Master thesis on Rational Homotopy Theory
https://github.com/Jaxan/Rational-Homotopy-Theory
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11 lines
777 B
11 lines
777 B
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\subsection{Polynomial Forms}
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There is a general way to construct functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA} \to \CDGA$) and we want to extend to a contravariant functor $\sSet \to \CDGA_\k$. This will be done via Kan extensions.
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Given a category $\cat{C}$ and a functor $F: \DELTA \to \cat{C}$, then define the following on objects:
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\begin{align*}
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F_!(X) &= \colim_{\Delta[n] \to X} F[n] &\quad X \in \sSet \\
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F^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y) &\quad C \in \cat{C}
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\end{align*}
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A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimits. Applying $F$ on these diagrams, make it clear that $F_!$ is functorial.
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