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 ∈ 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 ∈ 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”.