@S.carmeli It looks like it is claimed (without proof) in Section 3.1 of arxiv.org/pdf/1210.0290.pdf

How far (in terms of stems) can mortals compute the homology of the Lambda algebra? How does it compare to the may spectral sequence for instance?

Mortal means without computer assistance and average brain capacity.

@Drew thanks! but it makes me even more curious if the proof is documented somewhere. I always considered this fact as sort of standard but now when I actually need it, I figured out that I never saw the written proof of that, though I think several sources assume it without a proof.

@S.carmeli does it follow from combining HA.4.8.5.11 with 4.8.5.16? (the right adjoint given in 4.8.5.11 is described in the course of the proof or the remark afterwards as extracting the endomorphism object). Those two together say that the assignment RMod_{(-)} is symmetric monoidal as a functor to C-modules with a distinguished object, and the right adjoint (necessarily lax symmetric monoidal) extracts the endomorphisms of the unit. So this induces the same adjunction on cats of algs

probably you are right, im not familiar with the notations in this section of HA. So basically the idea is that this monster Pr^Mon_/M is symmetric monoidal, and a commutative algebra there is a pair of a C-algebra M and its unit 1_M? cant it be a commutative algebra M and a general commutative algebra in M or something?

I have a dream that one day someone who really knows this stuff will write a user guide or something, it will be so helpful to people like me who don't know what a simplex is... ;-)

In a parallel universe Biden chose to swear on HTT in his oath for office and such a guide has already been distributed in pamphlets all around the world.

@SaalHardali Well, Whitehead computed out through stem 34, according to Curtis-Goerss-Mahowald-Milgram (people.math.rochester.edu/faculty/doug/otherpapers/cgmm.pdf second page).

It'll match up pretty poorly to the May SS if you just compare the stems people have computed to. That feels a bit unfair since the lambda algebra also gives other information, like unstable data.

I think Tangora got up to the 70 stem or so with the May SS.

On the other hand, the lambda algebra has been used to compute through the 5-line (Lin, Chen), so maybe you could say infinitely far.

@WilliamBalderrama 34 isn't that bad. What about multiplicative extensions in the may ss? Are these easy to resolve? I didn't run either of these methods very deep but I do find lambda algebra more appealing for manual calculations in that regard. I hate extension problems..

@SaalHardali I'm not sure. I've never really dived into the May SS, so I don't know how far people have computed products with it.

Apparently Isaksen (Stable Stems) could resolve all hidden extensions by tau, h0, h1, and h2 in the C-motivic May SS through the 70-stem. Perhaps someone did this with the classic May SS somewhere.

IIRC Tangora originally got interested in the lambda algebra to resolve some products, but I think that was already in the range where you need a computer.

@DylanWilson ok I think I got it, that was extremely helpful and saved me (us) a lot of pain! thank you!

@S.carmeli glad to hear it!

I posted an answer to an old question of mine that Rune recently explained to me in this room. There's another answer to the question on there, but I couldn't quite figure it out, so I ended up using Rune's.

@WilliamBalderrama I see. Thanks for the helpful info!

i just noticed something kind of wacky -- though maybe it shouldn't be surprising. given a lax-monoidal functor $F: C \to D$, just because it is strict on a pair $(C,C')$ and on a pair $(C',C'')$ doesn't mean it is strict on the triple $(C,C',C'')$. this feels like a categorical analog of something more familiar, vaguely akin to the idea e.g. that in a group, "x commutes with y" is not a transitive relation.

Anyone have a good way to intuitively or pictorially think about relative colimits and limits? I feel like it's something like... restricting the sorts of cocones/cones we allow?