der Universitat Munchen Set 7
Prof. Dr. B. Pareigis
Problem set for
Advanced Algebra
(25) Let(AB;p
A
;p
B
)bethe productof AandB inC. Thenthere
isa naturalisomorphism
Mor(-;AB)
= Mor
C
(-;A)Mor
C (-;B):
(26) LetC be a category with nite products. Show that there is a
bifunctor--:CC !C suchthat (--)(A;B) isthe object
of a product of A and B. We denote elements in the image of
this functor by AB :=(--)(A;B)and similarlyf g.
(27) LetF :C !D be an equivalence with respect to G : D ! C,
':GF
= Id
C
, and :FG
= Id
D
. Show that G :D ! C is an
equivalence. Showthat G isuniquely determined by F up toa
naturalisomorphism.
(28) (a) Given V 2K-Mod. ForA2K-Alg dene
F(A):=ff :V !Ajf K-linear;8v;w2V :f(v)f(w)=0g:
Show that this denes afunctor F :K-Alg !Set.
(b) Show that F has the algebra D(V) as constructed in Ex-
ercise 2.1 (3)asa representing object.
