# Category Theory: August 22

### Covered in Class

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.

Coproducts

Coproduct in Set (disjoint union).

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.

Prove that a coproduct is unique up to unique isomorphism.

Prove that an equalizer is unique up to unique isomorphism.

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.

Try some or all exercises in Pierce section 1.7.4.