Cleans up algebra appendix

thesis/chapters/Applications_And_Further_Topics.tex

@@ -100,4 +100,4 @@ In this section we will prove that the rational cohomology of an H-space is free
 Let $X$ be an H-space, then we have the induced map $\mu^\ast: H^\ast(X; \Q) \to H^\ast(X; \Q) \tensor H^\ast(X; \Q)$ on cohomology. Because homotopic maps are sent to equal maps in cohomology, we get $H^\ast(\mu(x_0, -)) = \id_{H^\ast(X; \Q)}$. Now write $H^\ast(\mu(x_0, -)) = (\counit \tensor \id) \circ H^\ast(\mu)$, where $\counit$ is the augmentation induced by $x_0$, to conclude that for any $h \in H^{+}(X; \Q)$ the image is of the form
 $$ H^\ast(\mu)(h) = h \tensor 1 + 1 \tensor h + \psi, $$
 for some element $\psi \in H^{+}(X; \Q) \tensor H^{+}(X; \Q)$.
-
+\todo{continue here}

thesis/notes/Algebra.tex

@@ -6,11 +6,13 @@ In this section $\k$ will be any commutative ring. We will recap some of the bas
 
 \section{Graded algebra}
 
-\begin{definition}
+\Definition{graded-module}{
-	A \emph{graded module} $M$ is a family of modules $\{M_n\}_{n\in\Z}$. An element $x \in M_n$ is called a \emph{homogeneous element} and said to be of \emph{degree $\deg{x} = n$}. We will often identify $M = \bigoplus_{n \in \Z} M_n$.
+	A module $M$ is said to be \Def{graded} if it is equiped with a decomposition
-\end{definition}
+	\[ M = \bigoplus_{n\in\Z} M_n. \]
+	An element $x \in M_n$ is called a \Def{homogeneous element} and said to be of \Def{degree} $\deg{x} = n$.
+}
 
-For an ordinary module $M$ we can consider the graded module $M[0]$ \emph{concentrated in degree $0$} defined by setting $M[0]_0 = M$ and $M[0]_n = 0$ for $i \neq 0$. If clear from the context we will denote this graded module by $M$. In particular $\k$ is a graded module concentrated in degree $0$.
+If $M$ is just any module, it always has the trivial grading given by $M_0 = M$ and $M_i = 0$ for $i \neq 0$, i.e. $M$ is \Def{concentrated in degree 0}. In particular $\k$ itself is a graded module concentrated in degree $0$.
 
 \begin{definition}
 	A linear map $f: M \to N$ between graded modules is \emph{graded of degree $p$} if it respects the grading and raises the degree by $p$, i.e.
@@ -83,28 +85,32 @@ Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i =
 \todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary cdga's.}
 
 
-\section{Homology}
+\Remark{orthogonal-definition}{
+	Note that all the above definitions (i.e. the definitions of graded objects, algebras, differentials) are orthogonal, meaning that any combination makes sense. However, keep in mind that we require the structures to be compatible. For example, an algebra with differential should satisfy the Leibniz rule (i.e. the differential should be a map of algebras).
+}
 
-Whenever we have a differential graded module we have $d \circ d = 0$, or put in other words: the image of $d$ is a submodule of the kernel of $d$. The quotient of the two graded modules will be of interest.
 
-\begin{definition}
+\section{Homology}
-	Given a differential graded modules $(M, d)$ we define the \emph{homology} of $M$ as: $H(M, d) = \ker(d) / \im(d)$.
-	It is naturally graded as follows:
-	$$ H(M, d)_i = H_i(M, d) = \ker(\restr{d}{M_i}) / d(M_{i+1}). $$
-	If $d$ has degree $+1$ we define the \emph{cohomology} as:
-	$$ H(M, d)^i = H^i(M, d) = \ker(\restr{d}{M^i}) / d(M^{i-1}). $$
-\end{definition}
 
-For differential graded algebras we can consider the (co)homology by forgetting the multiplicative structure. However this multiplication will actually pass to (co)homology:
+Whenever we have a differential module we have $d \circ d = 0$, or put in other words: the image of $d$ is a submodule of the kernel of $d$. The quotient of the two graded modules will be of interest. Note that the following definition depends on the differential $d$, however it is often left out from the notation.
 
-\begin{lemma}
+\Definition{homology}{
-	Let $(A, d)$ be a differential graded algebra. The kernel $\ker(d)$ is a subalgebra of $A$ and the image $d(A)$ is an ideal, so that the quotient
+	Given a differential module $(M, d)$ we define the \Def{homology} of $M$ as:
-	$$ H(A) = \ker(d) / \im(d) $$
+	$$ H(M) = \ker(d) / \im(d). $$
-	is a graded algebra.
+}
-\end{lemma}
+
-\begin{proof}
+If the module has more structure as discussed above, homology will preserve this.
-	\todo{Maybe just state this?}
+\Remark{homology-preserves-structure}{
-\end{proof}
+	Let $M$ be a differential module. Then homology preserves the following.
+	\begin{itemize}
+		\item If $M$ is graded, so is $H(M)$, where the grading is given by
+		\[ H(M)_i = \ker(\restr{d}{M_i}) / d(M_{i+1}) \]
+		\item If $M$ has an algebra strucutre, then so does $H(M)$, given by
+		\[ [z_1] \cdot [z_2] = [z_1 \cdot z_2] \]
+		\item If $M$ is a commutative algebra, so is $H(M)$.
+	\end{itemize}
+}
+Of course the converses need not be true. For example the singular cochain complex asociated to a space is a graded differential algebra which is \emph{not} commutative. However, by taking homology one gets a commutative algebra.
 
 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}.
 

thesis/notes/Model_Categories.tex

@@ -70,9 +70,7 @@ This means that once we choose weak equivalences and fibrations for a category $
 
 \Lemma{model-cats-pushouts}{
 	Let $\cat{C}$ be a model category. Consider the following two diagrams where $P$ is the pushout and pullback respectively.
-
 	\cdiagram{Model_Cats_Pushouts}
-
 	\begin{itemize}
 		\item If $i$ is a (trivial) cofibration, so is $j$.
 		\item If $p$ is a (trivial) fibration, so is $q$.
@@ -111,11 +109,9 @@ Of course the most important model category is the one of topological spaces. We
 
 In this thesis we often restrict to $1$-connected spaces. The full subcategory $\Top_1$ of $1$-connected spaces satisfies MC2-MC5: the 2-out-of-3 property, retract property and lifting properties hold as we take the \emph{full} subcategory, factorizations exist as the middle space is $1$-connected as well. However $\Top_1$ does not have all limits and colimits.
 
-\Lemma{topr-no-colimit}{
+\Remark{topr-no-colimit}{
 	Let $r > 0$ and $\Top_r$ be the full subcategory of $r$-connected spaces. The diagrams
-
 	\cdiagram{Topr_No_Coequalizer}
-
 	have no coequalizer and respectively no equalizer in $\Top_r$.
 }
 
@@ -231,7 +227,7 @@ The two notions (left resp. right homotopy) agree on nice objects. Hence in this
 }
 
 
-\section{The Homotopy Category \texorpdfstring{$\Ho(\cat{C})$}{Ho(C)}}
+\section{The Homotopy Category}
 A model category induces a homotopy category $\Ho(\cat{C})$, in which weak equivalences are isomorphisms and homotopic maps are equal. This category only depends on the category $\cat{C}$ and the class of weak equivalences.
 \todo{Definition etc}