Category Theory: August 8

Covered in Class

1. Epimorphisms and Isomorphisms

   • In Set, monic and epic correspond to injective and surjective.

   • In Mon, monomorphisms are injective functions but epimorphisms are more general;
     for example the injection  ℕ → ℤ is monic and epic but not a surjection.

2. Initial and Terminal Objects.

   • Proved that any two initial objects must be isomorphic.

Suggested Exercises

All exercises are optional; pick any that look interesting and feel free to create your own! I'll be around in the lab on Tuesday from 10-11 and from 12 to about 1:30 if anyone wants to discuss solutions; also I encourage you to work together and discuss the material with each other! Feel free to post also to the #category-theory Slack channel.

See also the exercises in Gilbert Bernstein's Machines, Monoids, Categories handout.

1. [The same as question 3 last week.] Given an arrow f : A→ B in Mon, let |f| : |A|→ |B| be the image in Set under the forgetful functor. Prove that if |f| is injective then f is monic and if |f| is surjective then f is epic. Try also to prove the converses of each statement, and note where the difficulty is. In the monic case it can be proved by creating a free monoid from |A| (see Example 2.3 in Awodey), and as shown in class the epic converse does not hold.

2. Prove that for any category if an arrow f is an isomorphism, it is also both a monomorphism and an epimorphism.

3. Try exercises 2, 3, 4 and 5 in Pierce, section 1.3.10.

4. Describe the initial and terminal objects (if any) in the following categories:

   • Set⊆

   • Mon

   • A category with a single object (explain what constraints on arrows might be appropriate).

   • Cat