Thesis: again more improvements

thesis/3_SimplicialAbelianGroups.tex

@@ -11,7 +11,7 @@ Before defining \emph{simplicial abelian groups}, we will first discuss the more
 There are two special kinds of maps in $\DELTA$, the so called \emph{face} maps and \emph{degeneracy} maps. The \emph{$i$-th face maps} $\delta_i: [n-1] \to [n]$ is the unique injective monotone function which \emph{omits} $i$. More precisely, it is defined for all $n \in \Np$ as (note that we do not explicitly denote $n$ in this notation)
 $$ \delta_i: [n-1] \to [n], k \mapsto \begin{cases} k & \text{if } k < i,\\ k+1 & \text{if } k \geq i, \end{cases} \hspace{1.0cm} 0 \leq i \leq n. $$
 
-The \emph{$i$-th degeneracy map} $\sigma_i: [n+1] \to [n]$ is the unique surjective monotone function which \emph{hits $i$ twice}. More precisely it is defined for all $n \in \N$ as:
+The \emph{$i$-th degeneracy map} $\sigma_i: [n+1] \to [n]$ is the unique surjective monotone function which \emph{hits $i$ twice}. More precisely it is defined for all $n \in \N$ as
 $$ \sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i,\\ k-1 & \text{if } k > i, \end{cases} \hspace{1.0cm} 0 \leq i \leq n. $$
 
 The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of such maps. So in a sense, these are all the maps we need to consider.
@@ -139,7 +139,7 @@ Recall that for any category $\cat{C}$ we have the $\mathbf{Hom}$-functor $\Hom{
 Note that $\Delta[-]: \DELTA \to \sSet$ is exactly the Yoneda embedding. In a moment we will see why the Yoneda lemma is useful to us, but let us first explicitly describe two examples of such standard simplices.
 
 \begin{example}
-	We will compute how $\Delta[0]$ look like. Note that $[0]$ is an one-element set, so for any set $S$, there is only one function $\ast : S \to [0]$. Hence $\Delta[0]_n = \{\ast\}$ for all $n$ and the face and degeneracy maps are necessarily the identity maps $\id : \{\ast\} \to \{\ast\}$. Thus, $\Delta[0]$ looks like:
+	We will compute how $\Delta[0]$ look like. Note that $[0]$ is an one-element set, so for any set $S$, there is only one function $\ast: S \to [0]$. Hence $\Delta[0]_n = \{\ast\}$ for all $n$ and the face and degeneracy maps are necessarily the identity maps $\id: \{\ast\} \to \{\ast\}$. Thus, $\Delta[0]$ looks like
 	$$ \Delta[0] :=
 	\begin{tikzpicture}[baseline=-0.5ex]
 	\matrix (m) [matrix of math nodes] { 
@@ -163,7 +163,7 @@ Note that $\Delta[-]: \DELTA \to \sSet$ is exactly the Yoneda embedding. In a mo
 
 	For $\Delta[1]_0$ we have to consider maps from $[0]$ to $[1]$, we cannot first apply degeneracy maps (there is no object $[-1]$). So this leaves us with the face maps: $\Delta[1]_0 = \{\delta_0, \delta_1\}$. For $\Delta[1]_1$ we of course have the identity function and two functions $\delta_0\sigma_0, \delta_1\sigma_0$. Now $\Delta[1]_2$ are the maps from $[2]$ to $[1]$.
 
-	We will compute the two face maps $d_0$ and $d_1$ from $\Delta[1]_1$ to $\Delta[1]_0$. Recall that the $\mathbf{Hom}$-functor in the first argument (the contravariant argument) works with precomposition. So this gives:
+	We will compute the two face maps $d_0$ and $d_1$ from $\Delta[1]_1$ to $\Delta[1]_0$. Recall that the $\mathbf{Hom}$-functor in the first argument (the contravariant argument) works with precomposition. So this gives
 	\begin{align*}
 		d_0(\id) &= \id \delta_0 = \delta_0 \\
 		d_0(\delta_0\sigma_0) &= \delta_0 \sigma_0 \delta_0 = \delta_0 \\
@@ -193,13 +193,13 @@ Note that $\Delta[-]: \DELTA \to \sSet$ is exactly the Yoneda embedding. In a mo
 \end{example}
 
 \subsection{Simplicial objects in arbitrary categories}
-Of course the definition of simplicial set can easily be generalized to other categories. For any category $\cat{C}$ we can consider the functor category $\cat{sC} = \cat{C}^{\DELTA^{op}}$. In this thesis we are interested in the category of simplicial abelian groups:
+Of course the definition of simplicial set can easily be generalized to other categories. For any category $\cat{C}$ we can consider the functor category $\cat{sC} = \cat{C}^{\DELTA^{op}}$. In this thesis we are interested in the category of \emph{simplicial abelian groups}:
 $$ \sAb = \Ab^{\DELTA^{op}}. $$
 So a simplicial abelian group $A$ is a collection of abelian groups $A_n$, together with face and degeneracy maps, which in this case means group homomorphisms $d_i$ and $s_i$ such that the simplicial equations hold.
 
 Note that the set of natural transformations between two simplicial abelian groups $A$ and $B$ is also an abelian group. The proof that $\sAb$ is a preadditive category is very similar to the proof we saw in Section~\ref{sec:Chain Complexes}. For two natural transformations $f,g: A \to B$ we simply define $f+g$ pointwise by $(f+g)_n = f_n + g_n$ and it is easily checked that this is a natural transformation.
 
-As we are interested in simplicial abelian groups, it would be nice to make these standard $n$-simplices into simplicial abelian groups. We have seen how to make an abelian group out of any set using the free abelian group functor. We can use this functor $\Z[-] : \Set \to \Ab$ to induce a functor $\Z^\ast[-] : \sSet \to \sAb$ as shown in the following diagram.
+As we are interested in simplicial abelian groups, it would be nice to obtain simplicial abelian groups associated to the standard $n$-simplices. We have seen how to make an abelian group out of any set using the free abelian group functor. We can use this functor $\Z[-]: \Set \to \Ab$ to induce a functor $\Z^\ast[-]: \sSet \to \sAb$ as shown in the following diagram.
 \begin{figure}[h!]
 	\begin{tikzpicture}
 		\matrix (m) [matrix of math nodes]{
@@ -268,10 +268,10 @@ telling us that we can regard $n$-simplices in $X$ as maps from $\Delta[n]$ to $
 \begin{proof}
 	By using the (non-additive) Yoneda lemma and the fact that $\Z$ is a left adjoint, we already have a natural bijection:
 	$$ \Hom{\sAb}{\Z[\Delta[n]]}{A} \iso \Hom{\sSet}{\Delta[n]}{U(A)} \iso U(A)_n = A_n. $$
-	The only thing that we need to check is that this bijection preserves the group structure. Recall that this bijection from $\Hom{\sAb}{\Z[\Delta[n]]}{A}$ to $A_n$ is given by (where $\id = \id_{[n]}$ is a generator in $\Z[\Delta[n]]$):
+	The only thing that we need to check is that this bijection preserves the group structure. Recall that this bijection from $\Hom{\sAb}{\Z[\Delta[n]]}{A}$ to $A_n$ is given by (where $\id = \id_{[n]}$ is a generator in $\Z[\Delta[n]]$)
 	$$ \phi(f) = f_n(\id) \in X_n \quad\text{ for } f: \Delta[n] \to X. $$
 
-	Now let $A$ be a simplicial abelian group and $f, g: \Z\Delta[n] \to A$ maps. Then we compute:
+	Now let $A$ be a simplicial abelian group and $f, g: \Z\Delta[n] \to A$ maps. Then we compute
 	$$ \phi(f) + \phi(g) = f_n(\id) + g_n(\id) = (f_n + g_n)(\id) = (f+g)_n(\id) = \phi(f+g), $$
 	where we regard $\id \in \Delta[n]$ as an element $\id \in \Z\Delta[n]$, we can do so by the unit of the adjunction. So this bijection is also a group homomorphism, hence we have an isomorphism $\Hom{\sAb}{\Z[\Delta[n]]}{A} \iso A_n$ of abelian groups.
 \end{proof}

thesis/4_Constructions.tex

@@ -4,8 +4,8 @@
 Comparing chain complexes and simplicial abelian groups, one sees a certain similarity. Both concepts are defined as sequences of abelian groups with certain structure maps. At first sight simplicial abelian groups seem to have a richer structure. There are many face maps as opposed to only a single boundary homomorphism. Nevertheless, as we will show in this section, these two concepts give rise to equivalent categories.
 
 \subsection{Unnormalized chain complex}
-Given a simplicial abelian group $A$, we have a family of abelian groups $A_n$. For every $n>0$ we define a group homomorphism $\del_n : A_n \to A_{n-1}$
+Given a simplicial abelian group $A$, we have a family of abelian groups $A_n$. For every $n>0$ we define a group homomorphism
-$$\del_n = d_0 - d_1 + \ldots + (-1)^n d_n.$$
+$$\del_n = d_0 - d_1 + \ldots + (-1)^n d_n: A_n \to A_{n-1}.$$
 \begin{lemma}
 	Using $A_n$ as the family of abelian groups and the maps $\del_n$ as boundary maps gives a chain complex.
 \end{lemma}
@@ -21,9 +21,18 @@ $$\del_n = d_0 - d_1 + \ldots + (-1)^n d_n.$$
 	In this calculation we did the following. We split the inner sum in two halves \refeqn{1} and we use the simplicial equations on the second sum \refeqn{2}. Then we do a shift of indices \refeqn{3}. By interchanging the roles of $i$ and $j$ in the second sum, we have two equal sums which cancel out. So indeed this is a chain complex.
 \end{proof}
 
-This construction defines a functor $M : \sAb \to \Ch{\Ab}$. And in fact we already used it in the construction of the singular chain complex, where we defined the boundary maps (on generators) as $\del(\sigma) = \sigma \circ d_0 - \sigma \circ d_1 + \ldots + (-1)^{n+1} \sigma \circ d_{n+1}$. We will briefly come back to this in Section~\ref{sec:Homotopy}.
+Thus, associated to a simplicial abelian group $A$ we obtain a chain complex $M(A)$ with $M(A)_n = A_n$ and the boundary operator as above. This construction defines a functor
+$$ M: \sAb \to \Ch{\Ab} $$
+by assigning $M(f)_n = f_n$ for a natural transformation $f: A \to B$. It follows from a nice calculation that $M(f)$ is indeed a chain map:
+\begin{align*}
+	f_{n-1} \circ \del &= f_{n-1} \circ (d_0 - d_1 + \ldots + (-1)^n d_n) \\
+		&= f_{n-1} \circ d_0 - f_{n-1} \circ d_1 + \ldots + (-1)^n f_{n-1} \circ d_n \\
+		&\eqn{1} d_0 \circ f_n - d_1 \circ f_n + \ldots + (-1)^n d_n \circ f_n \\
+		&= (d_0 - d_1 + \ldots + (-1)^n d_n) \circ f_n = \del \circ f_n,
+\end{align*}
+where we used naturality of $f$ in step \refeqn{1}. This functor is in fact already used in the construction of the singular chain complex, where we defined the boundary maps (on generators) as $\del(\sigma) = \sigma \circ d_0 - \sigma \circ d_1 + \ldots + (-1)^{n+1} \sigma \circ d_{n+1}$. We will briefly come back to this in Section~\ref{sec:Homotopy}.
 
-Let us investigate whether this functor can be used for our sought equivalence. For a functor from $\Ch{\Ab}$ to $\sAb$ we cannot simply take the same collection of abelian groups. This is due to the fact that the degeneracy maps should be injective. This means that for a simplicial abelian group $A$, if we know $A_n$ is non-trivial, then all $A_m$ for $m > n$ are also non-trivial.
+Let us investigate whether this functor $M$ can be part of an equivalence. For a functor from $\Ch{\Ab}$ to $\sAb$ we cannot simply take the same collection of abelian groups. This is due to the fact that the degeneracy maps should be injective. This means that for a simplicial abelian group $A$, if we know $A_n$ is non-trivial, then all $A_m$ for $m > n$ are also non-trivial.
 
 But for chain complexes it \emph{is} possible to have trivial abelian groups $C_m$, while there is a $n < m$ with $C_n$ non-trivial. Take for example the chain complex
 $$ C = \ldots \to 0 \to 0 \to \Z. $$
@@ -125,9 +134,9 @@ We can extend the above lemmas to a more general statement.
 
 \begin{lemma}
 	\label{le:decomp3}
-	For all $x \in X_n$ we can write $x$ as:
+	For all $x \in X_n$ we can write $x$ as
 	$$ x = \sum_\beta \beta^\ast (x_\beta), $$
-	for certain $x_\beta \in N(X)_p$ and $\beta : [n] \epi [p]$.
+	for certain $x_\beta \in N(X)_p$, where $\beta$ ranges over all surjective functions $\beta : [n] \epi [p]$.
 \end{lemma}
 \begin{proof}
 	We will proof this using induction on $n$. For $n=0$ the statement is clear because $N(X)_0 = X_0$.
@@ -157,7 +166,7 @@ And by considering $X_n$ as a whole we get:
 
 Using Corollary~\ref{cor:decomp} we can prove a nice categorical fact about $N$, which we will use later on.
 \begin{lemma}
-	The functor $N$ is fully faithful, i.e.:
+	The functor $N$ is fully faithful, i.e.
 	$$ N: \Hom{\sAb}{A}{B} \iso \Hom{\Ch{\Ab}}{N(A)}{N(B)} \quad A, B \in \sAb. $$
 \end{lemma}
 \begin{proof}
@@ -188,7 +197,7 @@ Now this is a functor, because it is a composition of functors. Furthermore it i
 $$ \Hom{A}{F \Z^{\ast} \Delta (-)}{-}: A \to \sAb $$
 where we are supposed to fill in the second argument first, leaving us with a simplicial abelian group.
 
-Now we know that $\Ch{\Ab}$ is an abelian group and we have actually two functors $C, N : \sAb \to \Ch{\Ab}$, so we now have functors from $\Ch{\Ab} \to \sAb$. Of course we will be interested in the one using $N$. So we define the functor:
+Now we know that $\Ch{\Ab}$ is an abelian group and we have actually two functors $M, N : \sAb \to \Ch{\Ab}$, so we now have functors from $\Ch{\Ab} \to \sAb$. Of course we will be interested in the one using $N$. So we define the functor:
 $$ K(C) = \Hom{\Ch{\Ab}}{N\Z^\ast\Delta[-]}{C} \in \sAb. $$
 This definition is very abstract, but luckily we can also give a more explicit definition. By writing it out for low dimensions we see:
 

thesis/5_Homotopy.tex

@@ -3,7 +3,7 @@
 
 We have already seen homology in chain complexes. We can of course now translate this notion to simplicial abelian groups, by assigning a simplicial abelian group $X$ to $H_n(N(X))$. But there is a more general notion of homotopy for simplicial sets, which is also similar to the notion of homotopy in topology. We will define the notion of homotopy groups for simplicial sets.
 
-When dealing with homotopy groups in a topological space $X$ we always need a base-point $\ast \in X$. This is also the case for simplicial sets. We will notate the chosen base-point of a simplicial set $X$ with $\ast \in X_0$. Note that it is a $0$-simplex, but in fact the base-point is present in all sets $X_n$, because we can consider its degenerate simplices $s_0(\ldots(s_0(\ast))\ldots) \in X_n$, we will also denote these elements as $\ast$. Of course in our situation we are concerned with simplicial abelian groups, where there is an obvious choice for the base-point given by the neutral element $0$.
+When dealing with homotopy groups in a topological space $X$ we always need a base-point $\ast \in X$. This is also the case for simplicial sets. We will notate the chosen base-point of a simplicial set $X$ with $\ast \in X_0$. More formally, a \emph{pointed simplicial set} $(X, \ast)$ is a simplicial set $X$ together with a $0$-simplex $\ast \in X_0$. By the Yoneda lemma this $0$-simplex corresponds to a map $\Delta[0] \to X$, and any simplex in the image will be denoted by $\ast$. Another way of saying this is that we denote the degenerate simplices $s_0(\ldots(s_0(\ast))\ldots) \in X_n$ as $\ast$. Of course in our situation we are concerned with simplicial abelian groups, where there is an obvious choice for the base-point given by the neutral element $0$.
 
 \subsection{Homotopy groups}
 \begin{definition}
@@ -58,7 +58,7 @@ Before proving this, one should have a look at figure~\ref{fig:simplicial_eqrel}
 \end{proof}
 
 \begin{definition}
-	Given a simplicial abelian group $X$, we define the $n$-th homotopy group as:
+	Given a simplicial abelian group $X$, we define the $n$-th homotopy group as
 	$$ \pi_n(X) = Z_n(X) / \sim. $$
 \end{definition}
 
@@ -83,7 +83,7 @@ Note that this is an abelian group, because $Z_n(X)$ is a subgroup of $X_n$, and
 \end{proof}
 
 \begin{corollary}
-	For a chain complex $C$ we have $H_n(C) \iso \pi_n(K(C))$
+	For a chain complex $C$ we have $H_n(C) \iso \pi_n(K(C))$.
 \end{corollary}
 \begin{proof}
 	By the established equivalence we have for any chain complex $C$:
@@ -91,9 +91,13 @@ Note that this is an abelian group, because $Z_n(X)$ is a subgroup of $X_n$, and
 \end{proof}
 
 \subsection{Topology}
-In Section~\ref{sec:Constructions} we saw that we can construct a functor $G: \cat{C} \to \sSet$ if we are provided a functor the other way around. If we can define a functor $F: \DELTA \to \Top$, then for any space $X$ we have a simplicial set $\Hom{\Top}{F-}{X}: \DELTA^{op} \to \Set$. In Section~\ref{sec:Chain Complexes}, we already defined the \emph{topological $n$-simplex} $\Delta^n$ and face maps $\delta^i : \Delta^n \mono \Delta^{n+1}$. We can similarly define degeneracy maps $s^i: \Delta^n \to \Delta^{n-1}$ as:
+In Section~\ref{sec:Chain Complexes}, we already defined the topological $n$-simplex $\Delta^n \in \Top$. We will now relate these spaces to the standard $n$-simplices $\Delta[n] \in \sSet$. We will define a functor $\Delta^-: \DELTA \to \Top$ as follows
-$$ s^i(x_0, \ldots, x_n) = (x_0, \ldots, x_i + x_{i+1}, \ldots, x_n) \in \Delta^{n-1}. $$
+\begin{gather*}
-The reader is invited to check the cosimplicial identities himself and conclude that we now have a functor $F: \DELTA \to \Top$, and hence we have a functor $S: \Top \to \sSet$ given by:
+	\Delta^-([n]) = \Delta^n = \{(x_0, x_1, \ldots, x_n) \in \R^{n+1} \I x_i \geq 0 \text{ and } x_0 + \ldots + x_n = 1 \}, \\
+	\Delta^-(\delta_i) (x_0, \ldots, x_n) = (x_0, \ldots, x_{i-1}, 0, x_{i}, \ldots, x_n), \\
+	\Delta^-(\sigma_i) (x_0, \ldots, x_n) = (x_0, \ldots, x_{i} + x_{i+1}, \ldots, x_n).
+\end{gather*}
+The definition of $\Delta^-(\delta_i)$ was already defined in Section~\ref{sec:Chain Complexes} as the face maps $\delta^i: \Delta^n \to \Delta^{n+1}$. So in addition we defined degeneracy maps. The reader is invited to check the cosimplicial identities himself and conclude that we now have a functor $\Delta^-: \DELTA \to \Top$. By composing this with the $\mathbf{Hom}$-functor we obtain a functor $S: \Top \to \sSet$ given by
 $$ \text{Sing}(X)_n = \Hom{\Top}{\Delta^n}{X}. $$
 
 Recall construction of the singular chain complex in Section~\ref{sec:Chain Complexes}: