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(b) Show that the category of sets has pullbacks

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(1)

der Universitat Munchen Set 10

Prof. Dr. B. Pareigis

Problem set for

Advanced Algebra

(37) Let f;g : X ! Y be two maps. Show that the set fx 2

Xjf(x)= g(x)g with the inclusionmap into X is an equalizer

of f;g :X !Y.

(38) (a) Let the commutativediagram

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 alsoa 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

isa coproduct of X and Y in the category of sets.

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

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