Master thesis on Rational Homotopy Theory
https://github.com/Jaxan/Rational-Homotopy-Theory
You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
161 lines
7.9 KiB
161 lines
7.9 KiB
|
|
\section{Model structure on \texorpdfstring{$\CDGA_\k$}{CDGA}}
|
|
\label{sec:model-of-cdga}
|
|
|
|
\TODO{First discuss the model structure on (co)chain complexes. Then discuss that we want the adjunction $(\Lambda, U)$ to be a Quillen pair. Then state that (co)chain complexes are cofib. generated, so we can cofib. generate CDGAs.}
|
|
|
|
In this section we will define a model structure on CDGAs over a field $\k$ of characteristic zero\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 $f$ is a quasi 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.
|
|
|
|
Note that with these classes, every cdga is a fibrant object.
|
|
|
|
\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 $H(T(n)) = \k$ \todo{Note that this only hold when characteristic = 0} 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{bousfield}. 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}
|
|
|
|
|
|
\subsection{Homotopy relation on \texorpdfstring{$\CDGA_\k$}{CDGA}}
|
|
Although the abstract theory of model categories gives us tools to construct a homotopy relation (\DefinitionRef{homotopy}), it is useful to have a concrete notion of homotopic maps.
|
|
|
|
Consider the free cdga on one generator $\Lambda(t, dt)$, this can be thought of as the (dual) unit interval. Indeed there is an isomorphism $\Lambda(t, dt) \iso \Apl_1$ and so we have maps for the two endpoint: $d_0, d_1: \Lambda(t, dt) \to \k \iso \Apl_0$. Given a cdga $X$ we will consider $d_0, d_1: \Lambda(t, dt) \tensor X \to \k \tensor X \iso X$.
|
|
|
|
\Definition{cdga_homotopy}{
|
|
We call $f, g: A \to X$ homotopic ($f \simeq g$) if there is a map
|
|
$$ h: A \to \Lambda(t, dt) \tensor X, $$
|
|
such that $d_0 h = g$ and $d_1 h = f$.
|
|
}
|
|
|
|
In terms of model categories, such a homotopy is a right homotopy and the object $\Lambda(t, dt) \tensor X$ is a path object for $X$. We can easily see that it is a very good path object, first note that $\Lambda(t, dt) \tensor X \to X \oplus X$ is surjective (for $(x, y) \in X \oplus X$ take $t \tensor x + 1 \tensor y$), secondly $\Apl_0 \to \Apl_1$ is a cofibration and so is $X \to \Lambda(t, dt) \tensor X$.
|
|
|
|
Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
|
|
|
|
\Lemma{cdga_homotopy}{
|
|
If $A$ is a cofibrant cdga and $f \simeq^r g: A \to X$, then $f \simeq g$ in the above sense.
|
|
}
|
|
\Proof{
|
|
Because $A$ is cofibrant, there is a very good homotopy $H$. Consider a lifting problem to construct a map $Path_X \to \Lambda(t, dt) \tensor X$.
|
|
}
|
|
|
|
\Corollary{cdga_homotopy_eqrel}{
|
|
For cofibrant $A$, $\simeq$ defines a equivalence relation.
|
|
}
|
|
\Definition{cdga_homotopy_classes}{
|
|
For cofibrant $A$ define the set of equivalence classes as:
|
|
$$ [A, X] = \Hom_{\CDGA_\k}(A, X) / \simeq. $$
|
|
}
|
|
|
|
The results from model categories immediately imply the following results.
|
|
\Corollary{cdga_homotopy_properties}{
|
|
Let $A$ be cofibrant.
|
|
\begin{itemize}
|
|
\item Let $i: A \to B$ be a trivial cofibration, then the induced map $i^\ast: [B, X] \to [A, X]$ is a bijection.
|
|
\item Let $p: X \to Y$ be a trivial fibration, then the induced map $p_\ast: [A, X] \to [A, Y]$ is a bijection.
|
|
\item Let $A$ and $X$ both be cofibrant, then $f: A \we X$ is a weak equivalence if and only if $f$ is a strong homotopy equivalence. Moreover, the two induced maps are bijections:
|
|
$$ f_\ast: [Z, A] \tot{\iso} [Z, X], $$
|
|
$$ f^\ast: [X, Z] \tot{\iso} [A, X]. $$
|
|
\end{itemize}
|
|
}
|
|
|
|
\Lemma{cdga_homotopy_homology}{
|
|
Let $f, g: A \to X$ be two homotopic maps, then $H(f) = H(g): HA \to HX$.
|
|
}
|
|
\Proof{
|
|
We only need to consider $H(d_0)$ and $H(d_1)$.
|
|
}
|