I've been trying to figure out exactly where the mistake in Ittay Weiss's construction of the BV tensor product on Dendroidal sets was.

The fact that it fails to be associative appears to come down to something subtle in Brian Day's thesis. maths.mq.edu.au/~street/DayPhD.pdf

Lemma 3.1.1 states that when we look at biclosed monoidal functors on the functor category [A,V], these are exactly in bijection with promonoidal structures on A, or more to the point, biclosed monoidal structures on [A,Set] correspond exactly to pro-monoidal structures on A in the case where the enrichment is just in sets

One direction of the bijection (the 'trace' direction) seems to only use the fact that A c F is a full dense subcategory, F is cotensored in V (Sets), and the monoidal structure on F is biclosed

But since Weiss's construction fails to give a biclosed monoidal structure on Psh(Ω), it must be the case that the trace of the BV tensor product of symmetric colored operads back to Ω is not a pro-monoidal structure

so either Day's lemma has a missing detail, or I'm missing some essential use of the fact that F is the functor category

in the 'trace' direction

The other option, maybe is that the BV tensor product of Operads isn't biclosed, but Andrea Gagna says he's convinced that it is=

@GijsHeuts Since you answered Andrea's other question about this, I'm wondering if you might know the answer.

It looks like he hasn't been around on MO in about a year

so I guess that's not going to happen

I copied it into an e-mail, but if anybody else is familiar with the whole BV tensor product mistake that led to all of those errata, I'm all ears

Also, if anyone wants, this question is basically the same question but with all of the context stripped out: mathoverflow.net/questions/287598/…

Is there any reference for the structure of $BPL$ and the motivic Adams-Novikov 2-term when working $p$-complete at an odd prime $p$? Namely, I'd like to know that $(BP_{\bullet}, BP_{\bullet} BP) \otimes \mathbb{Z} [\tau] \simeq (BPL_{\bullet, \bullet}, BPL_{\bullet, \bullet} BPL)$ and hence the motivic Adams-Novikov for the cofibre $C \tau$ of $\tau$ collapses.

There's plenty of references at $p = 2$ (Isaksen, Gheorghe, Hu-Kriz) and digging back into the proofs I see they're pretty formal so my guess is that this largely works the same at an odd prime, but perhaps someone has already written this down..?

Okay, apologies as I might have found it. Somehow in my search for the longest time I avoided the use of "motivic adams-novikov at odd primes" thinking that's too easy, but that actually immediately yields Stahn's manuscript which I should have known of. Apologies again.

^ Only time in history 'motivic adams-novikov at odd primes' was followed immediately by 'thinking that's too easy'