Download Problems for Category theory and homological algebra

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.
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(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.
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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.)
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