# Hopf joke? (HW from class on April 5)

What did 0 say to 8?

1. (Here’s my wild stab at it.)

Nice coproduct!

(Refer to page 9 of the lecture notes on the incidence coalgebra.)

2. Playing off the “nice belt” comment from yesterday’s class: “You’re no poset!”

(Violation of the antisymmetry criteria.)

3. “You ought to get blown up.”

Oh wait, it’s not supposed to be an algebraic geometry joke.

Julia: how are you checking the antisymmetry of the set 8? 😮

• I didn’t see this comment until much later, but I was imagining 8 to be a poset-looking-thing with a top element and a bottom element and two elements on the same level in the middle, both connected to one another, and antisymmetry specifically prohibits horizontal connections.

• @Julia: ah! When I think of an 8, it usually has just one point in the middle. That would be a poset, even a ranked one.

• Most definitely. It depends on how you draw it – like a sideways infinity or like a theta. (and we never see them drawn as thetas unless they’re on a numeric LED)

The context for this is when Federico told his Hopf joke in class, it implied that an 8 was a 0 with a belt. That’s where I got the visual from.

4. Here’s an unrelated Hopf joke:

What did Santa Claus say to the bialgebra $\mathbb{K} G$, where $G = 2^S$ for some finite set $S$, on Christmas day? “Hopf, Hopf, Hopf! You’ve been a bad bialgebra, so there’s no antipode for you!”