@ -19,7 +19,7 @@ We will briefly define categories and functors to fix the notation. We will not
Instead of writing $f \in\Hom{\cat{C}}{A}{B}$ we write $f: A \to B$, as many categories have functions as maps. For brevity we sometimes write $gf$ instead of $g \circ f$. We will need the category $\Set$ of sets with functions, the category $\Ab$ of abelian groups with group homomorphisms and the category $\Top$ of topological spaces and continuous maps.
Instead of writing $f \in\Hom{\cat{C}}{A}{B}$ we write $f: A \to B$, as many categories have functions as maps. For brevity we sometimes write $gf$ instead of $g \circ f$. We will need the category $\Set$ of sets with functions, the category $\Ab$ of abelian groups with group homomorphisms and the category $\Top$ of topological spaces and continuous maps.
\begin{definition}
\begin{definition}
A \emph{functor}$F$ from a category $\cat{C}$and to a category $\cat{D}$ consists of a function $F_0$ from the objects of $\cat{C}$ to the objects of $\cat{D}$ and a function $F_1$ from maps in $\cat{C}$ to maps in $\cat{D}$, such that
A \emph{functor}$F$ from a category $\cat{C}$ to a category $\cat{D}$ consists of a function $F_0$ from the objects of $\cat{C}$ to the objects of $\cat{D}$ and a function $F_1$ from maps in $\cat{C}$ to maps in $\cat{D}$, such that
\begin{itemize}
\begin{itemize}
\item for $f: A \to B$, we have $F_1(f): F_0(A)\to F_0(B)$,
\item for $f: A \to B$, we have $F_1(f): F_0(A)\to F_0(B)$,
\item$F_1(\id_A)=\id_{F_0(A)}$ and
\item$F_1(\id_A)=\id_{F_0(A)}$ and
@ -71,7 +71,7 @@ This now also gives a notion of isomorphisms between functors. It can be easily
For any category $\cat{C}$ we can define the $\mathbf{Hom}$-functor. It assigns to two objects in $\cat{C}$ the set of maps between them, i.e.
For any category $\cat{C}$ we can define the $\mathbf{Hom}$-functor. It assigns to two objects in $\cat{C}$ the set of maps between them, i.e.
We will show that it indeed gives a functor in the first argument, a similar proof works for the second argument. Let $f: A' \to A$ be a map in $\cat{C}$ and $g \in\Hom{\cat{C}}{A}{B}$, then $g \circ f \in\Hom{\cat{C}}{A'}{B}$. Hence the assignment $g \mapsto g \circ f$ is a map from $\Hom{\cat{C}}{A}{B}$ to $\Hom{\cat{C}}{A'}{B}$. Note that the direction of the map if reversed. Using associativity and identity it is easily checked that this assignment is functorial.
We will show that it indeed defines a functor in the first argument, a similar proof works for the second argument. Let $f: A' \to A$ be a map in $\cat{C}$ and $g \in\Hom{\cat{C}}{A}{B}$, then $g \circ f \in\Hom{\cat{C}}{A'}{B}$. Hence the assignment $g \mapsto g \circ f$ is a map from $\Hom{\cat{C}}{A}{B}$ to $\Hom{\cat{C}}{A'}{B}$. Note that the direction of the map if reversed. Using associativity and identity it is easily checked that this assignment is functorial.
\subsection{Equivalence}
\subsection{Equivalence}
Recall that an \emph{isomorphism} between categories $\cat{C}$ and $\cat{D}$ consists of two functors $F:\cat{C}\to\cat{D}$ and $G: \cat{D}\to\cat{C}$ such that
Recall that an \emph{isomorphism} between categories $\cat{C}$ and $\cat{D}$ consists of two functors $F:\cat{C}\to\cat{D}$ and $G: \cat{D}\to\cat{C}$ such that
@ -138,10 +138,10 @@ The first definition of adjunction is useful when dealing with maps, since it gi
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
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
$$\eta(s)= e_s. $$
$$\eta(s)= e_s. $$
And given any map $f: S \to U(A)$ for any abelian group $A$, we can define
And given any map $f: S \to U(A)$ for any abelian group $A$, we can define
It is clear that $U(\overline{f})\circ\eta= f$. We will leave the other details (naturality of $\eta$, $\overline{f}$ being a group homomorphism, and uniqueness w.r.t.~$U(\overline{f})\circ\eta= f$) to the reader.
It is clear that $U(\overline{f})\circ\eta= f$. We will leave the other details (naturality of $\eta$, $\overline{f}$ being a group homomorphism, and uniqueness w.r.t.~$U(\overline{f})\circ\eta= f$) to the reader.
By the other description of adjunctions we have $\Hom{\Ab}{\Z[S]}{T}\iso\Hom{\Set}{S}{U(T)}$, which exactly tells us that we can define a group homomorphism from $\Z[S]$ to $T$ by only specifying it on the generators $e_s, s \in S$. This fact is used throughout the thesis.
By the other description of adjunctions we have $\Hom{\Ab}{\Z[S]}{A}\iso\Hom{\Set}{S}{U(A)}$, which exactly tells us that we can define a group homomorphism from $\Z[S]$ to $A$ by only specifying it on the generators $e_s, s \in S$. This fact is used throughout this thesis.
@ -18,7 +18,7 @@ There are many variants to this notion. For example, there are also unbounded ch
In order to organize these chain complexes in a category, we should define what the maps are. The diagram above already gives an idea for this.
In order to organize these chain complexes in a category, we should define what the maps are. The diagram above already gives an idea for this.
\begin{definition}
\begin{definition}
Let $C$ and $D$ be chain complexes, with boundary operators $\del^C_n$ and $\del^D_n$ respectively. A \emph{chain map}$f: C \to D$ consists of a family of maps $f_n: C_n \to D_n$, such that they commute with the boundary operators: $f_n \circ\del^C_{n+1}=\del^D_{n+1}\circ f_{n+1}$ for all $n \in\N$, i.e. the following diagram commutes:
Let $C$ and $D$ be chain complexes, with boundary operators $\del^C_n$ and $\del^D_n$ respectively. A \emph{chain map}$f: C \to D$ consists of a family of group homomorphisms $f_n: C_n \to D_n$, such that they commute with the boundary operators: $f_n \circ\del^C_{n+1}=\del^D_{n+1}\circ f_{n+1}$ for all $n \in\N$, i.e. the following diagram commutes:
\begin{center}
\begin{center}
\begin{tikzpicture}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes]{
\matrix (m) [matrix of math nodes]{
@ -90,7 +90,7 @@ In general there is no inclusion in the other direction. This defect can be meas
\end{proof}
\end{proof}
\subsection{A note on abelian categories}
\subsection{A note on abelian categories}
The category $\Ch{\Ab}$ in fact is an \emph{abelian category}. We will only need a very specific property of this fact later on, and hence we will only prove this single fact. For the precise definition of an abelian category we refer to the book of Rotman about homological algebra \cite[Chapter~5.5]{rotman}. The notion of an abelian category is interesting if one wants to consider chain complexes over other objects than abelian groups, because $\Ch{\cat{C}}$ will be an abelian category whenever $\cat{C}$ is abelian.\footnote{However, this generality might not be so interesting from a categorical standpoint, as there is a fully faithful (exact) functor $F: \cat{C}\to\Ab$ for any (small) abelian category $\cat{C}$, called the \emph{Mitchell embedding}\cite{rotman}. This gives a way to proof categorical statements in $\cat{C}$ by proving the statement in $\Ab$.} The property we want to use later on is the following.
The category $\Ch{\Ab}$ in fact is an \emph{abelian category}. We will only need a very specific property of this fact later on, and hence we will only prove this single fact. For the precise definition of an abelian category we refer to the book of Rotman about homological algebra \cite[Chapter~5.5]{rotman}. The notion of an abelian category is interesting if one wants to consider chain complexes over other objects than abelian groups, because $\Ch{\cat{C}}$ will be an abelian category whenever $\cat{C}$ is an abelian category.\footnote{However, this generality might not be so interesting from a categorical standpoint, as there is a fully faithful (exact) functor $F: \cat{C}\to\Ab$ for any (small) abelian category $\cat{C}$, called the \emph{Mitchell embedding}\cite{rotman}. This gives a way to proof categorical statements in $\cat{C}$ by proving the statement in $\Ab$.} The property we want to use later on is the following.
\begin{definition}
\begin{definition}
A category $\cat{C}$ is \emph{preadditive} if the set of maps between two objects is an abelian group, such that composition is bilinear. In other words: the $\mathbf{Hom}$-functor has as its codomain $\Ab$:
A category $\cat{C}$ is \emph{preadditive} if the set of maps between two objects is an abelian group, such that composition is bilinear. In other words: the $\mathbf{Hom}$-functor has as its codomain $\Ab$:
@ -203,4 +203,4 @@ Let $S^k = \{ x \in \R^{n+1} \I ||x|| = 1 \}$ be the $k$-sphere. For example, $S
For $S^0$ the homology group $H_0(S^0)$ is isomorphic to $\Z\oplus\Z$, and all other homology groups are trivial.
For $S^0$ the homology group $H_0(S^0)$ is isomorphic to $\Z\oplus\Z$, and all other homology groups are trivial.
\end{example}
\end{example}
We can use the latter example to prove a fact about $\R^n$ quite easily ($n > 0$). Note that $\R^n -\{0\}$ is homotopic equivalent to $S^{n-1}$, so their homology groups are the same. As a consequence $\R^n -\{0\}$ has the same homology groups as $\R^m -\{0\}$, only if $n=m$. Now if $\R^n$ is homeomorphic to $\R^m$, then also $\R^n -\{0\}\iso\R^m -\{0\}$, so this only happens if $n=m$. This result is known as the \emph{invariance of dimension}.
We can use the latter example to prove a fact about $\R^n$ quite easily ($n > 0$). Note that $\R^n -\{0\}$ is homotopy equivalent to $S^{n-1}$, so their homology groups are the same. As a consequence $\R^n -\{0\}$ has the same homology groups as $\R^m -\{0\}$, only if $n=m$. Now if $\R^n$ is homeomorphic to $\R^m$, then also $\R^n -\{0\}\iso\R^m -\{0\}$, so this only happens if $n=m$. This result is known as the \emph{invariance of dimension}.
@ -51,7 +51,7 @@ We can now depict the category $\DELTA$ as in Figure~\ref{fig:delta_cat}. Note t
\label{fig:delta_cat}
\label{fig:delta_cat}
\end{figure}
\end{figure}
Although this is a very abstract definition, a more geometric intuition can be given. In $\DELTA$ we can regard $[n]$ as an abstract version of the $n$-simplex $\Delta^n$. The face maps $\delta_i$ are then exactly maps which point out how we can embed $[n-1]$ in $[n]$. This is visualized in Figure~\ref{fig:delta_cat_geom}. This picture shows the images of the face maps, for example the image of $\delta_3$ from $[2]$ to $[3]$ is the set $\{0,1,2\}$, which corresponds to the bottom face of the tetrahedron. The degeneracy maps are harder to visualize, one can think of them as ``collapsing'' maps, where two points are identified with each other. For example, this collapses a triangle into a line.
Although this is a very abstract definition, a more geometric intuition can be given. In $\DELTA$ we can regard $[n]$ as an abstract version of the $n$-simplex $\Delta^n$. The face maps $\delta_i$ are then exactly maps which point out how we can embed $[n-1]$ in $[n]$. This is visualized in Figure~\ref{fig:delta_cat_geom}. This picture shows the images of the face maps, for example the image of $\delta_3$ from $[2]$ to $[3]$ is the set $\{0,1,2\}$, which corresponds to the bottom face of the tetrahedron. The degeneracy maps are harder to visualize, one can think of them as ``collapsing'' maps. For example, this collapses a triangle into a line.
\begin{figure}
\begin{figure}
\includegraphics{delta_cat_geom}
\includegraphics{delta_cat_geom}
@ -67,7 +67,7 @@ This category $\DELTA$ will act as a prototype for these kind of geometric struc
(Or equivalently a contravariant functor $X: \DELTA\to\Set.$)
(Or equivalently a contravariant functor $X: \DELTA\to\Set.$)
\end{definition}
\end{definition}
The category $\sSet$ of all simplicial sets is the functor category $\Set^{\DELTA^{op}}$, where morphisms are natural transformations. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $X: \DELTA^{op}\to\Set$, keeping in mind that these should satisfy some relations which we will discuss next. Hence we can picture a simplicial set as done in Figure~\ref{fig:simplicial_set}. Comparing this to Figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $X$ is a contravariant functor.
The category $\sSet$ of all simplicial sets is the functor category $\Set^{\DELTA^{op}}$, where morphisms are natural transformations. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $X: \DELTA^{op}\to\Set$, keeping in mind that these should satisfy some relations which we will discuss next. Hence we can depict a simplicial set as done in Figure~\ref{fig:simplicial_set}. Comparing this to Figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $X$ is a contravariant functor.
\begin{figure}
\begin{figure}
\includegraphics{simplicial_set}
\includegraphics{simplicial_set}
@ -80,7 +80,7 @@ The category $\sSet$ of all simplicial sets is the functor category $\Set^{\DELT
Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain relations, these are the so called \emph{cosimplicial identities}.
Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain relations, these are the so called \emph{cosimplicial identities}.
\begin{lemma}
\begin{lemma}
The face and degeneracy maps in $\DELTA$ satisfy the cosimplicial identities
The face and degeneracy maps in $\DELTA$ satisfy the \emph{cosimplicial identities}:
\begin{align}
\begin{align}
\delta_j\delta_i &= \delta_i\delta_{j-1}, \hspace{1.5cm}\textnormal{ if } i < j,\\
\delta_j\delta_i &= \delta_i\delta_{j-1}, \hspace{1.5cm}\textnormal{ if } i < j,\\
\sigma_j\delta_i &= \delta_i\sigma_{j-1}, \hspace{1.5cm}\textnormal{ if } i < j,\\
\sigma_j\delta_i &= \delta_i\sigma_{j-1}, \hspace{1.5cm}\textnormal{ if } i < j,\\
@ -96,7 +96,7 @@ 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:
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{lemma}
\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
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 \emph{simplicial identities} hold:
\begin{align}
\begin{align}
d_i d_j &= d_{j-1} d_i, \hspace{1.5cm}\text{ if } i < j,\\
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,\\
d_i s_j &= s_{j-1} d_i, \hspace{1.5cm}\text{ if } i < j,\\
@ -149,7 +149,7 @@ Note that $\Delta[-]: \DELTA \to \sSet$ is exactly the Yoneda embedding. So a $m
\end{proof}
\end{proof}
\begin{example}
\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 $\Delta[0]$ looks 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
@ -38,11 +38,11 @@ If we want $M$ to be essentially surjective, there should exist a simplicial abe
\subsection{Normalized chain complex}
\subsection{Normalized chain complex}
To repair this defect we should be more careful. Given a simplicial abelian group, simply taking the same collection for our chain complex will not work (as shown above). Instead we are after some ``smaller'' abelian groups, and in some cases the abelian groups should completely vanish (as in the example above).
To repair this defect we should be more careful. Given a simplicial abelian group, simply taking the same collection for our chain complex will not work. Instead we are after some ``smaller'' abelian groups, and in some cases the abelian groups should completely vanish (as in the example above).
Given a simplicial abelian group $A$, we define abelian groups $N(A)_n$ as
Given a simplicial abelian group $A$, we define abelian groups $N(A)_n$ as
Now define group homomorphisms $\del: N(A)_n \to N(A)_{n-1}$ as
Now define group homomorphisms $\del: N(A)_n \to N(A)_{n-1}$ as
@ -56,18 +56,17 @@ $$ \del = d_0|_{N(A)_n}. $$
The chain complex $N(A)$ is called the \emph{normalized chain complex} of $A$.
The chain complex $N(A)$ is called the \emph{normalized chain complex} of $A$.
\begin{lemma}
\begin{lemma}
The above construction gives a functor $N: \sAb\to\Ch{\Ab}$. Furthermore $N$ is additive.
The above construction defines a functor $N: \sAb\to\Ch{\Ab}$. Furthermore $N$ is additive.
\end{lemma}
\end{lemma}
\begin{proof}
\begin{proof}
Given a map $f: A \to B$ of simplicial abelian groups, we consider the restrictions
Given a map $f: A \to B$ of simplicial abelian groups, we consider the restrictions
$$ f_n |_{N(A)_n}: N(A)_n \to B_n. $$
$$ f_n |_{N(A)_n}: N(A)_n \to B_n. $$
Because $f_n$ commutes with the face maps we get
Because $f_n$ commutes with the face maps we get
$$ d_i(f_n(x))= f_{n-1}(d_i(x))=0, $$
$$ d_i(f_n(x))= f_{n-1}(d_i(x))=0, $$
for $i>0$ and $x \in N(A)_n$. So the restriction also restricts the codomain, i.e.$f_n |_{N(A)_n}: N(A)_n \to N(B)_n$ is well-defined. Furthermore it commutes with the boundary operators, since $f$ itself commutes with all face maps. This gives functoriality $N(f): N(A)\to N(B)$.
for $i>0$ and $x \in N(A)_n$. So the restriction also restricts the codomain, in other words$f_n |_{N(A)_n}: N(A)_n \to N(B)_n$ is well-defined. Furthermore it commutes with the boundary operators, since $f$ itself commutes with all face maps. This gives functoriality $N(f): N(A)\to N(B)$.
Let $f, g: A \to B$ be two maps, then
Let $f, g: A \to B$ be two maps, then we prove additivity by
By recalling that in both categories addition of maps was defined pointwise, we have additivity of $N$.
\end{proof}
\end{proof}
\begin{example}
\begin{example}
@ -89,10 +88,10 @@ The chain complex $N(A)$ is called the \emph{normalized chain complex} of $A$.
\end{tikzpicture}.$$
\end{tikzpicture}.$$
where all face and degeneracy maps are identity maps. Clearly the kernel of $\id$ is the trivial group. So $N(\Z[\Delta[0]])_i =0$ for all $i > 0$. In degree zero we are left with $N(\Z[\Delta[0]])_0=\Z$. So we can depict the normalized chain complex by
where all face and degeneracy maps are identity maps. Clearly the kernel of $\id$ is the trivial group. So $N(\Z[\Delta[0]])_i =0$ for all $i > 0$. In degree zero we are left with $N(\Z[\Delta[0]])_0=\Z$. So we can depict the normalized chain complex by
$$ N(\Z[\Delta[0]])=\cdots\to0\to0\to\Z. $$
$$ N(\Z[\Delta[0]])=\cdots\to0\to0\to\Z. $$
So in this example we see that the normalized chain complex is really better behaved than the unnormalized chain complex given by $M(\Z[\Delta[0]]$.
So in this example we see that the normalized chain complex is really better behaved than the unnormalized chain complex given by $M(\Z[\Delta[0]])$.
\end{example}
\end{example}
To see what $N$ exactly does there are some useful lemmas. These lemmas can also be found in \cite[Chapter~VIII~1-2]{lamotke}, but in this thesis more detail is provided. Some corollaries are provided to give some intuition, or so summarize the lemmas, these results can also be found in \cite[Chapter~8.2-4]{weibel}. For the following lemmas let $X \in\sAb$ be an arbitrary simplicial abelian group and $n \in\N$. For these lemmas we will need the subgroups $D_n(X)\subset X_n$ of degenerate simplices, defined as:
To see what $N$ exactly does there are some useful lemmas. These lemmas can also be found in \cite[Chapter~VIII~1-2]{lamotke}, but in this thesis more detail is provided. Some corollaries are provided to give some intuition, or so summarize the lemmas, these results can also be found in \cite[Chapter~8.2-4]{weibel}. For the following lemmas let $X \in\sAb$ be an arbitrary simplicial abelian group and $n \in\N$. For these lemmas we will need the subgroups $D_n(X)\subseteq X_n$ of degenerate simplices, defined as:
$$ x = b + c = b - s_{k-1} d_k b + s_{k-1} d_k b + c = b' + c',$$
$$ x = b + c = b - s_{k-1} d_k b + s_{k-1} d_k b + c = b' + c',$$
with $b' \in P_n^{k-1}$ and $c' \in D_n(X)$.
with $b' \in P_n^{k-1}$ and $c' \in D_n(X)$.
Doing this inductively gives us $x = b + c$, with $b \in P_n^0= N(X)_n$ and $c \in D_n(X)$, which is what we had to prove.
Doing this inductively gives us $x = b + c$, with $b \in P_n^0= N(X)_n$ and $c \in D_n(X)$.
\end{proof}
\end{proof}
\begin{lemma}
\begin{lemma}
\label{le:decomp2}
\label{le:decomp2}
@ -219,7 +218,7 @@ Furthermore the degeneracy maps $s_i: K(C)_{n-1} \to K(C)_n$ are given by precom
Let $f: N\Z[\Delta[n]]\to C$ be a chain map then $f \in D_n(K(C))$ if and only if $f_r =0$ forall $r \geq n$.
Let $f: N\Z[\Delta[n]]\to C$ be a chain map then $f \in D_n(K(C))$ if and only if $f_r =0$ forall $r \geq n$.
\end{lemma}
\end{lemma}
\begin{proof}
\begin{proof}
If $f \in D_n(K(C))$ we have$f =\sum_{i=0}^n s_i(f^{(i)})$ for some maps $f^{(i)}: N\Z[\Delta[n-1]]\to C$. Since $N\Z[\Delta[n-1]]_r =0$ as there are no injections $[r]\mono[n-1]$, we have $f^{(i)}_r =0$ for all $r > n-1$.
If $f \in D_n(K(C))$ we can write $f$ as$f =\sum_{i=0}^n s_i(f^{(i)})$ for some maps $f^{(i)}: N\Z[\Delta[n-1]]\to C$. Since $N\Z[\Delta[n-1]]_r =0$ as there are no injections $[r]\mono[n-1]$, we have $f^{(i)}_r =0$ for all $r > n-1$.
For the other direction let $f: N\Z[\Delta[n]]\to C$ be a chain map and $f_r =0$ forall $r \geq n$. Define $f_m^{(i)}(\eta)= f_m(\delta_i \eta)$ for $\eta: [m]\mono[n]$. This gives a chain map $f^{(i)}: N\Z[\Delta[n-1]]\to C$ by a simple calculation:
For the other direction let $f: N\Z[\Delta[n]]\to C$ be a chain map and $f_r =0$ forall $r \geq n$. Define $f_m^{(i)}(\eta)= f_m(\delta_i \eta)$ for $\eta: [m]\mono[n]$. This gives a chain map $f^{(i)}: N\Z[\Delta[n-1]]\to C$ by a simple calculation:
\foreach\i/\j in {2/2, 3/1, 4/0}\draw[->] (m-1-\i) -- node {$f_\j$} (m-2-\i);
\foreach\i/\j in {2/2, 3/1, 4/0}\draw[->] (m-1-\i) -- node {$f_\j$} (m-2-\i);
\end{tikzpicture}
\end{tikzpicture}
\Bigg\}\iso C_0 = K'(C)_0, $$
\Bigg\}\iso C_0 = K'(C)_0, $$
because for $f_1, f_2, \ldots$ there is now choice at all, and for $f_0: \Z\to C_0$ we only have to choose an image for $1\in\Z$. In the next dimension we see
because for $f_1, f_2, \ldots$ there is no choice at all, and for $f_0: \Z\to C_0$ we only have to choose an image for $1\in\Z$. In the next dimension we see
@ -18,7 +18,7 @@ When dealing with homotopy groups in a topological space $X$ we always need a ba
We will call $y$ the \emph{homotopy} and notate $y: x \sim x'$.
We will call $y$ the \emph{homotopy} and notate $y: x \sim x'$.
\end{definition}
\end{definition}
Of course we would like $\sim$ to be an equivalence relation, however this is not true for all simplicial sets. For example there is in general no reason for symmetry, existence of a $1$-simplex $y$ from $x$ to $x'$ does not give us a $1$-simplex $y'$ from $x'$ to $x$. One can give an precise condition on when it is a equivalence relation, the so called \emph{Kan-condition}. In our case of simplicial abelian groups, however, we can prove directly that $\sim$ is an equivalence relation.
Of course we would like $\sim$ to be an equivalence relation, however this is not true for all simplicial sets. For example there is in general no reason for symmetry, existence of a homotopy from $x$ to $x'$ does not give us a homotopy from $x'$ to $x$. One can give an precise condition on when it is a equivalence relation, the so called \emph{Kan-condition}. In our case of simplicial abelian groups, however, we can prove directly that $\sim$ is an equivalence relation.
In figure~\ref{fig:simplicial_htp} it is shown why the definition of homotopy makes sense for $n=1$. Two homotopic $1$-simplices from $Z_n(X)$ are depicted in two ways. The first way only shows the structure we have, indicating what the boundaries are (as described by the face maps). In the second figure we collapsed all occurrences of $0$ into a single point. This way of drawing a homotopy should remind the reader of homotopy (between paths) in a topological space.
In figure~\ref{fig:simplicial_htp} it is shown why the definition of homotopy makes sense for $n=1$. Two homotopic $1$-simplices from $Z_n(X)$ are depicted in two ways. The first way only shows the structure we have, indicating what the boundaries are (as described by the face maps). In the second figure we collapsed all occurrences of $0$ into a single point. This way of drawing a homotopy should remind the reader of homotopy (between paths) in a topological space.
@ -36,7 +36,7 @@ In figure~\ref{fig:simplicial_htp} it is shown why the definition of homotopy ma
\end{figure}
\end{figure}
\begin{lemma}
\begin{lemma}
The relation $\sim$ as defined above is an equivalence relation on $Z_n(X)$. Furthermore it is compatible with addition.
For any simplicial abelian group $X$, the relation $\sim$ as defined above is an equivalence relation on $Z_n(X)$. Furthermore it is compatible with addition.
\end{lemma}
\end{lemma}
Before proving this, one should have a look at figure~\ref{fig:simplicial_eqrel}. In this figure we show what we want to proof in degree $n=0$ (i.e. the simplices of interest are points, and the homotopies are paths).
Before proving this, one should have a look at figure~\ref{fig:simplicial_eqrel}. In this figure we show what we want to proof in degree $n=0$ (i.e. the simplices of interest are points, and the homotopies are paths).
@ -59,7 +59,7 @@ Before proving this, one should have a look at figure~\ref{fig:simplicial_eqrel}
\begin{definition}
\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
$$\pi_n(X)= Z_n(X)/\sim. $$
$$\pi_n(X)= Z_n(X)/_\sim. $$
\end{definition}
\end{definition}
Note that this is an abelian group, because $Z_n(X)$ is a subgroup of $X_n$, and $\sim$ also defines a subgroup. It is relatively straight forward to prove that this definition coincides with the $n$-th homology group of the associated normalized chain complex.
Note that this is an abelian group, because $Z_n(X)$ is a subgroup of $X_n$, and $\sim$ also defines a subgroup. It is relatively straight forward to prove that this definition coincides with the $n$-th homology group of the associated normalized chain complex.
@ -79,7 +79,7 @@ Note that this is an abelian group, because $Z_n(X)$ is a subgroup of $X_n$, and
&= \{ d_0 y \I y \in X_{n+1}, d_i(y) = 0 \text{ for all } i > 0 \}\\
&= \{ d_0 y \I y \in X_{n+1}, d_i(y) = 0 \text{ for all } i > 0 \}\\
&= \{ x \in N(X)_n \I x \sim 0 \}
&= \{ x \in N(X)_n \I x \sim 0 \}
\end{align*}
\end{align*}
So we see that $\pi_n(X)= Z_n(X)/\sim=\ker(\del)/\im(\del)= H_n(N(X))$.
So we see that $\pi_n(X)= Z_n(X)/_\sim=\ker(\del)/\im(\del)= H_n(N(X))$.
\end{proof}
\end{proof}
\begin{corollary}
\begin{corollary}
@ -97,7 +97,7 @@ In Section~\ref{sec:Chain Complexes}, we already defined the topological $n$-sim
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
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 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
$$\text{Sing}(X)_n =\Hom{\Top}{\Delta^n}{X}. $$
$$\text{Sing}(X)_n =\Hom{\Top}{\Delta^n}{X}. $$
Recall construction of the singular chain complex in Section~\ref{sec:Chain Complexes}:
Recall construction of the singular chain complex in Section~\ref{sec:Chain Complexes}:
In this thesis we will study the Dold-Kan correspondence, a celebrated result which belongs to the field of homological algebra or simplicial homotopy theory. Abstractly, one version of the theorem states that there is an equivalence of categories
In this thesis we will study the Dold-Kan correspondence, a celebrated result which belongs to the field of homological algebra or simplicial homotopy theory. Abstractly, one version of the theorem states that there is an equivalence of categories
$$ K: \Ch{\Ab}\simeq\sAb :N, $$
$$ K: \Ch{\Ab}\simeq\sAb :N, $$
where $\Ch{\Ab}$ is the category of chain complexes and $\sAb$ is the category of simplicial abelian groups. This theorem was discovered by A.~Dold \cite{dold} and D.~Kan \cite{kan} independently in 1957. Objects of either of these categories have important invariants. A more refined statement of this equivalence tells us that there is an natural isomorphism between homology groups of chain complexes and homotopy groups of simplicial abelian groups. A bit more precise:
where $\Ch{\Ab}$ is the category of chain complexes and $\sAb$ is the category of simplicial abelian groups. This theorem was discovered by A.~Dold \cite{dold} and D.~Kan \cite{kan} independently in 1957. Objects of either of these categories have important invariants. A more refined statement of this equivalence tells us that there is a natural isomorphism between homology groups of chain complexes and homotopy groups of simplicial abelian groups. A bit more precise:
$$\pi_n(A)\iso H_n(N(A))\text{ for all } n \in\N. $$
$$\pi_n(A)\iso H_n(N(A))\text{ for all } n \in\N. $$
In the first section some definitions from category theory are recalled, which are especially important in Sections~\ref{sec:Simplicial Abelian Groups} and \ref{sec:Constructions}. In Section~\ref{sec:Chain Complexes} we will discuss the category of chain complexes and in the end of this section a motivation from algebraic topology will be given for these objects. Section~\ref{sec:Simplicial Abelian Groups} then continues with the second category involved, $\sAb$. This section start with a slightly more general notion and it will be illustrated to have a geometrical meaning. In Section~\ref{sec:Constructions} the correspondence will be defined and proven. In the last section (Section~\ref{sec:Homotopy}) the refined statement will be proven and in the end some more general notes about topology and homotopy will be given, justifying once more the beauty of this correspondence.
In the first section some definitions from category theory are recalled, which are especially important in Sections~\ref{sec:Simplicial Abelian Groups} and \ref{sec:Constructions}. In Section~\ref{sec:Chain Complexes} we will discuss the category of chain complexes and in the end of this section a motivation from algebraic topology will be given for these objects. Section~\ref{sec:Simplicial Abelian Groups} then continues with the other category involved, the category of simplicial abelian groups. This section starts with a slightly more general notion and it will be illustrated to have a geometrical meaning. In Section~\ref{sec:Constructions} the correspondence will be defined and proven. In the last section (Section~\ref{sec:Homotopy}) the refined statement will be proven and in the end some more general notes about topology and homotopy will be given, justifying once more the beauty of this correspondence.
Joshua Moerman
12 years ago