Summary: Rings and Algebras Problem set #8. Nov. 10, 2011.
1. A morphism in a category C is a monomorphism if fx = fy implies x = y for morphisms x, y in C. Similarly, g
is an epimorphism if xg = yg implies x = y.
a) Describe monomorphisms and epimorphisms in the category CS (the category with one object and a prescribed
monoid S as the only Homset).
b) Give an example of a monoid for which there are morphism in CS which are both monomorphisms and
epimorphisms but which are not isomorphisms. (Recall that a morphism f is an isomorphism if there is a
morphism g such that fg and gf both exist and are equal to the corresponding identity map.
2. Determine whether the notions of monomorphisms and injective maps (or epimorphisms and surjective maps)
coincide in the following categories:
a) AB; b) RING; c) TOP; d*) GRP.
3. Determine, which of the following correspondences define a functor with the obvious action on the morphisms:
a) GRP # GRP, G # G # ;
b) GRP # GRP, G # Z(G);
c) AB # AB, A # t(A), where t(A) is the torsion subgroup of A;
d) RNG # RNG, R # J(R).
4. A product of objects A i (i # I) in a category C is a system (A, # i ) with A # Ob C and # i # Hom C (A, A i ) so that
whenever we have a system of morphisms # i : B # A i then there exists a unique # : B # A so that # i = # # #
for each i. The coproduct of these objects is defined dually.
a) Prove that products and coproducts -- if they exist in a category C -- are unique, up to isomorphism.