diff --git a/.gitignore b/.gitignore index f46bf71..5b09961 100644 --- a/.gitignore +++ b/.gitignore @@ -5,3 +5,5 @@ build *sublime* +*.bak + diff --git a/thesis/images/A_Preserves_WCof.png b/thesis/images/A_Preserves_WCof.png new file mode 100644 index 0000000..c869dd0 Binary files /dev/null and b/thesis/images/A_Preserves_WCof.png differ diff --git a/thesis/images/Apl_Extension.png b/thesis/images/Apl_Extension.png new file mode 100644 index 0000000..8889989 Binary files /dev/null and b/thesis/images/Apl_Extension.png differ diff --git a/thesis/images/C_Extension.png b/thesis/images/C_Extension.png new file mode 100644 index 0000000..4dbb14d Binary files /dev/null and b/thesis/images/C_Extension.png differ diff --git a/thesis/images/Extend_Boundary_Form.png b/thesis/images/Extend_Boundary_Form.png new file mode 100644 index 0000000..50e2576 Binary files /dev/null and b/thesis/images/Extend_Boundary_Form.png differ diff --git a/thesis/images/Kan_Extension.png b/thesis/images/Kan_Extension.png new file mode 100644 index 0000000..7684766 Binary files /dev/null and b/thesis/images/Kan_Extension.png differ diff --git a/thesis/notes/A_K_Quillen_Pair.tex b/thesis/notes/A_K_Quillen_Pair.tex new file mode 100644 index 0000000..db59d24 --- /dev/null +++ b/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} \ No newline at end of file diff --git a/thesis/notes/Algebra.tex b/thesis/notes/Algebra.tex index 86415a9..1c6b2b5 100644 --- a/thesis/notes/Algebra.tex +++ b/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 $\Lambda : Ch^\ast(\Q) \to \CDGA_\Q$ diff --git a/thesis/notes/Model_Categories.tex b/thesis/notes/Model_Categories.tex index 4493333..d68c0a9 100644 --- a/thesis/notes/Model_Categories.tex +++ b/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 diff --git a/thesis/notes/Model_Of_CDGA.tex b/thesis/notes/Model_Of_CDGA.tex index f6ca0cc..dc4adfb 100644 --- a/thesis/notes/Model_Of_CDGA.tex +++ b/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} diff --git a/thesis/notes/Polynomial_Forms.tex b/thesis/notes/Polynomial_Forms.tex index f88e0a6..2bb7684 100644 --- a/thesis/notes/Polynomial_Forms.tex +++ b/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. \ No newline at end of file diff --git a/thesis/preamble.tex b/thesis/preamble.tex index 7af14ff..8595e71 100644 --- a/thesis/preamble.tex +++ b/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} +} \ No newline at end of file diff --git a/thesis/thesis.tex b/thesis/thesis.tex index d186020..4bea684 100644 --- a/thesis/thesis.tex +++ b/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