\paragraph{Preliminaries and Notation}

We assume the reader is familiar with category theory, basics from algebraic topology and the basics of simplicial sets. Some knowledge about differential graded algebra (or homological algebra) and model categories is also assumed, but the reader may review some facts on homological algebra in Appendix \ref{sec:algebra} and facts on model categories in Appendix \ref{sec:model_categories}.

We will fix the following notations and categories.
\begin{itemize}
	\item $\k$ will denote a field of characteristic zero. Modules, tensor products, \dots\, are understood as $\k$-vector spaces, tensor products over $\k$, \dots.
	\item $\Hom_{\cat{C}}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. The subscript $\cat{C}$ may occasionally be left out.
	\item $\Top$: category of topological spaces and continuous maps. We denote the full subcategory of $r$-connected spaces by $\Top_r$, this convention is also used for other categories.
	\item $\Ab$: category of abelian groups and group homomorphisms.
	\item $\sSet$: category of simplicial sets and simplicial maps. More generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$. We have the homotopy equivalence $|-| : \sSet \leftadj \Top : S$ to switch between topological spaces and simplicial sets.
	\item $\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (as defined in the appendix) and graded algebra  maps. As a shorthand we will refer to such an object as \emph{dga}. Furthermore $\CDGA_\k$ is the full subcategory of $\DGA_\k$ of commutative dga's (\emph{cdga}'s).
\end{itemize}