\def\Cat#1{\hbox{${\cal #1}$}}
\def\Hom{\hbox{Hom}}
\def\Top#1{\hbox{$\hbox{${\cal T\!\scriptstyle\cal OP}$}(#1)$}}
\def\Set{\hbox{${\cal S\!\scriptstyle\cal ET}$}}
\parindent0in
\parskip=\baselineskip
\hbox to\hsize{\hfil \bf Homework 1: Math 953 Spring 2005\hfil}
\vskip\baselineskip
\hbox to\hsize{\hfil Due January 21, 2005\hfil}
Let \Cat C and \Cat D be categories.
{\bf Definition}: A (covariant) {\it functor\/} $F:\Cat C\to\Cat D$ is
an assignment of an object $F(A)$ of \Cat D for each object
$A$ of $\Cat C$ and a collection of maps $F: \Hom(A,B)\to\Hom(F(A),F(B))$
for each pair of objects $A$ and $B$ of $\Cat C$, such that
$F(1_A)=1_{F(A)}$ and whenever $f\circ g$ is defined,
$F(f\circ g)=F(f)\circ F(g)$.
{\bf Definition}: A {\it contravariant functor\/} $F:\Cat C\to\Cat D$ is
an assignment of an object $F(A)$ of \Cat D for each object
$A$ of $\Cat C$ and a collection of maps $F: \Hom(A,B)\to\Hom(F(B),F(A))$
for each pair of objects $A$ and $B$ of $\Cat C$, such that
$F(1_A)=1_{F(A)}$ and whenever $f\circ g$ is defined,
$F(f\circ g)=F(g)\circ F(f)$. (Alternatively, a contravariant functor
$F:\Cat C\to\Cat D$ is just a functor $\Cat C^{\rm op}\to\Cat D$.)
{\bf Definition}: A functor $F:\Cat C\to\Cat D$ is
an {\it isomorphism\/} of categories if there is a functor
$G: \Cat D\to \Cat C$ such that $FG$ is the identity functor
on \Cat D and $GF$ is the identity functor on \Cat C.
{\bf Definition}: Let $f\in \Hom(B,A) $ be an arrow in a category \Cat C.
We say that $f$ is a {\it monomorphism\/} if for every object $C$,
there never are arrows $g,h\in \Hom(C,B)$ such
that $fg=fh$ but $g\ne h$.
(1) Let $G$ and $H$ be groups and let \Cat G (\Cat H, resp.) be the
category with a single object whose arrows are the elements of $G$
($H$, resp.), with the group law giving composition of arrows.
Show that there is a bijection between functors $F:\Cat G\to \Cat H$
and homomorphisms $f:G\to H$, and that $F$ is an isomorphism if and only
if its corresponding homomorphism $f$ is.
(2) Let $X$ and $Y$ be topological spaces
and let \Top X and \Top Y be the corresponding
categories. For each continuous map $f:X\to Y$
define a functor $F_f:\Top Y\to \Top X$ such that
$F_f$ is an isomorphism if $f$ is a homeomorphism.
If $F_f$ is an isomorphism, must $f$ be a homeomorphism?
Is every functor $F:\Top Y\to \Top X$ of the form $F_f$
for some $f$?
(3) Show that a monomorphism in the category \Set\/ of sets
is the same thing as an injective map.
(4) Define {\it epimorphism\/} in an arbitrary category
such that an epimorphism in the category \Set\/ of sets
is the same thing as a surjective map.
Compare epimorphisms in a category \Cat C with
monomorphisms in $\Cat C^{op}$.
\bye