\section{Cochain models for the \texorpdfstring{$n$}{n}-disk and \texorpdfstring{$n$}{n}-sphere}
\label{sec:cdga-basic-examples}
We will first define some basic cochain complexes which model the $n$-disk and $n$-sphere. $D(n)$ is the cochain complex generated by one element $b \in D(n)^n$ and its differential $c = d(b)\in D(n)^{n+1}$. $S(n)$ is the cochain complex generated by one element $a \in S(n)^n$ which differential vanishes (i.e. $da =0$). In other words:
We will first define some basic cochain complexes which model the $n$-disk and $n$-sphere. $D(n)$ is the cochain complex generated by one element $b \in D(n)^n$ and its differential $c = d(b)\in D(n)^{n+1}$. $S(n)$ is the cochain complex generated by one element $a \in S(n)^n$ which differential vanishes (i.e. $da =0$). In other words:
$$ D(n)= ... \to0\to\k\to\k\to0\to ... $$
$$ D(n)= ... \to0\to\k\to\k\to0\to ... $$
@ -43,11 +40,7 @@ The situation for $\Lambda S(n)$ is easier: when $n$ is even it is given by poly
We will prove this theorem in the next section. Note that the functors $\Lambda$ and $U$ thus form a Quillen pair with this model structure.
We will prove this theorem in the next section. Note that the functors $\Lambda$ and $U$ thus form a Quillen pair with this model structure.
\subsection{Why we need $\Char{\k}=0$ for algebras}
\subsubsection{Why we need $\Char{\k}=0$ for algebras}
The above Quillen pair $(\Lambda, U)$ fails to be a Quillen pair if $\Char{\k}= p \neq0$. We will show this by proving that the maps $\Lambda(j_n)$ are not weak equivalences for even $n$. Consider $b^p \in D(n)$, then by the Leibniz rule:
The above Quillen pair $(\Lambda, U)$ fails to be a Quillen pair if $\Char{\k}= p \neq0$. We will show this by proving that the maps $\Lambda(j_n)$ are not weak equivalences for even $n$. Consider $b^p \in D(n)$, then by the Leibniz rule:
$$ d(b^p)= p \cdot c b^{p-1}=0. $$
$$ d(b^p)= p \cdot c b^{p-1}=0. $$
So $b^p$ is a cocycle. Now assume $b^p = dx$ for some $x$ of degree $pn -1$, then $x$ contains a factor $c$ for degree reasons. By the calculations above we see that any element containing $c$ has a trivial differential or has a factor $c$ in its differential, contradicting $b^p = dx$. So this cocycle is not a coboundary and $\Lambda D(n)$ is not acyclic.
So $b^p$ is a cocycle. Now assume $b^p = dx$ for some $x$ of degree $pn -1$, then $x$ contains a factor $c$ for degree reasons. By the calculations above we see that any element containing $c$ has a trivial differential or has a factor $c$ in its differential, contradicting $b^p = dx$. So this cocycle is not a coboundary and $\Lambda D(n)$ is not acyclic.
We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should be reminded of the topological $n$-simplex defined as convex span.
We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should be reminded of the topological $n$-simplex defined as convex span.
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.
Consider the free cdga on one generator $\Lambda(t, dt)$, 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:
$$ d_0, d_1 : \Lambda(t, dt)\to\k$$
$$ d_0(t)=1, \qquad d_1(t)=0, $$
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 tensorproducts as $d_0, d_1: \Lambda(t, dt)\tensor X \to\k\tensor X \tot{\iso} X$.
\Definition{cdga_homotopy}{
We call $f, g: A \to X$ homotopic ($f \simeq g$) if there is a map
$$ h: A \to\Lambda(t, dt)\tensor X, $$
such that $d_0 h = g$ and $d_1 h = f$.
}
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. 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 cofibration.
Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
\Lemma{cdga_homotopy}{
If $A$ is a cofibrant cdga and $f \simeq^r g: A \to X$, then $f \simeq g$ in the above sense.
}
\Proof{
Because $A$ is cofibrant, there is a very good homotopy $H$. Consider a lifting problem to construct a map $Path_X \to\Lambda(t, dt)\tensor X$.
}
\Corollary{cdga_homotopy_eqrel}{
For cofibrant $A$, $\simeq$ defines a equivalence relation.
}
\Definition{cdga_homotopy_classes}{
For cofibrant $A$ define the set of equivalence classes as:
$$[A, X]=\Hom_{\CDGA_\k}(A, X)/\simeq. $$
}
The results from model categories immediately imply the following results.
\Corollary{cdga_homotopy_properties}{
Let $A$ be cofibrant.
\begin{itemize}
\item Let $i: A \to B$ be a trivial cofibration, then the induced map $i^\ast: [B, X]\to[A, X]$ is a bijection.
\item Let $p: X \to Y$ be a trivial fibration, then the induced map $p_\ast: [A, X]\to[A, Y]$ is a bijection.
\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:
$$ f_\ast: [Z, A]\tot{\iso}[Z, X], $$
$$ f^\ast: [X, Z]\tot{\iso}[A, X]. $$
\end{itemize}
}
\Lemma{cdga_homotopy_homology}{
Let $f, g: A \to X$ be two homotopic maps, then $H(f)= H(g): HA \to HX$.
A model category induces a homotopy category $\Ho(\cat{C})$, in which weak equivalences are isomorphisms and homotopic maps are equal. This category only depens on the category $\cat{C}$ and the class of weak equivalences.
\todo{Definition etc}
\subsection{Quillen pairs}
\subsection{Quillen pairs}
In order to relate model categories and their associated homotopy categories we need a notion of maps between them. We want the maps such that they induce maps on the homotopy categories.
In order to relate model categories and their associated homotopy categories we need a notion of maps between them. We want the maps such that they induce maps on the homotopy categories.
\section{Model structure on \texorpdfstring{$\CDGA_\k$}{CDGA}}
.\TODO{First discuss the model structure on (co)chain complexes. Then discuss that we want the adjunction $(\Lambda, U)$ to be a Quillen pair. Then state that (co)chain complexes are cofib. generated, so we can cofib. generate CDGAs.}
\label{sec:model-of-cdga}
\TODO{First discuss the model structure on (co)chain complexes. Then discuss that we want the adjunction $(\Lambda, U)$ to be a Quillen pair. Then state that (co)chain complexes are cofib. generated, so we can cofib. generate CDGAs.}
In this section we will define a model structure on CDGAs over a field $\k$ of characteristic zero\todo{Can $\k$ be a c. ring here?}, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations.
In this section we will define a model structure on CDGAs over a field $\k$ of characteristic zero\todo{Can $\k$ be a c. ring here?}, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations.
@ -109,53 +106,3 @@ where $i$ is the obvious inclusion $i(a) = a \tensor 1$ and $p$ maps (products o
\begin{corollary}
\begin{corollary}
[MC5b] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a cofibration and $p$ a trivial fibration.
[MC5b] A map $f: A \to X$ can be factorized as $f = pi$ where $i$ is a cofibration and $p$ a trivial fibration.
\end{corollary}
\end{corollary}
\subsection{Homotopy relation on \texorpdfstring{$\CDGA_\k$}{CDGA}}
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.
Consider the free cdga on one generator $\Lambda(t, dt)$, this can be thought of as the (dual) unit interval. Indeed there is an isomorphism $\Lambda(t, dt)\iso\Apl_1$ and so we have maps for the two endpoint: $d_0, d_1: \Lambda(t, dt)\to\k\iso\Apl_0$. Given a cdga $X$ we will consider $d_0, d_1: \Lambda(t, dt)\tensor X \to\k\tensor X \iso X$.
\Definition{cdga_homotopy}{
We call $f, g: A \to X$ homotopic ($f \simeq g$) if there is a map
$$ h: A \to\Lambda(t, dt)\tensor X, $$
such that $d_0 h = g$ and $d_1 h = f$.
}
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, first note that $\Lambda(t, dt)\tensor X \to X \oplus X$ is surjective (for $(x, y)\in X \oplus X$ take $t \tensor x +1\tensor y$), secondly $\Apl_0\to\Apl_1$ is a cofibration and so is $X \to\Lambda(t, dt)\tensor X$.
Clearly we have that $f \simeq g$ implies $f \simeq^r g$ (see \DefinitionRef{right_homotopy}), however the converse need not be true.
\Lemma{cdga_homotopy}{
If $A$ is a cofibrant cdga and $f \simeq^r g: A \to X$, then $f \simeq g$ in the above sense.
}
\Proof{
Because $A$ is cofibrant, there is a very good homotopy $H$. Consider a lifting problem to construct a map $Path_X \to\Lambda(t, dt)\tensor X$.
}
\Corollary{cdga_homotopy_eqrel}{
For cofibrant $A$, $\simeq$ defines a equivalence relation.
}
\Definition{cdga_homotopy_classes}{
For cofibrant $A$ define the set of equivalence classes as:
$$[A, X]=\Hom_{\CDGA_\k}(A, X)/\simeq. $$
}
The results from model categories immediately imply the following results.
\Corollary{cdga_homotopy_properties}{
Let $A$ be cofibrant.
\begin{itemize}
\item Let $i: A \to B$ be a trivial cofibration, then the induced map $i^\ast: [B, X]\to[A, X]$ is a bijection.
\item Let $p: X \to Y$ be a trivial fibration, then the induced map $p_\ast: [A, X]\to[A, Y]$ is a bijection.
\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:
$$ f_\ast: [Z, A]\tot{\iso}[Z, X], $$
$$ f^\ast: [X, Z]\tot{\iso}[A, X]. $$
\end{itemize}
}
\Lemma{cdga_homotopy_homology}{
Let $f, g: A \to X$ be two homotopic maps, then $H(f)= H(g): HA \to HX$.
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 \Def{Kan extensions}.
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 \Def{Kan extensions}.
Given a category $\cat{C}$ and a functor $F: \DELTA\to\cat{C}$, then define the following on objects:
Given a category $\cat{C}$ and a functor $F: \DELTA\to\cat{C}$, then define the following on objects:
@ -26,7 +23,7 @@ In our case where $F = \Apl$ and $\cat{C} = \CDGA_\k$ we get:
\cdiagram{Apl_Extension}
\cdiagram{Apl_Extension}
\subsection{The cochain complex of polynomial forms}
\subsubsection{The cochain complex of polynomial forms}
In our case we take the opposite category, so the definition of $A$ is in terms of a limit instead of colimit. This allows us to give a nicer description:
In our case we take the opposite category, so the definition of $A$ is in terms of a limit instead of colimit. This allows us to give a nicer description:
@ -43,7 +40,7 @@ where the addition, multiplication and differential are defined pointwise. Concl
\end{align*}
\end{align*}
\subsection{The singular cochain complex}
\subsubsection{The singular cochain complex}
Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's \todo{Choose: normalized or not?}:
Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's \todo{Choose: normalized or not?}:
$$ C_n = C^\ast(\Delta^n; \k). $$
$$ C_n = C^\ast(\Delta^n; \k). $$
@ -54,7 +51,7 @@ The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n
where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X)\we C^\ast(X)$ for every $X \in\sSet$. Furthermore this map preserves multiplication on the homology algebras.
where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X)\we C^\ast(X)$ for every $X \in\sSet$. Furthermore this map preserves multiplication on the homology algebras.
\subsection{Integration and Stokes' theorem for polynomial forms}
\subsubsection{Integration and Stokes' theorem for polynomial forms}
In this section we will prove that the singular cochain complex is quasi isomorphic to the cochain complex of polynomial forms. In order to do so we will define an integration map $\int_n : \Apl_n^n \to\k$, which will induce a map $\oint_n : \Apl_n \to C_n$. For the simplices $\Delta[n]$ we already showed the cohomology agrees by the acyclicity of $\Apl_n = A(\Delta[n])$ (\LemmaRef{apl-acyclic}).
In this section we will prove that the singular cochain complex is quasi isomorphic to the cochain complex of polynomial forms. In order to do so we will define an integration map $\int_n : \Apl_n^n \to\k$, which will induce a map $\oint_n : \Apl_n \to C_n$. For the simplices $\Delta[n]$ we already showed the cohomology agrees by the acyclicity of $\Apl_n = A(\Delta[n])$ (\LemmaRef{apl-acyclic}).
Some general notation: \todo{leave this out, or define somewhere else?}
\section*{Preliminaries}
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 assumed, but the reader may review this in the appendices.
Some notation:
\begin{itemize}
\item$\k$ will denote an arbitrary commutative ring (or field, if indicated at the start of a section).
\item$\cat{C}$ will denote an arbitrary category.
\item$\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category $\cat{C}$.
\item$\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. We may leave out the subscript $\cat{C}$.
\end{itemize}
Some categories:
\begin{itemize}
\begin{itemize}
\item$\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category.
\item$\Top$: category of topological spaces and continuos maps.
\item$\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$.
\item$\Ab$: category of abelian groups and group homomorphisms.
\item$\DELTA$: category of simplices (i.e. finite, non-empty ordinals) and order preserving maps.
\item$\sSet$: category of simplicial sets and simplicial maps (more generally we have the category of simplicial objects, $\cat{sC}$, for any category $\cat{C}$).
\item$\Ch{\k}, \CoCh{\k}$: category of non-negatively graded chain (resp. cochain) complexes and chain maps.
\item$\DGA_\k$: category of non-negatively differential graded algebras over $\k$ (these are cochain complexes with a multiplication) and graded algebra maps. As a shorthand we will refer to such an object as \emph{dga}.
\item$\CDGA_\k$: the full subcategory of $\DGA_\k$ of commutative dga's (cdga's).
Joshua Moerman
10 years ago