Mathematisches Institut WS 2001/02 der Universität München Set 10 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(37) Let f, g : X ⟶ Y be two maps. Show that the set {x ∈ X|f(x) = g(x)} with the inclusion map into X is an equalizer of f, g : X ⟶ Y.

(38) (a) Let the commutative diagram

[figure: B; C; g; P; A; p; q; f]

be a pullback (a limit) of the morphisms f : A ⟶ C and g : B ⟶ C. Assume that g is a monomorphism. Show that p is also a monomorphism.

(b) Show that the category of sets has pullbacks.

(39) Let X and Y be two sets. Show that the disjoint union X ˙∪Y is a coproduct of X and Y in the category of sets.

(40) Show that the category of sets has coequalizers.

Due date: Tuesday, 8.1.2002, 16:15 in Lecture Hall 138