Master thesis on Rational Homotopy Theory
https://github.com/Jaxan/Rational-Homotopy-Theory
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52 lines
3.0 KiB
52 lines
3.0 KiB
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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.
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Consider the free cdga on one generator $\Lambda(t, dt)$\todo{same as $\Lambda D(0)$}, where $\deg{t} = 0$, this can be thought of as the (dual) unit interval with endpoints $1$ and $t$. We define two \emph{endpoint maps} as follows:
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$$ d_0, d_1 : \Lambda(t, dt) \to \k $$
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$$ d_0(t) = 1, \qquad d_1(t) = 0, $$
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this extends linearly and multiplicatively. Note that it follows that we have $d_0(1-t) = 0$ and $d_1(1-t) = 1$. These two functions extend to tensor products as $d_0, d_1: \Lambda(t, dt) \tensor X \to \k \tensor X \tot{\iso} X$.
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\Definition{cdga_homotopy}{
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We call $f, g: A \to X$ homotopic ($f \simeq g$) if there is a map
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$$ h: A \to \Lambda(t, dt) \tensor X, $$
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such that $d_0 h = g$ and $d_1 h = f$.
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}
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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\todo{Refereer}. First note that $\Lambda(t, dt) \tensor X \tot{(d_0, d_1)} X \oplus X$ is surjective (for $(x, y) \in X \oplus X$ take $t \tensor x + (1-t) \tensor y$). Secondly we note that $\Lambda(t, dt) = \Lambda(D(0))$ and hence $\k \to \Lambda(t, dt)$ is a cofibration, by \LemmaRef{model-cats-coproducts} we have that $X \to \Lambda(t, dt) \tensor X$ is a (necessarily trivial) cofibration.
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Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
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\Lemma{cdga_homotopy}{
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If $A$ is a cofibrant cdga and $f \simeq^r g: A \to X$, then $f \simeq g$ in the above sense.
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}
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\Proof{
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Because $A$ is cofibrant, there is a very good homotopy $H$. Consider a lifting problem to construct a map $Path_X \to \Lambda(t, dt) \tensor X$.
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}
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\Corollary{cdga_homotopy_eqrel}{
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For cofibrant $A$, $\simeq$ defines a equivalence relation.
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}
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\Definition{cdga_homotopy_classes}{
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For cofibrant $A$ define the set of equivalence classes as:
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$$ [A, X] = \Hom_{\CDGA_\k}(A, X) / \simeq. $$
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}
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The results from model categories immediately imply the following results. \todo{Refereer expliciet}
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\Corollary{cdga_homotopy_properties}{
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Let $A$ be cofibrant.
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\begin{itemize}
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\item Let $i: A \to B$ be a trivial cofibration, then the induced map $i^\ast: [B, X] \to [A, X]$ is a bijection.
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\item Let $p: X \to Y$ be a trivial fibration, then the induced map $p_\ast: [A, X] \to [A, Y]$ is a bijection.
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\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:
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$$ f_\ast: [Z, A] \tot{\iso} [Z, X], $$
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$$ f^\ast: [X, Z] \tot{\iso} [A, X]. $$
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\todo{De eerste werkt ook als $i$ gewoon een w.e. is. (Gebruik factorizatie.)}
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\end{itemize}
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}
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\Lemma{cdga_homotopy_homology}{
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Let $f, g: A \to X$ be two homotopic maps, then $H(f) = H(g): HA \to HX$.
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}
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\Proof{
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We only need to consider $H(d_0)$ and $H(d_1)$. \todo{Bewijs afmaken}
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}
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