The n-Category Café

David wrote:

Hi John,

Hope you’re doing well!

I was just thinking about your (and James Dolan’s) definition of the zeta functors associated to a finite type scheme (from here), and I had a small thought which I figured you might find interesting.

I was thinking about the functional equation of the completed zeta functions; how might we complete the zeta functors in such a way that they satisfy a similar functional equation? I don’t know, but I do have an idea for what the transformation $s \mapsto 1 -s$ might mean in this context. I claim that it is given by the reduced suspension. Let me explain.

First, I’ll want to see the formal power $n^{-s}$ as the power

$$\left(\frac{1}{n}\right)^s,$$

which I can then categorify by finding a group $G$ with cardinality $n$ and considering $B G^S$. In the case of the Riemann zeta species, $n$ is the cardinality of a finite semisimple ring (a product of finite fields, the groupoid of which has cardinality $1$ for each $n$), and we can simply deloop the additive group of this ring. This gives us a Dirichlet functor $\zeta$

$$S \mapsto \sum_{k \; \text{ finite semisimple}} B k^S$$

which categorifies the Riemann zeta function when $S$ is a finite set.

Taking this point of view on the zeta functor, we can ask the question: what is the transformation $s \mapsto 1-s$? Here’s where we can look at the reduced suspension $\Sigma S_+$. The universal property of the reduced suspension says that maps $\Sigma S_+ \to Y$ correspond to points of the homotopy type

$$(x : Y) \times (y : Y) \times (S_+ \to x = y)$$

(or, more classically, maps from the terminal morphism $S_+ \to 1$ to $\mathsf{Path}(Y) \to Y \times Y$). Since homotopy cardinality is multiplicative for fibrations, that type has cardinality

$$y \cdot y \cdot \left(\frac{1}{y}\right)^{s + 1} = \left(\frac{1}{y}\right)^{s -1}$$

(when $S$ is a finite set of cardinality $s$).

Taking $Y = B k$ for $k$ finite semisimple of cardinality $n$, we see that $\Sigma S_+ \to B k$ has cardinality $n^{s -1} = n^{-(1 -s)}$. Therefore, I think the transformation $s \mapsto 1 - s$ in the functional equation may be categorified by $S \mapsto \Sigma S_+$. If this makes sense, it suggests that completing the zeta functors is a form of stabilization.

Cheers,
David

And then:

As for another eyebrow wiggle about the cardinality of $\Sigma S_+$ when $S$ is a finite set: we have that $\Omega \Sigma S_+ = \langle S \rangle$, the free group on $S$ generators. This is of course infinite, but it it is the group completion of the free monoid $\mathsf{List}(S)$ on $S$ generators. Since

$$\mathsf{List}(S) = 1 + S + S^2 + S^3 + \cdots,$$

it has cardinality $\frac{1}{1 - s}$.

Maybe it’s better to use the “free delooping” (aka weighted colimit of $1 \rightrightarrows 1$ by $S \rightrightarrows 1$) instead of the reduced suspension. This doesn’t change the above argument because we’re mapping into a groupoid, but now it is true that the Euler characteristic / cardinality of that category is $1 - s$.