### Classical vs Quantum Computation (Week 15)

#### Posted by John Baez

In this week’s class on Classical vs. Quantum Computation, we continued to work through an example of how typed $\lambda$-calculi give cartesian closed categories:

- Week 15 (Feb. 22) - The λ-theory of commutative rings and the cartesian closed category it generates: the "free cartesian closed category on a commutative ring object". What is a cartesian closed functor from this to Set? Guess: just a commutative ring! Blog entry.

Last week’s notes are here.

Despite my promise last week, we didn’t get to the ‘typed λ-calculus for high school calculus’ this time — the students were clearly struggling with the basics of categorical logic! I should have discussed Lawvere theories before these ‘typed λ-calculi’, since cartesian categories are a simpler context than cartesian closed categories for discussing these basics. But, I’m sort of committed to focusing on closed categories of various sorts in this class.