Math 426, Winter 2014, Term 1
Problems and Categories and Groupoids
Problem 1.
Prove that every group is naturally isomorphic to its opposite group. This means
the following: Let G be the category of groups, and F : G → G the functor mapping
a group to its opposite group. Then there exists an invertible natural transformation
θ : idG → F .
Problem 2.
Let C and B be categories. Let F : B → C be a functor which is fully faithful and
essentially surjective. Prove that F is an equivalence of categories.
Problem 3.
Prove that in a connected groupoid all automorphism groups are isomorphic to each
other.
Problem 4.
Let G be a group acting on a set X from the right. Construct a groupoid whose
set of objects is X, and whose set of morphisms is X × G. This groupoid is called
a transformation groupoid.
Problem 5.
A groupoid is called finite, if evry object has finite automorphism group, and the
set of isomorphism classes is finite. Let Γ be a finite groupoid. Prove that
#Γ =
X
[x]
1
# Aut(x)
is a well-defined rational number. The sum is over all isomorphism classes of Γ, for
an object x, its isomorphism class is denoted [x].
Prove that if Γ is the transformation groupoid associated to the action of a finite
group G on a finite set X, then
#Γ =
#X
.
#G
Problem 6.
Construct the Galois covering spaces of the figure eight which correspond to the
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dihedral group with 8 elements and the quaternionic group with 8 elements. Here
we use D8 = ha, b | a4 = 1, b2 = 1, baba = 1i and Q8 = ha, b | a2 = b2 , bab = ai.
Problem 7.
Find the maximal abelian cover of a bouguet of n circles.
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