The issue is that $F_i$ is not necessarily defined, i.e., if $i > n$.

I still think that I'm getting it right but I'm too tired to continue :)

That's a feeling I know well! Best of luck; I hope I helped a little bit.

is there a finite spectrum X such that the mod 2 cohomology of X is A(2)//A(1)?

how's that even an A-module?

sorry, i meant to ask for H^*(X) to be A(2)//A(1) as an A(2)-module

@FoscoLoregian I think it just falls out from the formula for Kan extensions, viewing the n-skeleton as a left Kan extension from simplices of dim $\leq n$

@RuneHaugseng sure it does (or it is my hope)

if you have a formula with the coend tell me

@skd I think that the question is whether it even has an A-module structure extending the A(2)-one

@FoscoLoregian You also need to say $i\le n$ in your union, otherwise the formula makes no sense.

the p.w. formula for the left Kan extension suggests that the sk_n of a truncated sSet should be the initial simplicial set whose truncation is the given one (if such a thing exists). I think it's pretty intuitive that adding nothing but degenerate simplices in degrees above n does the job.

I know that the coskeleton construction kills homotopy groups above the given degree for fibrant sSets. Is there an analogous requirement for the skeleton functor to remove all non-degenerate simplices?

I will think about it; being the coend-guy I prefer to be dumb and follow a chain of isos :-)

@DenisNardin yeah that's the first thing to check

@EricPeterson i think i got this sorted out. pretty sure Z[ - ] / (ker eps)^(* 2) is the functor i was after. i'm a little fuzzy on that denominator being "closed" under the diagonal, but i think it's because "closed" doesn't mean what one might first guess & not because this is wrong

thanks anyway c h @ t

@FoscoLoregian The left Kan extension is given at [m] by the colimit over $(\Delta_{\leq n}^{op})_{/[m]}$, which for $m > n$ is exactly a colimit over the degeneracies of simplices of dimension (\leq) n.