From 79518698632c037c41924ad888895ed9e486e49c Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Thu, 1 May 2014 12:33:03 +0200 Subject: [PATCH] Adds a stub form model structure on CDGAs and adds stuff about model cats --- thesis/CDGA_Model.tex | 108 +++++++++++++++++++++++++++++++++++++++++ thesis/Definitions.tex | 75 +++++++++++++++++++++++----- thesis/preamble.tex | 67 +++++++++++++------------ thesis/thesis.tex | 4 +- 4 files changed, 210 insertions(+), 44 deletions(-) create mode 100644 thesis/CDGA_Model.tex diff --git a/thesis/CDGA_Model.tex b/thesis/CDGA_Model.tex new file mode 100644 index 0000000..09b7bad --- /dev/null +++ b/thesis/CDGA_Model.tex @@ -0,0 +1,108 @@ +% -*- 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} + + diff --git a/thesis/Definitions.tex b/thesis/Definitions.tex index c1ee755..5334c6f 100644 --- a/thesis/Definitions.tex +++ b/thesis/Definitions.tex @@ -1,16 +1,16 @@ - +% -*- 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} + +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} -Conclude with (co)chain complexes and (co)chain (co)algebras. +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} diff --git a/thesis/preamble.tex b/thesis/preamble.tex index e8b469e..6976610 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -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}} diff --git a/thesis/thesis.tex b/thesis/thesis.tex index 5c3a1c4..0687a9a 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -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}