Thesis: more improvements

thesis/1_CategoryTheory.tex

@@ -109,12 +109,12 @@ With the notion of isomorphisms between functors we can generalize this, and onl
 	for any $X \in \cat{C}$ and $Y \in \cat{D}$. The functor $F$ is called the \emph{left adjoint} and $G$ the \emph{right adjoint}.
 \end{definition}
 
-Now there are different definitions of adjunctions, which are equivalent. We will not prove that these are equivalent. One can find the proof in for example in the books of Mac Lane \cite{maclane} or Awodey \cite{awodey}. A particular nice one is the following:
+There are different equivalent descriptions of adjunctions. A particular nice one will be recalled. For a proof of equivalence to the above definition we refer to books on category theory such as the one of Mac Lane \cite{maclane} or Awodey \cite{awodey}.
 
-\begin{definition}
-	An adjunction between two categories $\cat{C}$ and $\cat{D}$ consists of two functors $F:\cat{C} \to \cat{D}$, $G: \cat{D} \to \cat{C}$ and a natural transformation, called the \emph{unit}:
+\begin{lemma}
+	Given functors $F:\cat{C} \to \cat{D}$, $G: \cat{D} \to \cat{C}$ then $F$ is a left adjoint and $G$ a right adjoint if and only if there exists a natural transformation, called the \emph{unit}
 	$$ \eta : \id_\cat{C} \to GF. $$
-	Such that for any map $f: S \to G(A)$ (in $\cat{C}$), there is a unique map $\overline{f}: F(S) \to A$ (in $\cat{D}$) such that $G(\overline{f}) \circ \eta = f$. I.e.:
+	such that for any map $f: S \to G(A)$ (in $\cat{C}$), there is a unique map $\overline{f}: F(S) \to A$ (in $\cat{D}$) such that $G(\overline{f}) \circ \eta = f$. I.e.:
 \begin{center}
 	\begin{tikzpicture}
 		\matrix (m) [matrix of math nodes]{
@@ -129,7 +129,7 @@ Now there are different definitions of adjunctions, which are equivalent. We wil
 		(m-1-3) edge node[auto] {$ \overline{f} $} (m-2-3);
 	\end{tikzpicture}
 \end{center}
-\end{definition}
+\end{lemma}
 
 Note that by considering the identity map $\id : G(A) \to G(A)$ in $\cat{C}$, we get a uniquely determined map $\overline{\id}:FG(A) \to A$. This map $FG(A) \to A$ is in fact natural in $A$, this natural transformation is called the \emph{co-unit}
 $$ \eps: FG \to \id. $$
@@ -141,17 +141,20 @@ The first definition of adjunction is useful when dealing with maps, since it gi
 \begin{example}
 	\emph{(Free abelian groups)} There is an obvious functor $U: \Ab \to \Set$, which sends an abelian group to its underlying set, forgetting the additional structure. It is hence called a \emph{forgetful functor}. This functor has a left adjoint $\Z[-]: \Set \to \Ab$  given by the \emph{free abelian group functor}. For a set $S$ define
 	$$ \Z[S] = \{ \phi: S \to \Z \I \text{supp}(\phi) \text{ is finite}\}, $$
-	where $\text{supp}(\phi) = \{ s \in S \I \phi(s) \neq 0 \}$. The group structure on $\Z[S]$ is given by pointwise addition. One can think of elements of this abelian group as formal sums, namely:
-	$$ \text{for } \phi \in F(S),\, \phi = \sum_{x \in \text{supp}(\phi)}\phi(x) x, $$
-	in other words $\Z[S]$ consists of linear combinations of elements in $S$.
-
-	There is a map $\eta: S \to U\Z[S]$ given by:
-	$$ \eta(s)(t) =
+	where $\text{supp}(\phi) = \{ s \in S \I \phi(s) \neq 0 \}$. The group structure on $\Z[S]$ is given by pointwise addition. We can define an element $e_s \in \Z[S]$ for every element $s \in S$ as
+	$$ e_s(t) =
 	\begin{cases}
 		1 \text{ if } s = t \\
 		0 \text{ otherwise}
 	\end{cases}. $$
 
+	One can think of elements of this abelian group as formal sums, namely:
+	$$ \text{for } \phi \in F(S),\, \phi = \sum_{x \in \text{supp}(\phi)}\phi(x) e_x, $$
+	in other words $\Z[S]$ consists of linear combinations of elements in $S$. The functor $\Z[-]$ is defined on functions as follows. Let $f: S \to T$ be a function, then define
+	$$ \Z[f](\phi) = \sum_{x \in \text{supp}(\phi)} \phi(x) e_{f(x)} \quad\text{for all } \phi \in \Z[S]. $$
+	It is left for the reader to check that this indeed gives a group homomorphism and that the functor laws hold. There is a map $\eta: S \to U\Z[S]$ given by
+	$$ \eta(s) = e_s. $$
+
 	And given any map $f: S \to U(A)$ for any abelian group $A$, we can define
 	$$ \overline{f}(\phi) = \sum_{x \in \text{supp}(\phi)} \phi(x) \cdot f(x). $$
 

thesis/3_SimplicialAbelianGroups.tex

@@ -95,8 +95,8 @@ Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain relatio
 
 Note that these cosimplicial identities are ``purely categorical'', i.e. they only use compositions and identity maps. Because a simplicial set $X$ is a contravariant functor, dual versions of these equations hold in its image. For example, the first equation corresponds to $X(\delta_i)X(\delta_j) = X(\delta_{j-1})X(\delta_i)$ for $i < j$. This can be used for an explicit definition of simplicial sets. In this definition a simplicial set $X$ consists of a collection of sets $X_n$ together with face and degeneracy maps. More precisely:
 
-\begin{definition}
-	\emph{(Explicitly)} An simplicial set $X$ consists of a collection sets $X_n$ together with functions $d_i: X_n \to X_{n-1}$ and $s_i: X_n \to X_{n+1}$ for $0 \leq i \leq n$ and $n \in \N$, such that the simplicial identities hold
+\begin{lemma}
+	A simplicial set $X$ is equivalently specified by a collection sets $X_n$, $n \in \N$, together with functions $d_i: X_n \to X_{n-1}$ and $s_i: X_n \to X_{n+1}$ for $0 \leq i \leq n$ and $n \in \N$, such that the simplicial identities hold
 	\begin{align}
 		d_i d_j &= d_{j-1} d_i,  \hspace{1.5cm} \text{ if } i < j,\\
 		d_i s_j &= s_{j-1} d_i,  \hspace{1.5cm} \text{ if } i < j,\\
@@ -104,9 +104,11 @@ Note that these cosimplicial identities are ``purely categorical'', i.e. they on
 		d_i s_j &= s_j d_{i-1},  \hspace{1.5cm} \text{ if } i > j+1,\\
 		s_i s_j &= s_{j+1} s_i,  \hspace{1.5cm} \text{ if } i \leq j.
 	\end{align}
-\end{definition}
+\end{lemma}
 
-It is already indicated that a functor from $\DELTA^{op}$ to $\Set$ is determined when the images for the face and degeneracy maps in $\DELTA$ are provided. So this gives a way of restoring the first definition from this one. Conversely, we can apply functoriality to obtain the second definition from the first. So these definitions are the same. From now on we will denote $X([n])$ by $X_n$, $X(\sigma_i)$ by $s_i$ and $X(\delta_i)$ by $d_i$, whenever we have a simplicial set $X$. For any other map $\beta : [n] \to [p]$ we will denote the induced map by $\beta^\ast: X_p \to X_n$.
+It is already indicated that a functor from $\DELTA^{op}$ to $\Set$ is determined when the images for the face and degeneracy maps in $\DELTA$ are provided. So this gives a way of restoring the definition from this specification. Conversely, we can apply functoriality to obtain this specification from the definition. We will not give the proof in more detail. From now on we will use the following notation for a simplicial set $X$:
+$$ X_n = X([n]), \quad s_i = X(\sigma_i) \quad\text{and}\quad d_i = X(\delta_i). $$
+For any other map $\beta : [n] \to [p]$ we will denote the induced map by $\beta^\ast: X_p \to X_n$.
 
 When using a simplicial set to construct another object, it is often handy to use this second definition, as it gives you a very concrete objects to work with. On the other hand, constructing this might be hard (as you would need to provide a lot of details), in this case we will often use the more abstract definition.