Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, we will first look at some basic category theory. If one is already familier with these concepts, he or she can skip this section. We will introduce the notions of categories, functors, isomorphims, natural transformations, equivalences (between categories) and adjunctions.
Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, we will first look at some basic category theory. If one is already familier with these concepts, he or she can skip this section. We will introduce the notions of categories, functors, isomorphims, natural transformations, equivalences and adjunctions.
\subsection{Categories}
\begin{definition}
A \emph{category}$\cat{C}$ consists of a collection \emph{objects}, and for each two objects $A$ and $B$ in $\cat{C}$ there is a (possibly empty)\emph{set of maps}(or arrows) from $A$ to $B$, notated as $\Hom{\cat{C}}{A}{B}$, such that:
A \emph{category}$\cat{C}$ consists of a collection \emph{objects}, and for each two objects $A$ and $B$ in $\cat{C}$ there is a set of\emph{maps} from $A$ to $B$, notated as $\Hom{\cat{C}}{A}{B}$, such that:
\begin{itemize}
\item\emph{(Identity)}
$\id_A \in\Hom{\cat{C}}{A}{A}$ for all $A$ in $\cat{C}$,
@ -22,10 +22,10 @@ Note that the collection of objects may be a proper class instead of a set, howe
As the notation suggests maps can be thought of as functions, which is also the case in many examples.
\begin{example}
The category $\Set$ has a objects sets, and as maps ordinary functions. Of course we then have the identity function $\id_X(x)= x$ and composition as usual.
The category $\Set$ has as its objects sets, and as maps it has ordinary functions. Of course we then have the identity function $\id_X(x)= x$ and composition as usual.
\end{example}
\begin{example}
The category $\Ab$ has a objects abelian groups, and the maps between two objects are exactly the grouphomomorphisms. We know that the identity function is indeed a grouphomomorphism, and composing two grouphomomorpisms, gives indeed a new grouphomomorphism.
The category $\Ab$ has as objects abelian groups, and the maps between two objects are exactly the grouphomomorphisms. We know that the identity function is indeed a grouphomomorphism, and composing two grouphomomorpisms, gives indeed a new grouphomomorphism.
\end{example}
In fact almost any mathematical structure can be described as a category, we have: $\cat{Ring}$ for rings, $\cat{Vect}$ for $\R$-vectorspaces, $\cat{Set_{fin}}$ for finite sets, $\cat{Poset}$ for posets, etc. Of course we would also like to express relations between categories, for example every abelian group is also a set. This idea can be formulated by the notion of a functor.
@ -57,7 +57,7 @@ Given a category $\cat{C}$ and two objects $A, B \in \cat{C}$ we would like to k
\begin{definition}
A map $f: A \to B$ in a category $\cat{C}$ is an isomorphism if there is a map $g: B \to A$ such that:
$$ f \circ g =\id_B \text{ and } g \circ f = id_A.$$
$$ f \circ g =\id_B \text{ and } g \circ f =\id_A.$$
\end{definition}
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, ie. the grouphomomorphisms which are both injective and surjective.
@ -65,4 +65,6 @@ For example the cyclic group $\Z_4$ and the klein four-group $V_4$ are not isomo
@ -39,14 +39,27 @@ In order to see why we are interested in the construction of homology groups, we
In particular $\Delta^0$ is simply a point, $\Delta^1$ a line and $\Delta^2$ a triangle. There are nice inclusions $\Delta^n \mono\Delta^{n+1}$ which we need later on. For any $n \in\N$ we define:
\begin{definition}
For $i \in\{0, \ldots, n+1\}$ the $i$-th face map $\delta^i : \Delta^n \mono\Delta^{n+1}$ is defined as:
$$\delta^i (x_0, \ldots, x_n)=(x_0, \ldots, x_{i-1}, 0, x_{i+1}, \ldots, x_n)\text{ for all } x \in\Delta^n.$$
$$\delta^i (x_0, \ldots, x_n)=(x_0, \ldots, x_{i}, 0, x_{i+1}, \ldots, x_n)\text{ for all } x \in\Delta^n.$$
\end{definition}
Note that if we have any continuous map $\sigma : \Delta^{n+1}\to X$ we can precompose with a face map to get $\sigma\circ\delta^i : \Delta^n \to X$. This will be used for defining the boundary operator. We can make pictures of this, and when concerning continuous maps $\sigma : \Delta^{n+1}\to X$ we will draw the images in the space $X$, instead of functions.
For any space $X$, we will be interested in continuous maps $\sigma : \Delta^n \to X$, such a map is called a $n$-simplex. Note that if we have any continuous map $\sigma : \Delta^{n+1}\to X$ we can precompose with a face map to get $\sigma\circ\delta^i : \Delta^n \to X$, as shown in figure~\ref{fig:diagram_d}. This will be used for defining the boundary operator. We can make pictures of this, and when concerning continuous maps $\sigma : \Delta^{n+1}\to X$ we will draw the images in the space $X$, instead of functions.
\todo{Ch: Make some pictures here}
\begin{figure}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes]{
\Delta^{n+1}& X \\
\Delta^n &\\
};
\path[->]
(m-1-1) edge node[auto] {$\sigma$} (m-1-2)
(m-2-1) edge node[auto] {$\delta^i $} (m-1-1)
(m-2-1) edge node[auto] {$$} (m-1-2);
\end{tikzpicture}
\caption{The $n$-simplex $\sigma$ can be made into a $n+1$-simplex $\sigma\circ\delta^i$}
\label{fig:diagram_d}
\end{figure}
\todo{Ch: Define free abelian group}
\todo{Ch: Make some pictures here}
We now have enough tools to define the singular chain complex of a space $X$.
@ -65,7 +78,5 @@ This might seem a bit complicated, but we can pictures this in an intuitive way,
\end{figure}
\todo{Ch: Proposition: $C(X)\in\Ch{\cat{Ab}}$}
\todo{Ch: Example homology of some space}
\todo{Ch: Show that $\Ch{\Ab}$ is an ab. cat. At least show functoriality $\Hom{\Ch{\Ab}}{-}{-}$}
@ -5,13 +5,15 @@ There are generally two definitions of a \emph{simplicial abelian group}, an abs
\subsection{Abstract definition}
\begin{definition}
We define a category $\DELTA$, where the objects are the finite ordinals $[n]=\{0, \dots, n\}$ and maps are monotone increasing functions.
We define a category $\DELTA$, where the objects are the finite ordinals $[n]=\{0, \dots, n\}$for $n \in\N$and maps are monotone increasing functions: $\Hom{\DELTA}{[n]}{[p]}=\{ f : [n]\to[p]\I f(i)\leq f(j)\text{ for all } i < j \}$.
\end{definition}
There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \emph{degeneracy} maps, defined as (resp.):
$$\delta_i: [n]\to[n+1], k \mapsto\begin{cases} k &\text{if } k < i;\\ k+1&\text{if } k \geq i. \end{cases}\hspace{0.5cm}0\leq i \leq n+1, \text{ and}$$
$$\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{0.5cm}0\leq i \leq n$$
\begin{align*}
\delta_i: [n] \to [n+1], k &\mapsto\begin{cases} k &\text{if } k < i;\\ k+1 &\text{if } k \geq i. \end{cases}\hspace{0.5cm} 0 \leq i \leq n+1, \text{ and}\\
\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{0.5cm} 0 \leq i \leq n
\end{align*}
for each $n \in\N$. The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of these maps. \todo{sAb: Epi-mono factorization of $\DELTA$} So in a sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as in figure~\ref{fig:delta_cat}.
@ -21,7 +23,7 @@ for each $n \in \N$. The nice things about these maps is that every map in $\DEL
\label{fig:delta_cat}
\end{figure}
Althoug 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 maps face maps $\delta_i$ are then exactly maps which point out how we can embed $\Delta^n$ in $\Delta^{n+1}$. This is shown in figure~\ref{fig:delta_cat_geom}. This picutre 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 is 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 eachother. \todo{sAb: how to draw $\sigma_i$?}
Althoug 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 $\Delta^n$ in $\Delta^{n+1}$. This is shown in figure~\ref{fig:delta_cat_geom}. This picutre 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 is 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 eachother. \todo{sAb: how to draw $\sigma_i$?}
\begin{figure}
\includegraphics{delta_cat_geom}
@ -32,12 +34,12 @@ Althoug this is a very abstract definition, a more geometric intuition can be gi
This category $\DELTA$ will act as a protoype for these kind of geometric structures in other categories. This leads to the following definition.
\begin{definition}
An simplicial abelian group $A$ is a contravariant functor:
An \emph{simplicial abelian group}$A$ is a contravariant functor:
$$A: \DELTA\to\Ab.$$
(Or equivalently a covariant functor $A: \DELTA^{op}\to\Ab.$)
\end{definition}
So the category of all simplicial abelian groups, $\sAb$, is the functor category $\Ab^{\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 $A: \DELTA^{op}\to Ab$. And hence we can picture a simplicial abelian group as done in figure~\ref{fig:simplicial_abelian_group}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $A$ is a contravariant functor.
So the category of all simplicial abelian groups, $\sAb$, is the functor category $\Ab^{\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 $A: \DELTA^{op}\to\Ab$. And hence we can picture a simplicial abelian group as done in figure~\ref{fig:simplicial_abelian_group}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $A$ is a contravariant functor.
\begin{figure}
\includegraphics{simplicial_abelian_group}
@ -61,13 +63,13 @@ Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain equatio
\end{align}
\end{lemma}
\begin{proof}
By writing out the definitions given above.
By writing out the definitions given above.\todo{sAb: this is a bit rude, maybe write out some of it...}
\end{proof}
Because a simplicial abelien group $A$ is a contravariant functor, these equations (which only consist of compositions and identities) also hold in its image. For example the first equation would look like: $ A(\delta_i)A(\delta_j)= A(\delta_{j-1})A(\delta_i)$ for $ i < j$ (again note that $A$ is contravariant, and hence composition is reversed). This can be used for an explicit definition of simplicial abelien groups. In this definition a simplicial abelian group $A$ consists of a collection abelian groups $(A_n)_{n}$ together with face and degeneracy maps (which are grouphomomorphisms) such that the simplicial equations hold. More precisely:
Because a simplicial abelien group $A$ is a contravariant functor, these equations (which only consist of compositions and identities) also hold in its image. For example the first equation would look like: $ A(\delta_i)A(\delta_j)= A(\delta_{j-1})A(\delta_i)$ for $ i < j$. This can be used for an explicit definition of simplicial abelien groups. In this definition a simplicial abelian group $A$ consists of a collection abelian groups $A_n$ together with the face and degeneracy maps. More precisely:
\begin{definition}
\emph{(Explicitly)} An simplicial abelian group $A$ consists of a collection abelian groups $A_n$ together with face maps $d_i : A_n \to A_{n-1}$ and degenracy maps$s_i : A_n \to A_{n+1}$ for $0\leq i \leq n$ and $n \in\N$, such that:
\emph{(Explicitly)} An simplicial abelian group $A$ consists of a collection abelian groups $A_n$ together with grouphomomorphisms $d_i : A_n \to A_{n-1}$ and $s_i : A_n \to A_{n+1}$ for $0\leq i \leq n$ and $n \in\N$, such that:
\begin{align}
d_i d_j &= d_{j-1} d_i \hspace{0.5cm}\text{ if } i < j,\\
d_i s_j &= s_{j-1} d_i \hspace{0.5cm}\text{ if } i < j,\\
@ -77,7 +79,7 @@ Because a simplicial abelien group $A$ is a contravariant functor, these equatio
\end{align}
\end{definition}
It is already indicated that a functor from $\DELTA^{op}$ to $\Ab$ is determined when the images for the face and degeneracy maps in $\DELTA$ are provided. So gives this a way of restoring the first definition from this one. Conversely, we can apply functorialty to obtain the second definition from the first. So these definitions are the same \todo{sAb: is it ok not to prove this?}. So from now on we will denote $A([n])$ by $A_n$, $A(\sigma_i)$ by $s_i$ and $A(\delta_i)$ by $d_i$, whenever we have a simplicial abelien group $A$.
It is already indicated that a functor from $\DELTA^{op}$ to $\Ab$ 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 functorialty to obtain the second definition from the first. So these definitions are the same \todo{sAb: is it ok not to prove this?}. So from now on we will denote $A([n])$ by $A_n$, $A(\sigma_i)$ by $s_i$ and $A(\delta_i)$ by $d_i$, whenever we have a simplicial abelien group $A$.
When using a simplicial abelian group 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.
@ -133,7 +135,7 @@ Note that indeed $\Hom{\DELTA}{X}{[n]} \in \Set$, because the collection of morp
\end{tikzpicture}.$$
\end{example}
As we are interested in simplicial abelian group, it would be nice to make these $n$-simplices into simplicial abelian groups. We have seen how to make an abelian group out of any set using the free abelian group. We can use this functor $\Z[-] : \Set\to\Ab$ to induce a functor $\Z^\ast[-] : \sSet\to\sAb$ as shown in the diagram~\ref{fig:diagram_Z}.
As we are interested in simplicial abelian group, 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. We can use this functor $\Z[-] : \Set\to\Ab$ to induce a functor $\Z^\ast[-] : \sSet\to\sAb$ as shown in the diagram~\ref{fig:diagram_Z}.
Joshua Moerman
11 years ago