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Continues with Kan extensions. Adds stub about sing cochain. Proves (A,K) to be Quillen Pair (adds diagrams).

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Joshua Moerman 8 years ago
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      thesis/images/A_Preserves_WCof.png
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  7. 33
      thesis/notes/A_K_Quillen_Pair.tex
  8. 4
      thesis/notes/Algebra.tex
  9. 2
      thesis/notes/Model_Categories.tex
  10. 2
      thesis/notes/Model_Of_CDGA.tex
  11. 45
      thesis/notes/Polynomial_Forms.tex
  12. 8
      thesis/preamble.tex
  13. 3
      thesis/thesis.tex

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.gitignore

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*sublime*
*.bak

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thesis/notes/A_K_Quillen_Pair.tex

@ -0,0 +1,33 @@
\subsection{$A$ and $K$ form a Quillen pair}
We will prove that $A$ preserves cofibrations and trivial cofibrations. We only have to check this fact for the generating (trivial) cofibrations in $\sSet$. Note that the contravariance of $A$ means that a (trivial) cofibrations should be sent to a (trivial) fibration.
\begin{lemma}
$A(i) : A(\Delta[n]) \to A(\del \Delta[n])$ is surjective.
\end{lemma}
\begin{proof}
Let $\phi \in A(\del \Delta[n])$ be an element of degree $k$, hence it is a map $\del \Delta[n] \to \Apl^k$. We want to extend this to the whole simplex. By the fact that $\Apl^k$ is Kan and contractible we can find a lift $\overline{\phi}$ in the following diagram showing the surjectivity.
\cimage[scale=0.5]{Extend_Boundary_Form}
\end{proof}
\begin{lemma}
$A(j) : A(\Delta[n]) \to A(\Lambda^n_k)$ is surjective and a quasi isomorphism.
\end{lemma}
\begin{proof}
As above we get surjectivity from the Kan condition. To prove that $A(j)$ is a quasi isomorphism we pass to the singular cochain complex and use that $C^\ast(j) : C^\ast(\Delta[n]) \we C^\ast(\Lambda^n_k)$ is a quasi isomorphism. Consider the following diagram and conclude that $A(j)$ is surjective and a quasi isomorphism.
\cimage[scale=0.5]{A_Preserves_WCof}
\end{proof}
Since $A$ is a left adjoint, it preserves all colimits and by functoriality it preserves retracts. From this we can conclude the following corollary.
\begin{corollary}
$A$ preserves all cofibrations and all trivial cofibrations and hence is a left Quillen functor.
\end{corollary}
\begin{corollary}
$A$ and $K$ induce an adjunction on the homotopy categories:
$$ \Ho{\sSet} \leftadj \opCat{\Ho{\CDGA}}. $$
\end{corollary}

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thesis/notes/Algebra.tex

@ -67,8 +67,6 @@ A differential graded module $(M, d)$ with $M_i = 0$ for all $i < 0$ is a \emph{
$$ d_{M \tensor N} = d_M \tensor \id_N + \id_M \tensor d_N. $$
\end{definition}
\todo{Prove that this is in fact a differential?}
Finally we come to the definition of a differential graded algebra. This will be a graded algebra with a differential. Of course we want this to be compatible with the algebra structure, or stated differently: we want $\mu$ and $\eta$ to be chain maps.
\begin{definition}
@ -107,6 +105,8 @@ For differential graded algebras we can consider the (co)homology by forgetting
\todo{Maybe just state this?}
\end{proof}
Note that taking homology of a differential graded module (or algebra) is functorial. Whenever a map $f: M \to N$ of differential graded modules (or algebras) induces an isomorphism on homology, we say that $f$ is a \emph{quasi isomorphism}.
\TODO{Discuss:
\titem The Künneth theorem (especially in the case of fields)
\titem The tensor algebra $T : Ch^\ast(\Q) \to \DGA_\Q$ and free cdga $\Lambda : Ch^\ast(\Q) \to \CDGA_\Q$

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thesis/notes/Model_Categories.tex

@ -88,7 +88,7 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival
\TODO{Maybe some basic propositions (refer to Dwyer \& Spalinski):
\titem Over/under category (or simply pointed objects)
\titem If a map has LLP/RLP wrt fib/cof, it is a cof/fib
\titem If a map has LLP/RLP w.r.t. fib/cof, it is a cof/fib
\titem Fibs are preserved under pullbacks/limits
\titem Cofibrantly generated mod. cats.
\titem Small object argument

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thesis/notes/Model_Of_CDGA.tex

@ -8,7 +8,7 @@ In this section we will define a model structure on CDGAs over a field $\k$ of c
\begin{proposition}
There is a model structure on $\CDGA_\k$ where $f: A \to B$ is
\begin{itemize}
\item a \emph{weak equivalence} if $H(f)$ is an isomorphism,
\item a \emph{weak equivalence} if $f$ is a quasi isomorphism,
\item a \emph{fibration} if $f$ is an surjective and
\item a \emph{cofibration} if $f$ has the LLP w.r.t. trivial fibrations
\end{itemize}

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thesis/notes/Polynomial_Forms.tex

@ -5,7 +5,46 @@ There is a general way to construct functors from $\sSet$ whenever we have some
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}
F_!(X) &= \colim_{\Delta[n] \to X} F[n] & X \in \sSet \\
F^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y) & 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.
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. Secondly we see readily that $F^\ast$ is functorial. By using the definition of colimit and the Yoneda lemma (Y) we can prove that $F_!$ is left adjoint to $F^\ast$:
\begin{align*}
\Hom_\cat{C}(F_!(X), Y) &\iso \Hom_\cat{C}(\colim_{\Delta[n] \to X} F[n], Y) \iso \lim_{\Delta[n] \to X} \Hom_\cat{C}(F[n], Y) \iso \lim_{\Delta[n] \to X} F^\ast(Y)_n \\
&\stackrel{\text{Y}}{\iso} \lim_{\Delta[n] \to X} \Hom_\sSet(\Delta[n], F^\ast(Y)) \iso \Hom_\sSet(\colim_{\Delta[n] \to X} \Delta[n], F^\ast(Y)) \\
&\iso \Hom_\sSet(X, F^\ast(Y)).
\end{align*}
Furthermore we have $F_! \circ \Delta[-] = F$. In short we have the following:
\cimage[scale=0.5]{Kan_Extension}
In our case where $F = \Apl$ and $\cat{C} = \CDGA_\k$ we get:
\cimage[scale=0.5]{Apl_Extension}
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:
\begin{align*}
A(X) &= \lim_{\Delta[n] \to X} \Apl_n \stackrel{Y}{\iso} \lim_{\Delta[n] \to X} \Hom_\sSet(\Delta[n], \Apl) \iso \Hom_\sSet(\colim_{\Delta[n] \to X}\Delta[n], \Apl) \\
&= \Hom_\sSet(X, \Apl),
\end{align*}
where the addition, multiplication and differential are defined pointwise. Conclude that we have the following contravariant functors (which form an adjoint pair):
\begin{align*}
A(X) &= \Hom_\sSet(X, \Apl) & X \in \sSet \\
K(C)_n &= \Hom_{\CDGA_\k}(C, \Apl_n) & C \in \CDGA_\k.
\end{align*}
\subsection{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:
$$ C_n = C^\ast(\Delta^n; \k). $$
The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n \to \Delta^{n-1}$ induce face and degeneracy maps on the dga's $C_n$, turning $C$ into a simplicial dga. Again we can extend this to functors by Kan extensions
\cimage[scale=0.5]{C_Extension}
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.

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thesis/preamble.tex

@ -42,6 +42,7 @@
\newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category
\newcommand{\Hom}{\mathbf{Hom}}
\newcommand{\id}{\mathbf{id}}
\newcommand{\Ho}[1]{\cat{Ho(#1)}}
% Categories
\newcommand{\Set}{\cat{Set}} % sets
@ -123,3 +124,10 @@
% headings for a table
\newcommand*{\thead}[1]{\multicolumn{1}{c}{\bfseries #1}}
% simple way to center an image
\newcommand{\cimage}[2][]{
\begin{center}
\includegraphics[#1]{#2}
\end{center}
}

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thesis/thesis.tex

@ -1,4 +1,4 @@
\documentclass[a4paper, 12pt, draft]{amsart}
\documentclass[a4paper, 12pt]{amsart}
\input{style}
\input{preamble}
@ -26,6 +26,7 @@ Some general notation: \todo{leave this out, or define somewhere else?}
\input{notes/Model_Of_CDGA} \vspace{2cm}
\input{notes/CDGA_Of_Polynomials} \vspace{2cm}
\input{notes/Polynomial_Forms} \vspace{2cm}
\input{notes/A_K_Quillen_Pair} \vspace{2cm}
% \listoftodos