# Category Theory: August 29

### Covered in Class

Products and Coproducts

Product (meet) and coproduct (join) in Preorder/Poset.

Rel category in Generating Compiler Optimizations from Proofs.

Pushouts

Pushouts and coproducts are essentially the same in Poset.

Given disjoint sets, a pushout (with ⊥ in the upper left corner) or coproduct can represent an untagged union. The example given was the union of even and odd integers.

Given two sets, and their intersection (with inclusion functions into each), the pushout is the union (with inclusions from the two sets into the union). If the upper left corner is not the intersection, then one must start dealing with equivalence classes, as we will explore further next time.

### 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.

Read through section 3 of Tate's paper, look up pushouts and pullbacks in your favorite category theory book, and try to understand as much as you can before our final class on September 12. Note that next week (September 5) will be a bonus class on unrelated material.