@ -52,7 +52,7 @@ Given a category $\cat{C}$ and two objects $A, B \in \cat{C}$ we would like to k
\end{definition}
\end{definition}
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, ie. the group homomorphisms which are both injective and surjective.
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, ie. 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.
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 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 $GF =\id_\cat{C}$.
@ -80,7 +80,7 @@ Note that an isomorphism between to categories is now also defined. Two categori
For any two categories $\cat{C}$ and $\cat{D}$ we can form a category with functors $F: \cat{C}\to\cat{D}$ as objects and natural transformations as maps. This category is called the \emph{functor category} and is denoted by $\cat{D}^\cat{C}$.
For any two categories $\cat{C}$ and $\cat{D}$ we can form a category with functors $F: \cat{C}\to\cat{D}$ as objects and natural transformations as maps. This category is called the \emph{functor category} and is denoted by $\cat{D}^\cat{C}$.
\end{lemma}
\end{lemma}
\begin{proof}
\begin{proof}
We refer to MacLane or Awodey.
We refer to MacLane \cite{maclane}or Awodey\cite{awodey}.
\end{proof}
\end{proof}
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.
Joshua Moerman
12 years ago