Category Theory: August 15

Covered in Class

1. Products

   • Product in Set and generalization to any category. Mac Lane's Concepts and Categories in Perspective includes a brief personal history of his development of the subject and mentions products.

   • Proof that any two products are isomorphic up to unique isomorphism.

   • "Universal constructions" such as the product are terminal objects in an appropriately defined category.

2. Coproducts

   • Coproduct in Set (disjoint union); we defined it (as tagged pairs) but did not yet prove it is the coproduct.

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. Carefully define the appropriate "wedge category" and prove that the product is the terminal object in it. See Pierce 1.6.

2. Try any exercises in section 1.5.6 of Pierce. We will do exercises 4 and 5 in class next week.

3. Let (P, π₁, π₂) and (P, ρ₁, ρ₂) both be products of objects A and B in a category (note that the object P is the same in both triples). Prove that  π₁ = ρ₁∘ f and π₂ = ρ₂∘ f, where f : P→ P is an isomorphism of P to itself (known as an automorphism). Note that f could be the identity of P.