msc-thesis/thesis/notes/Homotopy_Theory_CDGA.tex

\Chapter{Homotopy Theory For cdga's}{HomotopyTheoryCDGA}

Recall that a cdga $A$ is a commutative differential graded algebra, meaning that
\begin{itemize}\itemsep0em
	\item it has a grading: $A = \bigoplus_{n\in\N} A^n$,
	\item it has a differential: $d: A \to A$ with $d^2 = 0$,
	\item it has a multiplication: $\mu: A \tensor A \to A$ which is associative and unital and
	\item it is commutative: $x y = (-1)^{\deg{x}\cdot\deg{y}} y x$.
\end{itemize}
And all of the above structure is compatible with each other (e.g. the differential is a derivation of degree $1$, the maps are graded, \dots). The exact requirements are stated in the appendix on algebra. An algebra $A$ is augmented if it has a specified map (of algebras) $A \tot{\counit} \k$. Furthermore we adopt the notation $A^{\leq n} = \bigoplus_{k \leq n} A^k$ and similarly for $\geq n$.

There is a left adjoint $\Lambda$ to the forgetful functor $U$ which assigns the free graded commutative algebras $\Lambda V$ to a graded module $V$. This extends to an adjunction (also called $\Lambda$ and $U$) between commutative differential graded algebras and differential graded modules. We denote the subspace of elements of wordlength $n$ by $\Lambda^n V$ (note that this has nothing to do with the grading on $V$).

In homological algebra we are especially interested in \emph{quasi isomorphisms}, i.e. maps $f: A \to B$ inducing an isomorphism on cohomology: $H(f): HA \iso HB$. This notions makes sense for any object with a differential.

We furthermore have the following categorical properties of cdga's:
\begin{itemize}\itemsep0em
	\item The finite coproduct in $\CDGA_\k$ is the (graded) tensor product.
	\item The finite product in $\CDGA_\k$ is the cartesian product (with pointwise operations).
	\item The equalizer (resp. coequalizer) of $f$ and $g$ is given by the kernel (resp. cokernel) of $f - g$. Together with the (co)products this defines pullbacks and pushouts.
	\item $\k$ and $0$ are the initial and final object.
\end{itemize}

\section{Cochain models for the $n$-disk and $n$-sphere}
\input{notes/CDGA_Basic_Examples}

\section{The Quillen model structure on \titleCDGA}
\input{notes/Model_Of_CDGA}

\section{Homotopy relations on \titleCDGA}
\input{notes/Homotopy_Relations_CDGA}

\section{Homotopy theory of augmented cdga's}
\input{notes/Homotopy_Augmented_CDGA}

\section{Homotopy groups of cdga's}
\input{notes/Homotopy_Groups_CDGA}