12:00 AM
it seems like there should be a map tmf -> ko which fits into a commutative square with MString -> MSpin, MString -> tmf, and MSpin -> ko. Is there such a map? (I've tried looking into the literature but haven't found any discussion of this)
bonus: if such a map does exist, is the induced map on the Adams SSes at the prime 2 given by the map Ext_{A(2)}(F2, F2) -> Ext_{A(1)}(F2, F2) (an extension of A(2)-modules is an extension of A(1)-modules)?

12 hours later…
12:10 PM
@ArunDebray there is such a map. one way to get it is to take the map Tmf-->KO, given by evaluation at the cusp, and then take connective covers.

It turns out we do get a bifibration by Lemma 4.8 in Shah's thesis.
Is there any way of replying to one's own posts?

2 hours later…
2:04 PM
Let $R \to S$ be a map of $\mathbb{E}_{\infty}$-ring spectra. I'm looking for a neat if and only if condition for when $R$ is a "small extension" of $S$ in the sense that it's a fiber of some derivation $S \to S \oplus M$ for some $S$-module $M$.

2:45 PM
I'm thinking of something along the lines of the following conditions (which might not be sufficient):
1. The canonical map from the colimit of the cech nerve is an equivalence $| R^{\times_S \bullet + 1} | \to S$
2. The canonical map $cofib(R \to R^{\times_{S} \bullet +1}) \to \mathbb{L}_{ R^{\times_S \bullet + 1} / R}$ is an equivalence.

2:55 PM
Oh wait I 'm not so sure but maybe in $(2)$ what I actually want is to put $\mathbb{L}_{R^{\times_S \bullet +1} / R} \otimes S$ rather than $\mathbb{L}_{R^{\times_S \bullet +1} / R}$ as the target of the map.
of course by equivalence i just mean levelwise in $\bullet$

3 hours later…
6:00 PM
@DylanWilson awesome; thank you!

4 hours later…
9:39 PM
@AdrianClough Yes, you need to preface your message with ":#CODE" where #CODE is the code of the message you want to link to (you can find it by clicking in "permalink")
@DenisNardin Demonstration

10:34 PM
@lemiller thanks for clarifying. however, i'm not sure what you mean by an "asymmetry" in the setup.
in any case, as a first remark, note that _any_ functor $Poly_k \to Alg_k$ admits a left kan extension to a functor $Alg_k \to Alg_k$, just because $Alg_k$ admits all (small) colimits (and $Poly_k$ is small and $Alg_k$ is locally small).
regarding your main question, i think you should first be more precise about the meaning of "$RF_A$". classically, people arrived at derived functors from what might today be considered an ad hoc point of view. for instance, if you have a functor out of a mo