INTRODUCTION TO CATEGORY THEORY, II Homework 4, due Tuesday Aug 21. More adjunctions. 1. (Riehl 4.5.v) Show that a morphism f : x ! y in C is a monomorphism if and only if the square (∗) 1 x x > x 1x f _ _ x > y f is a pullback. Conclude that right adjoints preserve monomorphisms and that left adjoints preserve epimorphisms. 2. For a group G, let [G; G] denote the commutator subgroup of G. That is, [G; G] is the group generated by the elements of the form ghg−1h−1, where g; h 2 G. Let F : Ab ! Group be a the forgetful functor. Let A: Group ! Ab be the functor taking each group G to G=[G; G] (called the abelianization of G), and each homomorphism f : H ! G to the induced homomorphism A(f): H=[H; H] ! G=[G; G]: (1) Show that [G; G] is a normal subgroup of G and that G=[G; G] is abelian. (2) Show that A is a left adjoint of F . 3. Let F : Ab ! Group be the forgetful functor. Does F have a right adjoint? 4. Let X be a fixed set. Show that the functor X × −: Set ! Set cannot have a left adjoint unless X is a one-point set. 5. Let I : Set ! Top be the right adjoint to the forgetful functor Top ! Set. Show that I has no right adjoint. 6. (Riehl, 4.5 vii, parts i and ii) Consider a reflective subcategory inclusion D ,!C with reflector L: C!D. (1) Show that ηL = Lη and that these natural transformations are isomor- phisms. (2) Show that an object c 2 C is in the essential image of the inclusion D ,!C, meaning that it is isomorphic to an object in the subcategory D, if and only if ηc is an isomorphism. Date: August 14, 2018. 1 2 INTRODUCTION TO CATEGORY THEORY, II 7. (Riehl, 4.5 vii, part iii) Show that the essential image of D consists of those objects c that are local for the class of morphisms that are inverted by L. That is, c is in the essential image if and only if the pre-composition functions f ∗ C(b; c) −! C(a; c) are isomorphisms for all maps f : a ! b in C for which Lf is an isomorphism in D. This explains why the reflector is also referred to as "localization"..