EXERCISES FOR MATHEMATICS 246AFALL 2010Hatcher’s book is the default source for references.I . Foundational materialI.1 : Categories and functors1. Definition. A morphism f : A → B in a category is a monomorphism if for allg, h : C → A we have that foh = fog only if h = g. Dually, a morphism f : A → B in a categoryis an epimorphism if for all u, v : B → D we have that uof = vof only if u = v.(a) Prove that a monomorphism in the category Set is 1 − 1 and an epimorphism in Set isonto. [Hint: Prove the contrapositives.](b) Prove that in the category of Hausdorff topological spaces (and continuous maps) a mor-phism f : A → B is an epimorphism if f(A) is dense in B.(c) Prove that the composite of two monomorphisms is a monomorphism and the compositeof two epimorphisms is an epimorphism.(d) A morphism r : X → Y in a category is called a retract if there is a morphism q : Y → Xsuch that qr = idX. For example, in the category of sets or topological spaces the diagonal mapdX: X → X × X is a retract with q = projection onto either factor. Prove that every retract is amonomorphism.(e) A morphism p : A → B in a category is called a retraction if there is a morphism s : B → Asuch that qor = idB. For example, if r and q are as in (d) then q is a retraction. Prove that everyretract is a monomorphism and every retraction is an epimorphism.2. Let A be a category, and let f : A → B be a morphism in A such thatMorph (f, C) : Morph (B, C) → Morph (A, C)is an isomorphism for all objects C in A. Prove that f is an isomorphism. [Hint: Choose C = B orA and consider the preimages of the identity elements.] Also prove the (relatively straightforward)converse.3. An object 0 is called an initial object in the category A if for each object A in A thereis a unique morphism 0 → A. An object 1 is a terminal object in A if for each object A there is aunique morphism A → 1.(a) Prove that the empty set is initial and every one point set is terminal in Set.(b) Prove that a zero-dimensional vector space is both initial and terminal in the categoryVec–F of vector spaces over a field F .1(c) Prove that every two initial objects in a category A are uniquely isomorphic (there is aunique isomorphism from one to the other), and similarly for terminal objects.(d) If A contains an object Z that is both initial and terminal (a null object), prove that foreach pair of objects A, B in A there is a unique morphism A → B that factors as A → Z → B.Also, if W is any other such object, prove that this composite equals the composite A → W → B.[Hint: Consider the unique maps from W to Z and vice versa.]4. Prove that a covariant functor takes retracts to retracts and retractions to retractions.State the corresponding result for contravariant functors.5. If E is a terminal object in the category A and f : E → X is a morphism in A, provethat f is a monomorphism (in fact, something stronger is true—what is it?).6. Let A = (N+, Morph , ϕ), where N+denotes the positive integers, Morph (p, q) denotesall p × q matrices with integer coefficients, andϕ : Morph (p, q) × Morph (q, r) → m(p, r)is matrix multiplication. Verify that A is a category.7. If f is a morphism in a category A, a morphism g (in the same category) is called aquasi-inverse for f if and only if fogof = f. Prove that every morphism that has a quasi-inverseis itself the quasi-inverse of some morphism in the category.8. In the category of sets, show that the Axioms of Choice implies that every mappinghas a quasi-inverse. Also, in the matrix category of Exercise 6, show that every matrix has aquasi-inverse. [Hint: Look at the associated linear transformations, and choose bases in a suitablemanner.]NOTE. In fact, there are canonical choices of quasi-inverses. See the following Wikipedia articlesfor further information on generalizations of matrix inverses:http://en.wikipedia.org/wiki/Moore-Penroseinversehttp://en.wikipedia.org/wiki/Group inversehttp://planetmath.org/encyclopedia/DrazinInverse.html9. Suppose that C is a category in which every map has a quasi-inverse. Prove thatevery monomorphism in C is a retract. Using this, give examples of mappings in the category oftopological spaces (and continuous mappings) which do not have quasi-inverses.10. Let A and B be small categories. Prove that one can define a product category A × Bwhose objects are given by ordered pairs (X, Y ), where X and Y are objects of A and B respectively,whose morphisms are given by ordered pairs (f, g) of morphisms f in A and g in B, and whosedomain, codomain and composition operations are given as follows:Domain(f, g) =Domain(f), Domain(g)Codomain(f, g) =Codomain(f), Codomain(g)(f1, g1)o(f0, g0) = (f1of0, g1og0)2Prove that A×B with these definitions of objects, morphisms, domains, codomains and compositionforms a category, and show that “projections onto the first and second coordinates” define covariantfunctors from this category into A and B respectively.11. Suppose that we are in a category C with morphisms f : X → Y and g : Y → Z. Provethat if any two of f, g and gof are isomorphisms, then so is the third.12. Let IC0be the category whose objects are open intervals in the real line and whosemorphisms are continuous mappings, and let IC1be the subcategory with the same objects, butwhose morphisms are maps with continuous first derivatives. Give an example of a morphism inIC1which is an isomorphism in IC0but not in IC1(hence subcategories are not necessarily closedunder taking inverses).13. Let {Xα} be an indexed family of objects in a category C. Then a categorical productof the Xαis given by an object P and morphisms pα: P → Xαsuch that for each indexed familyof maps fαfrom a fixed object Y into the objects Xα, there is a unique f : Y → P such thatpαof = fαfor all α. — All the standard examples of product constructions turn out to have thisproperty.(a) Prove that if (P, pα) and (Q, qα) are categorical products, then there is a unique isomor-phism h : Q → P such that qα= pαoh for all α. [Hint: The only morphism ϕ from P to itselfsatisfying pα= pαoϕ for all α is the identity.](b) Formulate the dual notion of coproduct in a category (a product in the opposite category),and state the dual of the conclusion in (a).(c) Show that the (external) direct sum is both a product and coproduct in VECFfor finitefamilies of vector spaces, and show that the coproduct can be viewed as a proper subspace of