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Do I understand correctly that this iterated K-theory is not quite the same as secondary K-theory, as treated by
Seems so
Note that there is a canonical map K◦K→K(2) from iterated K-theory to secondary K-theory, which is nontrivial [HSS17, Remark 6.23]. Indeed, it is expected that secondary K-theory is a substantially richer invariant than iterated K-theory
Probably worth its own entry then.
This is a very good question! I’d recommend the sketch paper ’Chern character, loop spaces, and derived algebraic geometry’ by Toën and Vezzozi: what is clear is that both secondary K-theory and elliptic cohomology are related to categorifications of K-theory; both can be thought of in terms of categorified vector spaces of some kind (2-vector bundles/dg-categories/etc). And elliptic cohomology is supposed, by red-shift, to be the algebraic K-theory of topological K-theory. Thus the two are closely related. Toën and Vezzozi suggest that maybe secondary K-theory will turn out to be some kind of ’algebraic elliptic cohomology’, which seems a reasonable guess.
It will be very interesting if/when people take the next step up the categorical ladder, because things are much harder there. Up to dimension 2, one has a kind of guide from étale cohomology: one can more or less construct the first and second étale cohomology groups geometrically (there is a Picard stack, and there is probably a derived Brauer 2-stack) using essentially 1-categorical gadgets (Azumaya algebras in the case of second étale cohomology), so one has a guide for how things should look. There is no such thing known for the third étale cohomology group and onwards.
it is expected that secondary K-theory is a substantially richer invariant than iterated K-theory
Sounds like a grant application. What’s the evidence?
The Introduction of this article discusses secondary K-theory, and there’s a comparison map right at the end (Remark 6.23), but I can’t see anything explicit on the substantial extra richness:
If one believes that secondary K-theory is something like ’algebraic elliptic cohomology’, then the difference should be something like the difference between topological and algebraic K-theory. Of course that is very vague :-).
In the definition of secondary K-theory per now, one is only really seeing a shadow of what should be a ’categorified cohomology theory/spectrum’.
If there is a comparison map between algebraic and topological K-theory, and the associations in #7 are right, we’d expect a map from secondary to iterated K-theory. I’ve only seen mentioned one the other way, the ’canonical’ one in #2.
If one believes that secondary K-theory is something like ’algebraic elliptic cohomology’
On what basis would you believe this? At the face of it this sounds wild. I’d be happy to learn about it if its true, but what’s the evidence?
Notice BDR fell into this trap previously with prematurely claiming that categorified vector bundles present elliptic cohomology – despite the glaring hint that no elliptic curve would make an appearance. Later they changed terminology to claim a “form of elliptic cohomology” to mean something “essentially of chromatic filtration 2”.
Notice that instead of connecting to elliptic cohomology, the interesting result they eventually got, after Richter joined in, was that – after group completion – the categorified vector bundles presented iterated K-theory. So in that case, instead of the categorified theory being “substantially richer”, it ended up – after waving a homotopy-theory wand over it, at least – being equivalent to iterated K-theory.
That’s a really interesting result. It might be more widely known had it not been served with a red herring.
Re #5: if $F$ is a field, then $K_0(K(F))=\mathbb{Z}$ whereas $K_0^{(2)}(F)$ contains the Brauer group of $F$. I don’t really know anything about the higher homotopy groups, however. It’s not inconceivable that they would agree.
The only explicit connection I know between secondary/iterated K-theory and elliptic cohomology is the chromatic complexity, as Urs mentioned.
Yes, I was just using the ’elliptic’ terminology for ease of reference to Toën-Vezzosi, the point as I mentioned is really to do with different ways of categorifying vector bundles, topological vs algebraic. I think there is a decent chance of some kind of comparison map, David, when formulated in the correct context.
I have added the following previously missing clause to the first lines of the Idea-section:
When $R$ happens to be a connective $E_\infty$-ring spectrum, then also the representing spectrum $K(R)$ of its algebraic K-theory is a connective $E_\infty$-ring spectrum (Schwänzl & Vogt 1994, Thm. 1, EKMM 97, Thm. 6.1)
and I added the following pointer, to go with this:
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