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Stream: theory: algebraic topology

Topic: Secondary algebraic K-theory

Reuben Stern (they/them) (May 26 2020 at 05:00):

John Baez said:

I really like this stuff! These days it would probably be subsumed by some $(\infty,1)$-categorical work, but there's a certain charm to the stripped-down $(2,1)$-categorical approach.

I like this stuff, too! Hoyois, Safronov, Scherotzke, and Sibilla even use $(\infty, 2)$-categorical geometry in their work on the Grothendieck-Riemann-Roch theorem for secondary algebraic K-theory.

Reuben Stern (they/them) (May 26 2020 at 05:09):

Yes! Well, there are actually a few different definitions of secondary algebraic K-theory, which haven't yet been proven equivalent or inequivalent..... the easiest thing to say (and the irony of the word "easy" here isn't lost on me) is that for a field $k$, the secondary Grothendieck group $K^{(2)}_0(k)$ is defined to be the free abelian group on equivalence classes of $k$-linear pretriangulated dg-categories, modulo the equivalence relation where $[\mathcal{A}] + [\mathcal{C}] = [\mathcal{B}]$ when there exists a split Verdier quotient sequence $\mathcal{A} \leftrightarrows \mathcal{B} \leftrightarrows \mathcal{C}$. This receives a map, for instance, from the Grothendieck group of varieties $K_0(\mathsf{Var}_{/k})$.

Reuben Stern (they/them) (May 26 2020 at 05:17):

Let me preface by saying this is unfortunately full of higher-categorical jargon, that I don't know how to avoid.

Reuben Stern (they/them) (May 26 2020 at 05:17):

A lot can be found in work of Toën--Vezzosi (for instance), where $K^{(2)}(k)$ is defined as the $K$-theory spectrum of a certain Waldhausen structure on $\mathsf{Cat}^\mathsf{sat}_\infty(k)$, the $\infty$-category of fully-dualizable, idempotent-complete, stable $k$-linear $\infty$-categories. Let me note that this Waldhausen structure uses non-trivial information about the $(\infty, 2)$-categorical enhancement of $\mathsf{Cat}^\mathsf{sat}_\infty(k)$.

Reuben Stern (they/them) (May 26 2020 at 05:25):

The other main definition is, as far as I know, due to Hoyois-Scherotzke-Sibilla, and is even more explicitly $(\infty, 2)$-categorical. Namely, they introduce a construction of "noncommutative motives" $\mathsf{Mot}(\mathcal{E})$, where $\mathcal{E}$ is an $(\infty, 2)$-category "enriched in $k$-linear stable $\infty$-categories" (I won't make this precise). In the case where $\mathcal{E} = \mathsf{Cat}_\infty^\mathsf{sat}(k)$, this category $\mathsf{Mot}(\mathsf{Cat}_\infty^\mathsf{sat}(k)) =: \mathsf{Mot}^\mathsf{sat}(k)$ is an $\infty$-categorical version of Kontsevich's triangulated category of noncommutative motives. By construction, $\mathsf{Mot}^\mathsf{sat}(k)$ is a stable $\infty$-category, so it makes sense to take its algebraic K-theory spectrum. We thus have the definition $K^{(2)}(k) := K(\mathsf{Mot}^\mathsf{sat}(k))$.

Reuben Stern (they/them) (May 26 2020 at 05:31):

At the moment, it is much easier to work with this $K$ of $\mathsf{Mot}$ definition because it is very general and the "two-step" aspect makes it somewhat easier to understand (the Hoyois-Safronov-Scherotzke-Sibilla categorified GRR paper illustrates this nicely, in that they "compose" the ordinary GRR and a categorified version to get a secondary GRR).

Now, in the world of primary algebraic K-theory, stable $\infty$-categories are nice because they come with canonical Waldhausen structures, and so we can take their K-theory. We do not as of right now know what the analog is for secondary K-theory, i.e., which $(\infty, 2)$-categories come equipped with "canonical" Waldhausen structures in the same way that $\mathsf{Cat}_\infty^\mathsf{sat}(k)$ does.

Reuben Stern (they/them) (May 26 2020 at 05:31):

I'm super sorry to spam this thread, @John Baez; if someone wishes to talk about this further, I'd be happy to in a different stream or thread.

Reuben Stern (they/them) (May 26 2020 at 07:01):

There are a few motivations. I think the first historically comes from chromatic homotopy theory and the "redshift" conjecture. For those who don't know, chromatic homotopy theory involves studying stable homotopy types (that is, spectra) via a filtration (called the chromatic filtration) that is somehow defined in terms of the height filtration of formal group laws. In particular, it makes sense to talk about a spectrum having "chromatic height $n$", which is understood to be a measure of complexity of the associated cohomology theory. It is a classic problem to find "geometrically-defined" cohomology theories of height $n$ for each $n \geq 0$. For small $n$, we have these: singular cohomology with complex coefficients — represented by the Eilenberg-MacLane spectrum $H\mathbb{C}$ — has height 0, and complex topological K-theory — represented by the spectrum $KU$ — has height 1.

We note that these two theories are defined by saying what their cocycles are: in the case of $H\mathbb{C}$, they are sheaves of complex numbers, and in the case of $KU$, they are sheaves of complex vector spaces.

The program of categorification says that the $n = 2$ case of the above should be something like "sheaves of complex 2-vector spaces", where a complex 2-vector space is a $\mathbb{C}$-linear stable $\infty$-category. So it is reasonable to ask: does this form a height-2 theory? There's also an important connection between the 0 and 1 levels: the algebraic K-theory spectrum $K(\mathbb{C})$ is very closely related to $KU$. (I apologize for being vague here; I don't perfectly understand the story myself [but I'm working on it!].) So one might wonder, whatever this height-2 theory is, is it related to $K(KU)$, or $K(K(\mathbb{C}))$? Or, perhaps, it is most natural to consider $K^{(2)}(\mathbb{C})$, which receives a map $K(K(\mathbb{C})) \to K^{(2)}(\mathbb{C})$ that is not injective.

There is a conjecture called chromatic redshift, which loosely claims that if $R$ is a spectrum of chromatic height $n$, then $K(R)$ has height $n+1$. The other buzzword to throw out is "elliptic cohomology"; see Baas-Dundas-Rognes.

The other motivation is more recent, but more convincing in my opinion. In the same way that people have come to understand schemes by studying their $\infty$-categories $\mathsf{QCoh}(X)$ and $\mathsf{Perf}(X)$, some people have come to understand that other, more complicated geometric objects may be understood by studying their $(\infty, 2)$-categories $\mathsf{QCoh}^{(2)}(X) := \mathsf{ShvCat}(X)$ and $\mathsf{Perf}^{(2)}(X) := \mathsf{ShvCat}^\mathsf{sat}(X)$ of quasicoherent sheaves of stable $\infty$-categories or perfect complexes of sheaves of stable $\infty$-categories. In the same way that people use the algebraic K-theory of a scheme to extract certain information from $\mathsf{Perf}(X)$ to be more easily used (since the entire category itself is exactly as intractable as the original scheme), we may use secondary algebraic K-theory to extract information from $\mathsf{Perf}^{(2)}(X)$. It is expected that $K^{(2)}$ will see aspects of a scheme not detectable by primary K-theory, but may still be computable.

Reuben Stern (they/them) (May 26 2020 at 07:13):

I've done a somewhat poor job explaining all of this; it's 3am my time and I'm fighting insomnia. No, you haven't missed anything — elliptic cohomology does have height 2. I think the question is about a geometric model for elliptic cohomology in the same way that singular cohomology and complex topological K-theory have "cochain-level" descriptions as I mentioned.

Reuben Stern (they/them) (May 26 2020 at 07:15):

I know the one :P

Morgan Rogers (he/him) (May 26 2020 at 13:58):

Rongmin Lu said:

Since we're in the "history of ideas" stream (we need an admin to move threads between streams), I'm going to ask a historical question. What's the motivation for coming up with secondary algebraic K-theory?

Where do you think this topic would fit best?

Notification Bot (May 26 2020 at 17:37):

This topic was moved here from #learning: history of ideas > Secondary algebraic K-theory by John Baez

Morgan Rogers (he/him) (Jun 01 2020 at 16:14):

I've been meaning to create an algebraic topology stream for a while. I figured that K-theory should technically come under that umbrella, so here we are.