Does anyone know a way to construct a homotopy commutative KO-algebra structure on KU, without appealing to geometric or fixed point considerations? What I had in mind was using the Wood cofiber sequence $\Sigma^1 KO \to KO \to KU$. Applying - \otimes_{KO} KU, and using the \pi_1(KU) = 0, gives a left unital multiplication KU \otimes_{KO} KU ---> KU, but it seems non-trivial (due to non-uniqueness) to product a homotopy commutative and associative multiplication this way.

Can somebody help out with this reference request of a friend of mine? He and I talked about this stuff and it's pretty straightforward to prove, so this really just about finding a written account. mathoverflow.net/questions/271906/…

@ArunDebray: Maybe I should have asked before making one, but are pull requests the most convenient way for you to receive typo corrections in the equivariant homotopy course notes?

@OmarAntolín-Camarena Yes, that was very convenient. Thanks for submitting it!

Cool. I might have some more later.

Hi. Just a thing that have been bugging me. In Lurie's definition (Def 2.1.1.10 in HA) of a coloured $\infty$-operad, he's assuming that the operad is symmetric, right? If so, then this seems to contradict his choice of what an ordinary (1-categorial) coloured operad means (Def 2.1.1.1 in HA), which does not assume any kind of action by the symmetric group.

@user40276 The symmetric group action is encoded in point (3) of Definition 2.1.1.1

@EspenNielsen Ah!Ok. Thank you. The index set $I$ is not ordered. I don't know why I've assumed it.