Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, let us begin by recalling some basic category theory. The reader who is already familiar with these concepts, is invited to skip this section. We will introduce the notions of categories, functors, isomorphisms, natural transformations, equivalences, adjunctions and the Yoneda lemma.
Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, let us begin by recalling some basic category theory. The reader who is already familiar with these concepts, is invited to skip this section. We will recall the notions of categories, functors, isomorphisms, natural transformations, equivalences, adjunctions and the Yoneda lemma.
We will briefly define categories an functors to fix the notation. We will not provide many examples or intuition in these concepts. For a more elaborated exposition one should have a read in \cite{awodey} or \cite{maclane}. The more complicated definitions will be discussed in a bit more detail.
We will briefly define categories and functors to fix the notation. We will not provide many examples or intuition in these concepts. For a more elaborated exposition one should have a read in \cite{awodey} or \cite{maclane}. The more complicated definitions will be discussed in a bit more detail.
\subsection{Categories}
\begin{definition}
A \emph{category}$\cat{C}$ consists of a collection of \emph{objects}, a set of \emph{maps}$\Hom{\cat{C}}{A}{B}$ for each two objects $A, B \in\cat{C}$ and a binary operator named \emph{composition}$-\circ-:\Hom{\cat{C}}{B}{C}\times\Hom{\cat{C}}{A}{B}$ such that
A \emph{category}$\cat{C}$ consists of a collection of \emph{objects}, a set of \emph{maps}$\Hom{\cat{C}}{A}{B}$ for each two objects $A, B \in\cat{C}$ and a binary operator \emph{composition}
\item$\circ$ is associative, i.e. $h \circ(g \circ f)=(h \circ g)\circ f$ and
\itemcomposition is associative, i.e. $h \circ(g \circ f)=(h \circ g)\circ f$ and
\item there exists an neutral element $\id_A \in\Hom{\cat{C}}{A}{A}$ for all $A$ in $\cat{C}$, i.e.
$$\id_B \circ f = f = f \circ\id_A. $$
\end{itemize}
\end{definition}
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 $g \circ f$. There is a category $\Set$ of sets with functions, a category $\Ab$ of abelian groups with group homomorphisms, a category $\Top$ of topological spaces and continuous maps, and many more.
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}
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
@ -28,7 +30,7 @@ Instead of writing $f \in \Hom{\cat{C}}{A}{B}$ we write $f: A \to B$, as many ca
For a category $\cat{C}$ we denote the \emph{opposite} category by $\cat{C}^{op}$. The opposite category consists of the same objects, but the maps and composition are reversed. A \emph{contravariant functor}$F$ from $\cat{C}$ to $\cat{D}$ is a functor $F: \cat{C}^{op}\to\cat{D}$.
Note that the composition of two functors is again a functor, and that we always have an identity functor, sending each object to itself and each map to itself. This gives rise to a category $\cat{Cat}$ of \emph{small} categories. Note that we need some kind of \emph{smallness} to avoid set-theoretical issues, because we require the collection of maps between objects to be a set, whereas the collection of objects is not necessarily a set. However we will not be interested in these set-theoretic issues, and hence skip the definition of small.
Note that the composition of two functors is again a functor, and that we always have an identity functor, sending each object to itself and each map to itself. This gives rise to a category $\cat{Cat}$ of \emph{small} categories. Note that we need some kind of \emph{smallness} to avoid set-theoretical issues. However we will not be interested in these set-theoretical issues, and hence skip the definition of small.
\subsection{Isomorphisms}
Given a category $\cat{C}$ and two objects $A, B \in\cat{C}$ we would like to know when those objects are regarded as the same, according to the category. This will be the case when there is an isomorphism between the two.
@ -41,7 +43,7 @@ Given a category $\cat{C}$ and two objects $A, B \in \cat{C}$ we would like to k
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, i.e. the group homomorphisms which are both injective and surjective.
For example the cyclic group $\Z_4$ and the Klein four-group $V_4$ are not isomorphic in $\Ab$, but if we regard only the sets $\Z_4$ and $V_4$, then they are (because there is a bijection). So it is good to note that whether two objects are isomorphic really depends on the category we are working in.
Note that an isomorphism between to categories is now also defined. Two categories $\cat{C}$ and $\cat{D}$ are isomorphic if there are functors $F$ and $G$ such that $ FG =\id_\cat{D}$ and $GF =\id_\cat{C}$.
Note that an isomorphism between two categories is now also defined. Two categories $\cat{C}$ and $\cat{D}$ are isomorphic if there are functors $F$ and $G$ such that $ FG =\id_\cat{D}$ and $GF =\id_\cat{C}$.
\subsection{Natural transformations}
@ -125,24 +127,21 @@ 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 $\text{supp}(\phi)=\{ s \in S \I\phi(s)\neq0\}$. 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
where $\text{supp}(\phi)=\{ s \in S \I\phi(s)\neq0\}$. The group structure on $\Z[S]$ is given by pointwise addition. We can define a generator$e_s \in\Z[S]$ 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:
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
One can think of elements of this abelian group as formal sums, namely by writing $\phi\in\Z[S]$ as $\phi=\sum_{x \in\text{supp}(\phi)}\phi(x) e_x$. 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
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.
A \emph{(non-negative) chain complex of abelian groups}$C$ is a collection of abelian groups $C_n$, $n \in\N$, together with group homomorphisms $\del_n: C_n \to C_{n-1}$, which we call \emph{boundary homomorphisms}, such that $\del_n \circ\del_{n+1}=0$ for all $n \in\Np$.
A \emph{(non-negative) chain complex of abelian groups}$C$ is a collection of abelian groups $C_n$, $n \in\N$, together with group homomorphisms $\del_n: C_n \to C_{n-1}$, which we call \emph{boundary operators}, such that $\del_n \circ\del_{n+1}=0$ for all $n \in\Np$.
\end{definition}
Thus graphically a chain complex $C$ can be depicted by the following diagram:
@ -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.
\begin{definition}
Let $C$ and $D$ be chain complexes, with boundary maps $\del^C_n$ 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 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:
\begin{center}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes]{
@ -38,13 +38,13 @@ Note that if we have two such chain maps $f:C \to D$ and $g:D \to E$, then the l
$\Ch{\Ab}$ is the category of chain complexes of abelian groups with chain maps.
\end{definition}
Note that we will often drop the indices of the boundary morphisms, since it is often clear in which degree we are working. The boundary operators give rise to certain subgroups, because all groups are abelian, subgroups are normal subgroups.
Note that we will often drop the indices of the boundary operators, since it is often clear in which degree we are working. The boundary operators give rise to certain subgroups, because all groups are abelian, subgroups are normal subgroups.
\begin{definition}
\label{def:cycles}
Given a chain complex $C$ we define the following subgroups:
\begin{itemize}
\item the subgroup of \emph{$n$-cycles}: $Z_n(C)=\ker(\del_n: C_n \to C_{n-1})\nsubgrp C_n$, and
\item the subgroup of \emph{$n$-cycles}: $Z_n(C)=\ker(\del_n: C_n \to C_{n-1})\nsubgrp C_n$,
\item the subgroup of \emph{$0$-cycles}: $Z_0(C)= C_0$, and
\item the subgroup of \emph{$n$-boundaries}: $B_n(C)=\im(\del_{n+1}: C_{n+1}\to C_n)\nsubgrp C_n$.
\end{itemize}
@ -104,7 +104,7 @@ Clearly the category $\Ab$ is preadditive, since we can add group homomorphisms
\begin{proof}
We can add chain maps level-wise. Given two chain maps $f, g: C \to D$, we define $f+g$ as:
$$(f+g)_n = f_n + g_n, $$
where we use the fact that $\Ab$ is preadditive. Note that $f+g$ is also a chain map, since it commutes with the boundary operator. The bilinearity of composition follows level-wise from the fact that $\Ab$ is preadditive.
where we use the fact that $\Ab$ is preadditive. Note that $f+g$ is also a chain map, since it commutes with the boundary operators. The bilinearity of composition follows level-wise from the fact that $\Ab$ is preadditive.
\end{proof}
Of course given two preadditive categories $\cat{C}$ and $\cat{D}$, not every functor will preserve this extra structure.
@ -117,7 +117,7 @@ Of course given two preadditive categories $\cat{C}$ and $\cat{D}$, not every fu
\subsection{The singular chain complex}
\label{sec:singular}
In order to see why we are interested in the construction of homology groups, we will look at an example from algebraic topology. We will see that homology gives a nice invariant for spaces. We will form a chain complex from a topological space $X$. In this section we will not be very precise, as it will only act as a motivation. However the intuition might be very useful later on, and so pictures are provided to give meaning to this construction.
In order to see why we are interested in the construction of homology groups, we will look at an example from algebraic topology. We will see that homology gives a nice invariant for spaces. We will construct a chain complex for any topological space. In this section we will not be very precise, as it will only act as a motivation. However the intuition might be very useful later on, and so pictures are provided to give meaning to this construction.
\begin{definition}
The \emph{topological $n$-simplex}$\Delta^n$, $n \in\N$, is the set
@ -185,7 +185,7 @@ A direct consequence of being a functor is that homeomorphic spaces have isomorp
In the remainder of this section we will give the homology groups of some basic spaces. For most spaces it is hard to calculate the homology groups from the definitions above. One generally proves these results by using theorems from algebraic topology or homological algebra, which are beyond the scope of this thesis. The first example can be calculated from the definitions above, however the proof is not included as the example is only included as a motivation.
\begin{example}
The homology of the one-point space $\ast$ is given by:
The homology of the one-point space $\ast$ is given by
@ -5,7 +5,7 @@ Before defining \emph{simplicial abelian groups}, we will first discuss the more
\subsection{Abstract definition}
\begin{definition}
We define a category $\DELTA$, where the objects are the finite ordinals $[n]=\{0 < \dots < n\}$ for $n \in\N$ and maps are monotone functions: $\Hom{\DELTA}{[n]}{[m]}=\{ f : [n]\to[m]\I f(i)\leq f(j)\text{ for all } i < j \}$.
We define a category $\DELTA$, where the objects are the \emph{finite ordinals}$[n]=\{0 < \dots < n\}$ for $n \in\N$ and maps are \emph{monotone functions}: $\Hom{\DELTA}{[n]}{[m]}=\{ f : [n]\to[m]\I f(i)\leq f(j)\text{ for all } i < j \}$.
\end{definition}
The category $\DELTA$ is sometimes referred to as the \emph{category of finite ordinals} or the \emph{cosimplicial index category}. 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)
@ -138,7 +138,15 @@ 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. So a $m$-simplex in $\Delta[n]$ is nothing more than a monotone function $[m]\to[n]$. 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.
Note that $\Delta[-]: \DELTA\to\sSet$ is exactly the Yoneda embedding. So a $m$-simplex in $\Delta[n]$ is nothing more than a monotone function $[m]\to[n]$. In a moment we will see why the Yoneda lemma is useful to us, but let us first describe which functions are non-degenerate. Recall that a simplex is degenerate if it lies the image of $s_i$ for some $i$. In the simplicial set $\Delta[n]$ the degeneracy maps $s_i$ are given by precomposing with $\sigma_i$ (by definition of the $\mathbf{Hom}$-functor).
\begin{lemma}
\label{le:standard_nondeg}
The non-degenerate $m$-simplices in $\Delta[n]$ are precisely injective monotone functions $[m]\mono[n]$.
\end{lemma}
\begin{proof}
Given a $m$-simplex $x \in\Delta[n]_m$, using the epi-mono factorization we can write it as $x =\delta\sigma: [m]\to[n]$, where $\delta$ is injective and $\sigma$ surjective. It is now easily seen that $x$ is degenerate if and only if $\sigma\neq\id$. In other words a $m$-simplex $x \in\Delta[n]_m$ is non-degenerate if and only if $x: [m]\to[n]$ is injective. Note that for $m>n$ no such injective monotone functions exist and for $m=n$ there is a unique one, namely $\id_{[n]}$.
\end{proof}
\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
@ -160,8 +168,6 @@ Note that $\Delta[-]: \DELTA \to \sSet$ is exactly the Yoneda embedding. So a $m
Note that the only non-degenerate simplex is the unique $0$-simplex.
\end{example}
We can characterize the non-degenerate simplices for the other standard $n$-simplices as well. Recall that a simplex is degenerate if it lies the image of $s_i$ for some $i$. In the simplicial set $\Delta[n]$ the degeneracy maps $s_i$ are given by precomposing with $\sigma_i$ (by definition of the $\mathbf{Hom}$-functor). Given a $m$-simplex $x \in\Delta[n]_m$, using the epi-mono factorization we can write it as $x =\delta\sigma: [m]\to[n]$, where $\delta$ is injective and $\sigma$ surjective. It is now easily seen that $x$ is degenerate if and only if $\sigma\neq\id$. In other words a $m$-simplex $x \in\Delta[n]_m$ is non-degenerate if and only if $x: [m]\to[n]$ is injective. Note that for $m>n$ no such injective monotone functions exist and for $m=n$ there is a unique one, namely $\id_{[n]}$.
\begin{example}
$\Delta[1]$ is a bit more interesting, but still not too complicated. We will describe the first three sets $\Delta[1]_0$, $\Delta[1]_1$ and $\Delta[1]_2$. We can use the fact that any monotone function $f: [n]\to[m]$ is a composition of first applying degeneracy maps, and then face maps, i.e.: $f: [n]\tot{\sigma_{i_0}\cdots\sigma_{i_M}}[k]\tot{\delta_{j_0}\cdots\delta_{j_N}}[m]$, where $k \leq m, n$.
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.
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 operator. 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
Using $A_n$ as the family of abelian groups and the maps $\del_n$ as boundary maps gives a chain complex.
Using $A_n$ as the family of abelian groups and the maps $\del_n$ as boundary operators gives a chain complex.
\end{lemma}
\begin{proof}
We already have a collection of abelian groups together with maps, so the only thing to prove is $\del_{n-1}\circ\del_n =0$. This can be done with a calculation.
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}
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. Following the book \cite{goerss} we will call the chain complex $M(X)$ the \emph{Moore complex} or \emph{unnormalized chain complex} of $X$. This construction defines a functor
Thus, associated to a simplicial abelian group $A$ we obtain a chain complex $M(A)$ with $M(A)_n = A_n$ and the boundary operators as above. Following the book \cite{goerss} we will call the chain complex $M(X)$ the \emph{Moore complex} or \emph{unnormalized chain complex} of $X$. 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*}
@ -30,13 +30,12 @@ by assigning $M(f)_n = f_n$ for a natural transformation $f: A \to B$. It follow
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}.
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 operators (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 $M$ 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
Let us investigate whether this functor $M$ can be part of an equivalence. If $M$ would be part of an equivalence, it would be \emph{essentially surjective}, meaning that for any chain complex $C$ there exists a simplicial abelian group $A$ such that $M(A)\iso C$. For example take the following chain complex
$$ C =\ldots\to0\to0\to\Z. $$
Now if we would construct a (non-trivial) simplicial abelian group $K(C)$ from this chain complex, we now know that $K(C)_n$ is non-trivial for all $n \in\N$. This means that $M(K(C))_n$ is non-trivial for all $n \in\N$. For an equivalence we require a (natural) isomorphism: $M(K(C))\tot{\iso} C$, this in particular means an isomorphism in each degree $n > 0$: $0\neq M(K(C))_n \tot{\iso} C_n =0$, which is not possible. So the functor $M$, as defined as above, will not give us the equivalence we wanted, although it is a very nice functor.
If we want $M$ to be essentially surjective, there should exist a simplicial abelian group $A$ with $A_0\iso\Z$ and $A_0\iso0$. Recall that the degeneracy maps are injective. This contradicts as there is no injective map $\Z\mono0$. So it is easily seen that $M$ cannot be part of an equivalence, although it is a nice functor.
\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).
@ -64,7 +63,7 @@ The chain complex $N(A)$ is called the \emph{normalized chain complex} of $A$.
$$ f_n |_{N(A)_n}: N(A)_n \to B_n. $$
Because $f_n$ commutes with the face maps we get
$$ d_i(f_n(x))= f_{n-1}(d_i(x))=0, $$
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 operator, 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, 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 $N(f): N(A)\to N(B)$.
@ -212,7 +211,7 @@ Many functors to $\sAb$ can be shown to have this description.\footnote{And also
K(C) = \Hom{\Ch{\Ab}}{N\Z[\Delta[-]]}{C}.
\end{gather*}
This is a very abstract definition so we will first discuss what a chain map $N\Z[\Delta[n]]\to C$ looks like. Recall that the non-degenerate $m$-simplices of $\Delta[n]$ are exactly injective maps $\eta: [m]\mono[n]$for all $m$. So $N\Z[\Delta[n]]$ consists of linear combinations of those non-degenerate simplices, as $N$ precisely gives us the non-degenerate elements. Note that $N\Z[\Delta[n]]_m$ are free groups, since $\Z[\Delta[n]]_m$ are free. In other words, when defining a chain map $N\Z[\Delta[n]]\to C$ it is sufficient to define it on the generators, i.e. on the injections $\eta: [m]\mono[n]$. This fact is used throughout the following proofs.
This is a very abstract definition so we will first discuss what a chain map $N\Z[\Delta[n]]\to C$ looks like. Recall that the non-degenerate $m$-simplices of $\Delta[n]$ are exactly injective maps $\eta: [m]\mono[n]$(Lemma~\ref{le:standard_nondeg}). So $N\Z[\Delta[n]]$ consists of linear combinations of those non-degenerate simplices, as $N$ precisely gives us the non-degenerate elements. Note that $N\Z[\Delta[n]]_m$ are free groups, since $\Z[\Delta[n]]_m$ are free. In other words, when defining a chain map $N\Z[\Delta[n]]\to C$ it is sufficient to define it on the generators, i.e. on the injections $\eta: [m]\mono[n]$. This fact is used throughout the following proofs.
Furthermore the degeneracy maps $s_i: K(C)_{n-1}\to K(C)_n$ are given by precomposition of the induced map $\sigma_{i\ast}: N\Z[\Delta[n]]\to N\Z[\Delta[n-1]]$ which in their turn are given by postcomposition. More precisely this gives $s_i(f)_m(\eta)= f_m(\sigma_i \eta)$ for any $f \in K(C)_{n-1}$ and $\eta: [m]\mono[n]$. We will now have a closer look at the degenerate elements of $K(C)$.
\begin{lemma}
@ -224,7 +223,7 @@ Furthermore the degeneracy maps $s_i: K(C)_{n-1} \to K(C)_n$ are given by precom
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:
where we used that $f$ is a chain map at \refeqn{1} and the definition of the boundary map of $N(-)$\emph{and} the definition of face maps in $\Delta[-]$ at \refeqn{2}.
where we used that $f$ is a chain map at \refeqn{1} and the definition of the boundary operator of $N(-)$\emph{and} the definition of face maps in $\Delta[-]$ at \refeqn{2}.
Now let $\eta: [m]\mono[n]$ and $\eta\neq\id$ (we already know $f(\id_{[n]})=0$ by assumption) then by the epi-mono factorization we have $\eta=\delta_{i_a}\cdots\delta_{i_1}$ with $a>0$, so
@ -261,7 +260,7 @@ We now have enough lemmas to prove the main equivalence quite easily. The most i
0 \quad\text{otherwise}
\end{cases}\\
\end{align*}
Clearly $\phi_n(g)= x$ by definition and $g$ is a chain map as we defined it to commute with the boundary map. For proving injectivity consider $g \in\ker(\phi_n)$ then for trivial reasons we have $f_r =0$ for all $r > n$ and $f_n(\id_{[n]})=0$ gives $f_n =0$. Applying Lemma~\ref{le:degen_k} gives us $f \in D_n(K(C))$, but $f \in N(K(C))_n$. So by using Corollary~\ref{cor:NandD} we get $f =0$. Thus $\phi_n$ is an isomorphism, which gives us $NK(C)\iso C$.
Clearly $\phi_n(g)= x$ by definition and $g$ is a chain map as we defined it to commute with the boundary operators. For proving injectivity consider $g \in\ker(\phi_n)$ then for trivial reasons we have $f_r =0$ for all $r > n$ and $f_n(\id_{[n]})=0$ gives $f_n =0$. Applying Lemma~\ref{le:degen_k} gives us $f \in D_n(K(C))$, but $f \in N(K(C))_n$. So by using Corollary~\ref{cor:NandD} we get $f =0$. Thus $\phi_n$ is an isomorphism, which gives us $NK(C)\iso C$.
We now have established two natural isomorphisms $\id_\sAb\iso KN$ and $NK \iso\id_\Ch{\Ab}$. Hence we have an equivalence $\Ch{\Ab}\simeq\sAb$.
because again we can choose $f_1$ anyway we want, which gives us $C_1$. But then we are forced to choose $f_0(x, x)=\del(f_1(x))$ for all $x \in\Z$, so we are left with choosing an element $c \in C_0$ for defining $f(1,-1)= c$. Adding this gives $C_1\oplus C_0$.
where the last functor is the \emph{unnormalized chain complex}. All the categories involved have a notion of homotopy. In topological spaces this is the known notion where $f, g:X \to Y$ are homotopic if there exists a homotopy $H:I \times X \to Y$ with the appropriate properties. In simplicial sets (or simplicial abelian groups) we only saw the notion of homotopy groups, but there exists a more general notion of homotopy, as discussed in the overview of Friedman \cite{friedman}. And finally in chain complexes we saw homology groups, but this category also has a more general notion of chain homotopy, which can be found in any book on homological algebra such as in the book of Rotman \cite{rotman}.
Joshua Moerman
12 years ago