The algebra and geometry of higher categories – II

Hi all! I wanted to take off from last time, and talk about the geometry of higher categories, which, as I had mentioned previously, is still in its infancy. Recall in the previous post we talked about stable $(\infty,2)$-categories, $(\infty,2)$-operads, and $2$-rings.

What is classical chromatic homotopy? Recall that the premise of chromatic homotopy theory is that the structure of the stable homotopy category is controlled by the quasicoherent sheaves over the moduli stack of formal groups. Somewhat more precisely, the spectrum $\mathrm{MU}$ should be viewed as a map $\mathrm{Sp}\to\mathrm{QCoh}(\mathscr{M}_\mathrm{FG})$. The Landweber exact functor theorem states that flat affine maps to $\mathscr{M}_\mathrm{FG}$ give rise to homology theories. Since the $(\infty,2)$-category of stable $\infty$-categories generalizes the notion of spectra, one can hope that there is an analogue of the moduli stack of formal groups which plays a role similar to the ordinary stack $\mathscr{M}_\mathrm{FG}$.

So, let’s define the notion of an $\mathbf{E}_\infty$-Hopf algebroid. First, though, what is a Hopf algebroid? It’s basically a pair of graded commutative $k$-algebras $(A,\Gamma)$ such that for any graded commutative $k$-algebra $C$, the pair $(\mathrm{Hom}(A,C),\mathrm{Hom}(\Gamma,C))$ is a groupoid (so $\mathrm{Hom}(A,C)$ is like the collection of objects and $\mathrm{Hom}(\Gamma,C)$ is the collection of maps between these objects). This can be encoded as conditions involving a bunch of maps between $A$ and $\Gamma$. Awesome.

Now the reason you should care about Hopf algebroids is because to every Hopf algebroid one can associate a stack, given by the stack $[\mathrm{Spec}(\Gamma)\rightrightarrows\mathrm{Spec}(A)]$. This is a Deligne-Mumford stack (i.e. it’s nice). Recall the moduli stack $\mathscr{M}_\mathrm{FG}$ of formal groups. This is actually a Deligne-Mumford stack, and comes from a Hopf algebroid $(L,W)$, where $L$ is the Lazard ring and $W$ is the polynomial ring $L[b_1,\cdots]$ over $L$ where $|b_i|=2i$. (This is the construction of the moduli stack of formal group laws via Hopf algebroids which I had mentioned we’d go into in the post on chromatic homotopy theory.)

But if you’re a homotopy theorist there’s another reason to care about Hopf algebroids: if $E$ is a flat ring spectrum, i.e., $E_\ast(E\otimes X)\simeq E_\ast(E)\otimes_{E_\ast}E_\ast(X)$, then $(E_\ast,E_\ast(E))$ is a Hopf algebroid. And guess what? Many of the interesting ring spectra are flat, including $\mathrm{MU}$! Although this is pretty cool, of what relevance is it to $\mathscr{M}_\mathrm{FG}$? Recall Quillen’s theorem that $\mathrm{MU}_\ast\cong L$, where $L$ is the Lazard ring. There’s another incredible result, proved by two cool guys, Landweber (of the Landweber exact functor theorem) and Novikov, which says that $\mathrm{MU}_\ast\mathrm{MU}$ is isomorphic to $W$ (as above). Since $\mathrm{MU}$ is flat, we can consider the Hopf algebroid $(\mathrm{MU}_\ast,\mathrm{MU}_\ast\mathrm{MU})$, and because of the above discussion, we see that this is isomorphic to $(L,W)$! So the Deligne-Mumford stack associated to $(\mathrm{MU}_\ast,\mathrm{MU}_\ast\mathrm{MU})$ is the moduli stack of formal group laws!

Having said all of this, we can go back to what we wanted to define: the notion of a $\mathbf{E}_\infty$-Hopf algebroid. This isn’t too hard to define by looking at the commutative diagrams which define a Hopf algebroid. The stack associated to a $\mathbf{E}_\infty$-Hopf algebroid is a derived Deligne-Mumford stack. Indeed, it’s too natural of an assumption to believe that this is false. Why? Given a Hopf algebroid, you can take its étale spectrum, and taking the coequalizer gives you something locally equivalent to the étale spectrum of a $\mathbf{E}_\infty$-ring! (For those readers not familiar with the étale spectrum stuff: think of the étale spectrum of a $\mathbf{E}_\infty$-ring as giving an “affine” derived Deligne-Mumford stack instead of an affine derived scheme.)

Let’s now define an analogue of the Lazard Hopf algebroid (namely the Hopf algebroid $(\mathrm{MU}_\ast,\mathrm{MU}_\ast\mathrm{MU})$, which is equivalent to the Hopf algebroid which has the Lazard ring). The analogue of $\mathrm{MU}$ is the $2$-ring $\mathrm{Mod}_\mathrm{MU}$, so we look at the pair $(\pi_\ast\mathrm{Mod}_\mathrm{MU},\pi_\ast\mathrm{Mod}_{\mathrm{MU}}\otimes\mathrm{Mod}_\mathrm{MU})\cong(\pi_\ast\mathrm{Mod}_\mathrm{MU},\pi_\ast\mathrm{Mod}_{\mathrm{MU}\otimes\mathrm{MU}})$. The resulting Deligne-Mumford stack is defined to be the derived moduli stack $\mathscr{M}_\mathrm{FG}$ of derived formal groups.

Many results from ordinary chromatic homotopy theory translate over to this new “derived” chromatic homotopy theory. It’s pretty cool that to every $2$-ring we can associate a quasicoherent sheaf over the above derived Deligne-Mumford stack, so we get a functor $\mathrm{Alg}_{/\mathbf{E}_2[\infty]}(\mathrm{Cat}^\mathrm{st}_\infty)\supset 2\mathrm{Ring}\to\mathrm{QCoh}(\mathscr{M}_\mathrm{FG})$ induced by $\pi_\ast\mathrm{Mod}_\mathrm{MU}$. That’s exactly the case in ordinary chromatic homotopy theory! This is awesome.

One of the main applications of this, which is formally unexplored is how this can be used to study transchromatic homotopy theory. In this formalism, every ordinary formal group law of arbitrary height gives rise to a derived formal group (law) of (derived) height one. Consequently, studying the stratification of the derived moduli stack of derived formal groups will give input into how the different layers of the ordinary moduli stack of formal groups fit together (because they do so in the derived stack). Via “twisting” the definitions around (i.e. using essentially equivalent ones but ones for which not a lot of additional work is required to prove stuff), one can also reach an analogue of the Landweber exact functor theorem.

Oh, hey, wait. There’s another thing that’s really cool, which I remembered from a class that I took on this in January at UCLA taught by my mentor, Professor Marcy Robertson. First, recall the construction of the derived Deligne-Mumford stack of elliptic curves. We’ve got the ordinary Deligne-Mumford stack, $(\mathscr{M}_{1,1},\mathscr{O}_{\mathscr{M}_{1,1}})$ (no, I’m not an algebraic geometer (or even a mathematician), I just use $\mathscr{M}_{1,1}$ instead of $\mathscr{M}_{ell}$ because I think it subtly hints at generalizations). We can take the étale topos of this moduli stack, and we end up with a derived Deligne-Mumford stack denoted $(\mathscr{M}^\mathrm{Der},\mathscr{O}_{\mathscr{M}^\mathrm{Der}})$ where $\pi_0\mathscr{O}_{\mathscr{M}^\mathrm{Der}}=\mathscr{O}_{\mathscr{M}_{1,1}}$. (If you don’t understand why we get a derived Deligne-Mumford stack, see above for an explanation of what the word “étale” means – basically, it tells you: work in stacks, not schemes.)

What follows is a subset of the main points of $\mathrm{tmf}$ I got from the course at UCLA and my own readings (eg the Talbot notes, which are really good): using $\mathscr{M}_{1,1}$ we’ve been able to define the interesting spectrum known as $\mathrm{tmf}$. We define the sheaf $\mathscr{O}^{top}$ by saying something like this: look at a flat affine map $\mathrm{Spec}(R)\to\mathscr{M}_{1,1}$. Then if the map $\mathscr{M}_{1,1}\to\mathscr{M}_\mathrm{FG}$ is flat, by the Landweber exact functor theorem, the sheaf $\mathscr{O}^{top}$ associates to each elliptic curve $\mathrm{Spec}(R)\to\mathscr{M}_{1,1}$ this $\mathbf{E}_\infty$-ring. The global sections of this sheaf of spectra over $\mathscr{M}_{1,1}$ is defined to be “the” spectrum of topological modular forms (there’s actually like three different ones, called $\mathrm{TMF}$, $\mathrm{Tmf}$, and $\mathrm{tmf}$ (I can imagine how hard it’ll be to say something about these three spectra)).

Ok, so now we’ve got the derived moduli stack of derived formal groups, the derived moduli stack of elliptic curves, and the derived Landweber exact functor theorem (when you read that, did you think “(derived Landweber) exact functor theorem” or “derived (Landweber exact functor theorem)”?). So maybe we can do a similar thing to get “derived $\mathrm{tmf}$“? (That’d be really cool!)

I hope to be able to write down formal proofs of all these statements soon. But for now, I guess this post concludes the two-series posts on the algebra and geometry of higher categories.

Have fun,
Sanath