From 58c1e6c8ba3164e393344c6b5804db2946263aff Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Fri, 16 May 2014 10:40:04 +0100 Subject: [PATCH] More about model cats --- thesis/1_Algebra.tex | 5 +++-- thesis/2_Model_Cats.tex | 20 +++++++++++++++++--- 2 files changed, 20 insertions(+), 5 deletions(-) diff --git a/thesis/1_Algebra.tex b/thesis/1_Algebra.tex index b3dedba..efcc7ed 100644 --- a/thesis/1_Algebra.tex +++ b/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} diff --git a/thesis/2_Model_Cats.tex b/thesis/2_Model_Cats.tex index ff3dfb3..4d129d7 100644 --- a/thesis/2_Model_Cats.tex +++ b/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} \ No newline at end of file +\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}