@AlexanderCampbell oh thanks. Actually yeah I've got fibrant objects anyway so that's great.

Hrm, okay here's something which I think might be blazingly obvious, but I just want to check myself on it: given a map f:X→Pic(𝕊) I get a Thom spectrum with a Thom diagonal Δ:Mf→MfΛX₊. However, by restricting to each point x∈X, I get a bunch of "trivial" Thom diagonals 𝕊≃Mx→MxΛS^0≃Mx. I was sort of thinking that the Thom diagonal Δ should be the colimit of those trivial ones, but that can't be true can it?

Because those trivial ones are all equivalences, and the Thom diagonal itself is not.

So is the colimit of that just the identity Mf→Mf?

so things like the map S[t] -> S that sends t to 2 is kind of confusing and i feel like, at best, there are a bunch of facts that you can learn to make it less confusing

one is that left multiplication by t and right multiplication by t are both "2" as a maps of spectra S -> S. but: right multiplication by t is not "2" as a map of left S[t]-modules

e.g. you can calculate End_{S[t]} (S), whose coefficient ring is S_*[β]/(β^2) where β is in degree -1. the image of t in this ring is (2 + η β). so even though left & right multiplication by t are the same on the underlying spectrum, if you know one of them then you can see that the other one is different.

alternatively, still thinking of S as a bimodule, there is the following: (t-2)*1 = 0 and 1*(t-2) = 0 in S_*, but these aren't the "same" identity. there's a nullhomotopy of (t-2)*1 and a nullhomotopy of 1*(t-2) but those nullhomotopies aren't "the same".

(this is phrasable as <t-2,1_S,t-2> contains η if you like brackets)

and that's a little weird, but it's indicative of the fact that (t-2) isn't strictly central in S[t] (just like t is, but 2 isn't).

and maybe all of this has to do with the fact that, as an S[t]-module, S has a 0-dimensional cell and a 1-dimensional cell enforcing t=2. the 0-dimensional cell controls everything that we see on homotopy groups, but the 1-dimensional cell plays no less of a role when we start smashing it with things & the enforcer starts to play a more visible role.

you can cook up examples of differential graded algebras where similar funky things happen with exotic elements. but somehow the expectation is that 2 should be kind of inert & we get a little surprised when it's not

Is there a nice presentation of $BP^* BP$ as an associative algebra in $BP_{\ast}$-bimodules? (meaning writing it as a quotient of the free associative algebra on some bimodule modulo some sub-bimodule of relations). If finiteness issues are a problem maybe I should ask for a presentation of $BP_{\ast} BP$ as a co-associative co-algebra in bi-modules.

Hi, its me trying to get help with references again: does anyone knows where the fact that Mod_{(-)}(C):calg(C)-->calg(Mod_C(Prl)) is left adjoint to the functor of (C-linear) endomorphisms of the unit? It seems like many very closely related statements are in Higher algebra but we couldn't find exactly that.