der Universitat Munchen Set 7
Prof. Dr. B. Pareigis
Problem set for
Quantum Groups and Noncommutative Geometry
(25) Letthe set Z together with the multiplication m:Z Z !Z be a monoid.
Show that the unit element e2Z is uniquely determined.
Let (Z;m) be a group. Show that also the inverse i : Z ! Z is uniquely
determined.
Showthat unitelementandinverses ofgroupsarepreserved bymapsthat are
compatiblewith the multiplication.
(26) Findan example of monoidsY and Z and a map f :Y !Z with f(y
1 y
2 )=
f(y
1 )f(y
2
)for all y
1
;y
2
2Y,but f(e
Y )6=e
Z .
(27) Let (G;m) be a group in C and i
X
: G(X) ! G(X) be the inverse for all
X 2C. Show that i is anatural transformation.
Show that the Yoneda Lemma provides a morphism S : G ! G such that
i
X
=Mor
C
(X;S)=S(X) forall X 2C.
Formulateand prove properties of S of the type Sid=:::.
(28) LetC bea category with nite products. Show that amorphismf :G !G 0
inC is a homomorphismof groupsif and onlyif
GG G
- m
G 0
G 0
G 0
- m
0
? ff
? f
commutes.
