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.
12 lines
777 B
12 lines
777 B
10 years ago
|
|
||
|
|
\subsection{Polynomial Forms}
|
||
|
|
|
||
|
|
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.
|
||
|
|
|
||
|
|
Given a category $\cat{C}$ and a functor $F: \DELTA \to \cat{C}$, then define the following on objects:
|
||
|
|
\begin{align*}
|
||
|
|
F_!(X) &= \colim_{\Delta[n] \to X} F[n] &\quad X \in \sSet \\
|
||
|
|
F^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y) &\quad C \in \cat{C}
|
||
|
|
\end{align*}
|
||
|
|
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.
|