More about model cats
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@ -41,7 +41,7 @@ The graded modules together with graded maps of degree $0$ form the category $\g
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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 $0$.
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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 $0$.
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\end{definition}
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\end{definition}
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Again these objects and maps form a category, denoted as $\grAlg{\k}$.
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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 $\mu$.
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\begin{definition}
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\begin{definition}
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A graded algebra $A$ is \emph{commutative} if for all $x, y \in A$
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A graded algebra $A$ is \emph{commutative} if for all $x, y \in A$
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@ -57,7 +57,7 @@ Again these objects and maps form a category, denoted as $\grAlg{\k}$.
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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. $d_N f = f d_M$.
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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. $d_N f = f d_M$.
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\end{definition}
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\end{definition}
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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.
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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} $+1$.
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The tensor product of two differential graded modules is again a differential graded module if we define the differential as follows. \todo{Define this}
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The tensor product of two differential graded modules is again a differential graded module if we define the differential as follows. \todo{Define this}
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@ -72,6 +72,7 @@ It is not hard to see that this definition precisely defines the monoidal object
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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}.
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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}.
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\todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary CDGAs.}
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\subsection{Homology}
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\subsection{Homology}
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@ -7,7 +7,7 @@
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\newcommand{\Cof}{\mathfrak{Cof}}
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\newcommand{\Cof}{\mathfrak{Cof}}
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\begin{definition}
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\begin{definition}
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A \emph{model category} is a category $\cat{C}$ together with three subcategories:
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A \emph{(closed) model category} is a category $\cat{C}$ together with three subcategories:
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\begin{itemize}
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\begin{itemize}
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\item the class of weak equivalences $\W$,
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\item the class of weak equivalences $\W$,
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\item the class of fibrations $\Fib$ and
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\item the class of fibrations $\Fib$ and
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@ -59,7 +59,7 @@
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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.
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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.
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\end{definition}
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\end{definition}
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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:
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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:
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\begin{center}
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\begin{center}
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\begin{tikzpicture}
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\begin{tikzpicture}
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@ -86,4 +86,18 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival
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\end{tikzpicture}
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\end{tikzpicture}
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\end{center}
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\end{center}
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\todo{maybe some basic propositions}
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\todo{Maybe some basic propositions:
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> Over/under category (or simply pointed objects) \\
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> If a map has LLP/RLP wrt fib/cof, it is a cof/fib \\
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> Fibs are preserved under pullbacks/limits \\
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}
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\todo{Define homotopy category}
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\todo{Cofibrantly generated mod cats?}
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\todo{Small obj. argument?}
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\subsection{Quillen pairs}
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In order to relate model categories and their associated homotopy categories we need a notion of maps between them.
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\todo{Definition etc}
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