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 introduce the notions of categories, functors, isomorphisms, natural transformations, equivalences, adjunctions and the Yoneda lemma.
\subsection{Categories}
\subsection{Categories}
\todo{CT: Where to start...?}
\begin{definition}
\begin{definition}
A \emph{category}$\cat{C}$ consists of a collection of \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:
A \emph{category}$\cat{C}$ consists of a collection of \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}
\begin{itemize}
@ -85,6 +86,7 @@ Note that an isomorphism between to categories is now also defined. Two categori
This now also gives a notion of isomorphisms between functors. It can be easily seen that a isomorphism between two functors is a natural transformation which is an isomorphism pointwise. Such a natural transformation is called a natural isomorphism.
This now also gives a notion of isomorphisms between functors. It can be easily seen that a isomorphism between two functors is a natural transformation which is an isomorphism pointwise. Such a natural transformation is called a natural isomorphism.
\todo{CT: Hom-functor}
\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:
$$ FG =\id_\cat{D}\text{ and }\id_\cat{C}= GF. $$
$$ FG =\id_\cat{D}\text{ and }\id_\cat{C}= GF. $$
@ -133,7 +135,7 @@ Now there are different definitions of adjunctions, which are equivalent. We wil
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}:
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}:
$$\eps: FG \to\id. $$
$$\eps: FG \to\id. $$
It can be shown that an equivalence $F: \cat{C}\tot{\simeq}\cat{D}$ is both a left and right-adjoint. We skecth the proof of $F$ being a left-adjoint. Clearly we already have the natural transformation $\eta: \id_\cat{C}\to GF$. To construct $\overline{f}$ from $f: S \to G(A)$ we can apply the functor $F$, to get $F(S)\to FG(A)$, using the other natural isomorphism we get $F(S)\to FG(A)\to A$. We leave the details to the reader.
It can be shown that an equivalence $F: \cat{C}\tot{\simeq}\cat{D}$ is both a left and right-adjoint. We sketch the proof of $F$ being a left-adjoint. Clearly we already have the natural transformation $\eta: \id_\cat{C}\to GF$. To construct $\overline{f}$ from $f: S \to G(A)$ we can apply the functor $F$, to get $F(S)\to FG(A)$, using the other natural isomorphism we get $F(S)\to FG(A)\to A$. We leave the details to the reader.
The first definition of adjunction is useful when dealing with maps, since it gives an bijection between the $\mathbf{Hom}$-sets. However the second definition is useful when proving a certain construction is part of an adjunction, as shown in the following example.
The first definition of adjunction is useful when dealing with maps, since it gives an bijection between the $\mathbf{Hom}$-sets. However the second definition is useful when proving a certain construction is part of an adjunction, as shown in the following example.
@ -158,7 +160,6 @@ The first definition of adjunction is useful when dealing with maps, since it gi
\end{example}
\end{example}
\subsection{Yoneda}
\subsection{Yoneda}
\todo{CT: Hom-functor}
So far we have only encountered definitions from category theory. However there is a very important lemma by Yoneda. This lemma gives a nice way to construct certain natural transformations.
So far we have only encountered definitions from category theory. However there is a very important lemma by Yoneda. This lemma gives a nice way to construct certain natural transformations.
\begin{definition}
\begin{definition}
@ -172,5 +173,6 @@ So far we have only encountered definitions from category theory. However there
$$\mathbf{Nat}(y(C), F)\iso F(C), $$
$$\mathbf{Nat}(y(C), F)\iso F(C), $$
which is natural in both $F$ and $C$, where $\mathbf{Nat}(G, G')$ denotes the set of natural transformation between $G$ and $G'$, in other words $\mathbf{Nat}=\mathbf{Hom}_{\Set^{\cat{C}^{op}}}$.
which is natural in both $F$ and $C$, where $\mathbf{Nat}(G, G')$ denotes the set of natural transformation between $G$ and $G'$, in other words $\mathbf{Nat}=\mathbf{Hom}_{\Set^{\cat{C}^{op}}}$.
\end{lemma}
\end{lemma}
\todo{CT: Prove Yoneda? I guess not...}
We will use this lemma when we discuss simplicial abelian groups.
We will use this lemma when we discuss simplicial abelian groups.
@ -166,7 +166,7 @@ Note that this is also the definition of the Yoneda embedding $\Delta[n] = y[n]$
\subsection{Other simplicial objects}
\subsection{Other simplicial objects}
Of course the abstract definition of simplicial abelian group 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 $\sAb=\Ab^{\DELTA^{op}}$ of simplicial abelian groups. 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.
Of course the abstract definition of simplicial abelian group 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 $\sAb=\Ab^{\DELTA^{op}}$ of simplicial abelian groups. 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:ChainComplexes}. For two natural transformations $f,g: A \to B$ we simply define $f+g$ pointwise: $(f+g)_n = f_n + g_n$.
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:ChainComplexes}. For two natural transformations $f,g: A \to B$ we simply define $f+g$ pointwise: $(f+g)_n = f_n + g_n$.
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. 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 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 following diagram.
Joshua Moerman
12 years ago