# The algebra and geometry of higher categories

Hi all! Today I want to talk about (the first part of) something which I’ve been trying to do (I’d call it my research, but it’s really just a bunch of kinda-fleshed-out ideas I’ve had). I’d say preliminary reading would be the posts “An intro to chromatic homotopy theory” and “Marked Simplicial Sets – I” (which really isn’t too much).

Recall that a marked anodyne map is a map with the left lifting property with respect to any Cartesian fibration $X\to S$. Now, a fibrant object in the Cartesian model structure on $(\mathrm{Set}^+_\Delta)_{/S}$ is an object of the form $X^\natural$ where $X\to S$ is a Cartesian fibration. What one should get from this is that a fibrant object in $(\mathrm{Set}^+_\Delta)_{/\Delta^0}$ is basically an $\infty$-category (because it’s Quillen equivalent to the Joyal model structure, which models $\infty$-categories, etc.). Cool.

But now let me ask you: what is a marked simplicial set? Well, it’s just a simplicial set with chosen $1$-simplices. Can we be like Grothendieck and generalize? Sure! Define a scaled simplicial set to be a simplicial set with chosen $2$-simplices. It turns out that you can define a model structure on the category of scaled simplicial sets, much like the Joyal model structure, such that the fibrant objects in this model structure are precisely the $(\infty,2)$-categories.

Now, let’s go back to the $\infty$-category picture. There’s the notion of a stable $\infty$-category. A basic review is here (I don’t know why the text is bolded like that, it’s weird). The idea is that these things generalize spectra in the following way. Let’s go back to classic homological algebra. How are abelian categories like abelian groups? The category $\mathrm{Ab}$ of abelian groups should be the prototypical example of an abelian category, and indeed it is. So the idea is that a stable $\infty$-category should be like the $\infty$-category of spectra.

What about the $(\infty,2)$-category of stable $\infty$-categories? That should give us insight into the notion of a stable $(\infty,2)$-category, right? The answer is YES! What I did was define this $(\infty,2)$-category of stable $\infty$-categories, and study certain properties of this thing. These properties are analogous to the “ordinary” $\infty$-categorical properties of the $\infty$-category of spectra, so we say that an $(\infty,2)$-category satisfying these properties is a stable $(\infty,2)$-category. And there we are – a definition of stability in higher higher (not a typo) category theory!

Awesome. Now, we can take $\mathbf{E}_\infty$-algebra objects (commutative algebra objects) of the $\infty$-category of spectra, and bam! we end up with (I guess by definition…) $\mathbf{E}_\infty$-ring spectra. These are what we called (commutative) ring spectra in the post on chromatic homotopy theory. So they’re obviously important.

But now, we ask, wait, what about commutative algebra objects in the stable $(\infty,2)$-category of stable $\infty$-categories? Wouldn’t those be “derived” ring spectra? Let’s take a step back and look at the big picture:

(This is from my slides for my talk at JMM, which is basically about whatever’s in this post.) So we can ask, wait, $2$-rings are ?-algebra objects in ? The second question mark has a (seemingly) obvious candidate (with one pitfall we’ll mention later), namely the stable $(\infty,2)$-category of stable $\infty$-categories. Now, $\mathrm{Comm}^\otimes$ is an ordinary operad, and $\mathbf{E}_\infty^\otimes$ is an $\infty$-operad, so, maybe, we should ask for $(\infty,2)$-operads. Ok, but what model of $\infty$-operads should we generalize?

There are three well-known models, namely: Lurie’s stuff (pretty algebraic and really complicated, so we can’t use this), Moerdijk-Weiss stuff (combinatorial, perhaps easier to generalize), and Barwick’s complete Segal operads (simplicial, easiest to generalize). We generalize Barwick’s model because, well, we already have the technology to generalize – the theory of scaled simplicial sets! This actually isn’t too hard to do, and we can safely say that a model for $(\infty,2)$-operads is in place. Yay! Via a mildly complicated definition we can also generalize the commutative $\infty$-operad to the commutative $(\infty,2)$-oeprad, which we call $\mathbf{E}_2[\infty]$. One cool thing is that $\mathbf{E}_\infty$, when viewed as an $(\infty,2)$-operad, sits inside $\mathbf{E}_2[\infty]$!

So, now, we say, let’s just define a $2$-ring to be a $\mathbf{E}_2[\infty]$-algebra object in the stable $(\infty,2)$-category of stable $\infty$-categories. But this doesn’t work! We need the $\infty$-category to also be presentable. Why? Because otherwise it wouldn’t be a $\mathrm{Sp}$-module, and consequently wouldn’t fit in that nice little table up there. Having worked out the definition, we can ask: can we do anything with this?

Maybe algebraic geometry? We just need to find a way of getting a spectrum from a stable $\infty$-category. But this isn’t too hard! Again we go back to the “ordinary” $\infty$-category case. There’s a $t$-structure on the stable $\infty$-category of spectra, whose (homotopy category of the) heart is simply $\mathrm{Ab}$. This map is what we commonly know as $\pi_0$ of a spectrum. So analogously, define the notion of a $t$-structure on a stable $(\infty,2)$-category, and do something similar to get an underlying spectrum from a stable $\infty$-category. Let $\mathscr{C}$ be a $2$-ring, and define $\mathrm{Spec}(\mathscr{C})$ to be $\mathrm{Spec}(\pi_0\mathscr{C})$. For example, I’ve provided a rather informal proof that $\pi_0\mathrm{Mod}_R\simeq R$, so $\mathrm{Spec}(\mathrm{Sp})$, for example, contains information about the stable homotopy category, and can be drawn as follows (those $K(n)_{(p)}$ things are called the $n$th Morava K-theories at the prime $p$; I’ll probably write a post about these things soon, because not only do they behave like fields (e.g. they are “intermediate characteristic” as the below picture shows), but they are also very related to formal group laws):

Pretty cool, huh? I may talk about the second part later (which is about the chromatic homotopy theory of $2$-rings, and is the geometry part of the title), but it’s still in its infancy, and nothing’s actually concrete yet.

Anyway, hope you enjoy 2016/(4/1)=504 ! Have fun,
SKD