Category Theory: August 22

Covered in Class

1. Products

   • The product is "minimal" with respect to objects that make the diagram commute. For example for A = {1,2} and B = {3,4}, if P = {(1,3}, {1,4}, {2,3}} then for some objects C there will be no h : C → P that works, and if P = {(1,3}, {1,4}, {2,3}, {2,4}, {5,6}} there are multiple arrows that work. 

2. Coproducts

   • Coproduct in Set (disjoint union).

3. Equalizers

   • Equalizer in Set.

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. Prove that a coproduct is unique up to unique isomorphism.

2. Prove that an equalizer is unique up to unique isomorphism.

3. Define the coequalizer of a pair of arrows and prove it is unique up to unique isomorphism. Construct the coequalizer in Set and prove its correctness. 

4. Try some or all exercises in Pierce section 1.7.4.