I feel like the misère form of a non-loopy game should still be non-loopy, but it might not make a huge difference.

No, leading to a terminal node. Otherwise the misère version of the plain zero game would be dud which is a draw.

OK, so that leaves the question of Hartogs numbers of proper classes in ZF unclear. Oh well.

Oh, that's a shame. So there's no model of this in ZF. Well, that answers all my questions on the topic thoroughly: the class of atoms can have any Hartogs number that's an infinite successor cardinal, and with replacement instead of collection it can also be an infinite limit cardinal. I'm glad to see someone else has asked the question.

$\aleph_0$ is out because replacement implies collection in the absence of a proper class of urelements.

@AlecRhea Does that include proper classes definable from set parameters?

@JamesHanson Yep

Oh, cool! So global choice is weaker than limitation of size. This raises the question of whether the Hartogs number of a proper class can be $\aleph_1$ even without urelements.

I just realized — if you do that, that automatically implies nontriviality of both irreps.

There is a representation of $\mathfrak{sl}_2\times\mathfrak{sl}_2$ on $\mathbb{C}^3$ such that neither factor acts trivially. They just act the same. To fix this, additionally require that there is a vector in the span of $Lv$ that isn't in the span of $Rv$ for every nonzero $v$, and vice versa.

Interestingly, though, Env and Reader are basically monad/comonad ways of doing the same thing, while the other pairs do different things.