree $0$:
A \emph{graded algebra} consists of a graded module $A$ together with two maps of degree $0$:
$$ \mu: A \tensor A \to A \quad\text{ and }\quad \eta: k \to A $$
such that $\mu$ is associative and $\eta$ is a unit for $\mu$.
A graded map between two graded algebra will be called \emph{graded algebra map} if the map is compatible with the multiplication and unit.
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 $\grAlg{\k}$.
\begin{definition}
A graded algebra $A$ is \emph{commutative} if for all $x, y \in A$
$$ xy = (-1)^{\deg{x}\deg{y}}yx. $$
$$ xy = (-1)^{\deg{x}\deg{y}} yx. $$
\end{definition}
\todo{Add a remark about the signs somewhere}
\subsection{Differential graded algebra}
Now define differentials... and the categories $\cat{DGA}_\k, \cat{CGDA}_\k$.
Note that a monoidal object of differential graded modules is the same as a graded algebra with a differential.
\begin{definition}
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}
Conclude with (co)chain complexes and (co)chain (co)algebras.
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}
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:
$$ d(xy) = d(x) y + (-1)^{\deg{x}} x d(y) \quad\text{ for all } x, y \in A. $$
\end{definition}
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}.
\subsection{Model categories}
\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 $p \in \Fib$, 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 $\cof$ $\fib$}
\todo{define (co)fibrant objects}
\todo{maybe some basic propositions}

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thesis/preamble.tex
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@ -1,12 +1,13 @@
% -*- root: thesis.tex -*-
% clickable tocs
\usepackage{hyperref}
% use english
\usepackage[english, british]{babel}
% floating figures
\usepackage{float}
\usepackage{listings}
\usepackage{tikz}
\usetikzlibrary{matrix, arrows, decorations}
\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto}
@ -25,44 +26,48 @@
% for the fib arrow
\usepackage{amssymb}
% mathbb for lowercase
% mathbb for lowercase bbs
\usepackage{bbm}
% for slanted text/symbols
\usepackage{slantsc}
% Some basic objects
\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\Np}{{\mathbb{N}^{>0}}} % positive numbers
\newcommand{\Z}{\mathbb{Z}} % integers
\newcommand{\R}{\mathbb{R}} % reals
\renewcommand{\k}{\mathbbm{k}} % default ground ring
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator*{\tensor}{\otimes}
\DeclareMathOperator*{\bigtensor}{\bigotimes}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Np}{{\mathbb{N}^{>0}}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\renewcommand{\k}{\mathbbm{k}}
\newcommand{\cat}[1]{\mathbf{#1}}
\newcommand{\Set}{\cat{Set}}
\newcommand{\sSet}{\cat{sSet}}
\newcommand{\Top}{\cat{Top}}
\newcommand{\DELTA}{\cat{\Delta}}
\newcommand{\grMod}[1]{\cat{gr-{#1}Mod}}
\newcommand{\grAlg}[1]{\cat{gr-{#1}Alg}}
\newcommand{\Hom}[3]{\mathbf{Hom}_{#1}(#2, #3)}
% Basic category stuff
\newcommand{\cat}[1]{\mathbf{#1}} % the category of ...
\newcommand{\Hom}{\mathbf{Hom}}
\newcommand{\id}{\mathbf{id}}
\newcommand{\I}{\,\mid\,}
% Categories
\newcommand{\Set}{\cat{Set}} % sets
\newcommand{\sSet}{\cat{sSet}} % simplicial sets
\newcommand{\Top}{\cat{Top}} % topological spaces
\newcommand{\DELTA}{\cat{\Delta}} % the simplicial cat
\newcommand{\grMod}[1]{\cat{gr\mbox{-}{#1}Mod}} % graded modules over a ring
\newcommand{\grAlg}[1]{\cat{gr\mbox{-}{#1}Alg}} % graded algebras over a ring
\newcommand{\DGA}{\cat{DGA}} % differential graded algebras
\newcommand{\CDGA}{\cat{CDGA}} % commutative dgas
\newcommand{\cof}{\hookrightarrow} % cofibration
\newcommand{\fib}{\twoheadrightarrow} % fibration
\newcommand{\we}{\tot{\simeq}} % weak equivalence
% Notation and operators
\newcommand{\I}{\,\mid\,} % seperator in set notation
\newcommand{\del}{\partial} % boundary
\newcommand{\iso}{\cong} % isomorphic
\newcommand{\eq}{\sim} % homotopic
\newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} % arrow with name
\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} % mapsto with name
\newcommand{\cof}{\hookrightarrow} % cofibration
\newcommand{\fib}{\twoheadrightarrow} % fibration
\newcommand{\we}{\tot{\simeq}} % weak equivalence
\renewcommand{\deg}[1]{|{#1}|}
\DeclareMathOperator*{\colim}{colim}
\DeclareMathOperator*{\tensor}{\otimes}
\DeclareMathOperator*{\bigtensor}{\bigotimes}
\renewcommand{\deg}[1]{{|{#1}|}}
% restriction of a function
\newcommand\restr[2]{{% we make the whole thing an ordinary symbol
\left.\kern-\nulldelimiterspace % automatically resize the bar with \right
#1 % the function
@ -70,6 +75,7 @@
\right|_{#2} % this is the delimiter
}}
% todos
\newcommand{\todo}[1]{
\addcontentsline{tdo}{todo}{\protect{#1}}
$\ast$ \marginpar{\tiny $\ast$ #1}
@ -85,4 +91,5 @@
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
% headings for a table
\newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}}

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thesis/thesis.tex
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@ -1,4 +1,4 @@
\documentclass[a4paper, 11pt]{amsart}
\documentclass[a4paper, 12pt, draft]{amsart}
\input{style}
\input{preamble}
@ -9,8 +9,10 @@
\begin{document}
\maketitle
\tableofcontents
\input{Definitions} \newpage
\input{CDGA_Model} \newpage
\nocite{*}
\bibliographystyle{alpha}