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Problems for Category theory and homological algebra February 3, 2016 Milan Lopuhaä Notation for some categories: Category Set G-Set Grp Ab Ring Pos Top Haus Top∗ Veck Objects Sets G-Sets (see problem 1.6) Groups Abelian groups Rings with 1 Partially ordered sets Topological spaces Hausdorff topological spaces Pointed topological spaces k-vector spaces Morphisms Maps G-equivariant maps Group homomorphisms Group homomorphisms (Unitary) Ring homomorphisms Monotone functions Continuous maps Continuous maps Pointed continuous maps k-linear maps 1. (a) Prove that the composition of two monomorphisms is again a monomorphism. (b) Prove that the composition of two epimorphisms is again an epimorphism. (c) Let f , g be morphisms such that g ◦ f is a monomorphism. Prove that f is a monomorphism. (d) Let f , g be morphisms such that g ◦ f is an epimorphism. Prove that g is an epimorphism. 2. (a) Show that the embedding Z → Q is an epimorphism in Ring. (b) An (additively written) abelian group A is called divisible if for each a ∈ A and each n ∈ Z>0 there exists an element b ∈ A such that nb = a. The category of divisible abelian groups (with group homomorphisms as morphisms) is denoted Div. Show that the quotient map Q → Q/Z is a monomorphism in Div. (c) Show that in Top the monomorphisms are the injective continuous maps whereas the epimorphisms are the surjective continuous maps. (d) Show that in Haus a continuous map is an epimorphism if and only if its image is dense. 1 (e) (Bonus exercise) Prove that a homomorphism f : G → H in Grp is a monomorphism if and only if f is injective, and is an epimorphism if and only if f is surjective. 3. Let G be a group. A G-set is a set X together with an action of G on X. A G-equivariant map between G-sets is a map f : X → Y such that f (g · x) = g · f (x) for all g ∈ G and x ∈ X. (a) Prove that there exists a category G-Set whose objects are the G-sets and whose morphisms are the G-equivariant maps. (b) Do products and coproducts exist in G-Set ? If so, describe them concretely. 4. Describe products, coproducts, monomorphisms and epimorphisms in the category Pos. 5. Let n be a squarefree integer, i.e., there is no prime number p such that p2 | n. We define a category Cn as follows. The set of objects is ob Cn = d ∈ Z≥1 d divides n , and the set of morphisms between two objects a and b is Z/mZ if b = ma for some integer m, HomCn (a, b) = ∅ otherwise. (a) Let a, b, c ∈ ob Cn such that c = kb and b = ma. Let f : a → b and g : b → c be two morphisms. Let (g ◦ f ) : a → c be a morphism such that (g ◦ f ) ≡ f mod m and (g ◦ f ) ≡ g mod k. Show that g ◦ f exists and is unique, and that Cn is a category with this composition (and with the unique element of Z/Z as the identity of every object). (b) Let a, b ∈ ob Cn . Show that a × b exists if and only if gcd(a, b) = 1. (c) Describe the monomorphisms and epimorphisms in Cn . (d) Show that the categories Cn and Cnop are isomorphic. 6. Prove that there exists a functor F : Ring → Grp such that for all rings R we have F (R) = R× , the unit group of R. 7. Show that there does not exist a functor F : Grp → Ab such that F (G) is the center of G for every group G. Hint: consider S2 −→ S3 −→ S2 . 8. (a) Let X be a topological space. Let the relation X on X be given by x X y if and only if x ∈ {y}. Show that X is a partial ordering on X. (b) Let Pos be the category of partially ordered sets. Show that there exists a functor F : Top → Pos such that F (X) = (X, X ) for all X ∈ Top. 2 9. (a) Let A, B ∈ Ab. Show that the direct product A × B, together with the projection maps, is the categorical product of A and B in Ab. (b) Let A, B ∈ Ab. Show that the direct product A × B, together with the inclusion maps A → A × B and B → A × B given by a 7→ (a, eB ) and b 7→ (eA , b), is also the categorical coproduct of A and B in Ab. (c) Give examples of abelian groups A and B such that the direct product A × B is not the categorical coproduct of A and B in Grp. (Prove that the example you give has the required property.) 3