The n-Category Café

Woohoo, thanks! Suddenly the subject of analytic number theory makes a lot more sense to me.

I only just noticed something about Dirichlet series versus ordinary generating series. The ordinary generating series (or function) of a sequence $\alpha$ is $$\sum_{n = 1}^\infty \alpha(n) x^n.$$ In the expression for Dirichlet series, it would make no essential difference if we changed $s$ to $-s$. (As far as I know — which isn’t very far — it’s just a matter of convention.) And writing $x = -s$, the Dirichlet series for $\alpha$ becomes $$\sum_{n = 1}^\infty \alpha(n) n^x.$$ These two expressions are pleasingly symmetric!

Tom wrote:

I only just noticed something about Dirichlet series versus ordinary generating series.

I noticed that too, once. It started out seeming just ‘cute’. But then it revealed some deeper aspects, and James Dolan and I wrote a paper about it, called Zeta functions. I’ll quote the beginning to whet your interest, or maybe at least someone’s interest:

In this paper we begin categorifying the theory of zeta functions. As a precursor, we must understand how to get a Dirichlet series from a species. This is a very nice counterpart to the usual recipe for computing a formal power series from a species, namely its generating function.

Briefly, a species is any type of structure that one can put on finite sets: for example, a coloring, or ordering, or tree structure. Suppose $F$ is some such structure. If we denote the set of $F$-structures on the $n$-element set by $F(n)$, the generating function of $F$ is the formal power series $$|F|(x) = \sum_{n \ge 0} \frac{|F(n)|}{n!} x^n \, .$$ On the other hand, the Dirichlet series associated to $F$ is
$$\overline{F}(s) = \sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s} \, .$$ Far from being ad hoc tricks, the generating function and the Dirichlet series fit into a nice unified theory, which we shall explain here.

Of course calling the exponent ‘$-s$’ is just a convention; we could call it ‘$x$’ if we wanted. The more important thing is that generating functions get along with the Cauchy product of species, while Dirichlet series get along with the Dirichlet product.

The ‘Cauchy product’ of species is the one you probably knowif you’ve ever heard of species and how to multiply them. It goes sort of like this:

$$(F \cdot_{C} G)(n) = \sum_{k + k' = n} F(k) \times G(k')$$

The ‘Dirichlet product’ goes sort of like this:

$$(F \cdot_{D} G)(n) = \sum_{k \times k' = n} F(k) \times G(k')$$

I say ‘sort of like this’ because it’s easy to misinterpret these formulas unless you read our paper. But the point is that the Cauchy product involves the usual $+$ operation on finite sets, while the Dirichlet product involves the usual $\times$ operation.

And, the Cauchy product gets along with generating functions:

$$|F \cdot_{C} G| = |F| |G|$$

while the Dirichlet product gets along with the Dirichlet series:

$$\overline{F \cdot_{D} G} = \overline{F} \overline{G}$$

(For know-it-alls: the groupoid of finite sets is a rig category in an obvious way, so it has two monoidal structures, so the category of presheaves on this groupoid gets two monoidal structures, thanks to Day convolution. These presheaves are called ‘species’, and these two monoidal structures on species are called the Cauchy and Dirichlet product.)

Tobias wrote:

Woohoo, thanks! Suddenly the subject of analytic number theory makes a lot more sense to me.

That’s how I felt when Jim figured out how to understand Hasse-Weil zeta function of a scheme using this viewpont. All that stuff about Dirichlet series and L-functions is a lot less ad hoc than it sounds at first. And ‘the Hasse-Weil zeta function of a scheme’ is a lot less scary than it sounds at first: we explain it in our paper.

Hi Tom,

all this Moebius stuff is very interesting! I look forward to hear more about it.

But here is just some advertisement:

In case anybody is interested, I have on my web page a short introduction to Moebius inversion — very basic stuff, but including a neat conceptual proof of Euler’s product expansion of Riemann’s zeta function. (It’s from a seminar talk last year which I meant to type up. The second part was about Leroux-Menni theory, but that’s not in the note (yet).)

See “Incidence Hopf Algebras” on the page
http://mat.uab.cat/~kock/upcoming.html

Cheers,
Joachim.

Maxim, I’m not sure you’re answering Tobias’s question. At least, I don’t see an answer in your comment. What Tobias means by a diagram is (as he says) a finite category $C$ together with a functor $D: C \to FinProb$. Can you give a precise definition of the entropy of $D$?

Urs, this puzzles me:

I know. ;-)

I had the impression that in homotopy theory, $B G$ had denoted the classifying space of $G$ (I mean an actual topological space) since the dawn of time?

No, that’s what it means in topology. In homotopy theory it means the object presented by that topological space in the homotopy category $Ho(Top) \simeq Ho(\infty Grpd)$ (or better in the $\infty$-category $\infty Grpd$ of which this is the decategorification). Under the fully faithful inclusion

$$Ho(Grpd) \simeq Ho(Top_{1Types}) \hookrightarrow Ho(Top) \simeq Ho(\infty Grpd)$$

it is identified precisely with the one-object groupoid given by $G$.

(Here we are regarding $G$ as a discrete group. We can generalize this discussion to the case where $G$ is some cohesive group and $\infty$-groupoids are generalized to cohesive $\infty$-groupoids, but let’s not do that right now.)

the notation $\mathbf{B}G$ for the topos of $G$-sets is used in Mac Lane and Moerdijk’s topos theory book and in Moerdijk’s book on classifying spaces and classifying toposes.

Yes, and that’s good: this topos is the topos-incarnation of both, the one-object groupoid $\{\bullet \stackrel{g}{\to} \bullet | g \in G\}$ and the topological space $|N \{\bullet \stackrel{g}{\to} \bullet | g \in G\}|$:

$$\mathbf{B}G := Sh(\{\bullet \stackrel{g}{\to} \bullet | g \in G\}) \simeq_{whe} Sh(|N \{\bullet \stackrel{g}{\to} \bullet | g \in G\}|) \,.$$

So this is not a different use from the one I am advocating above, but in line with it.

It should be

$$Cat_* \stackrel{\overset{B}{\leftarrow}}{\underset{\Omega}{\to}} Monoid \,.$$

The forgetful functor $Cat_* \to Cat$ can be left implicit or not, depending on taste and context. I think that’s not the issue.

First let $X$ be a category. In your preferred system of notation, does $B X$ mean anything?

Only if $X$ is equipped with monoidal structure. Then it means the one-object 2-category with $(B X)(\ast,\ast) \simeq X$. Analogously for $N X$ and $|X|$.

As I said before, I’d be happy with writing $|B M|$ for $|N B M|$. We can safely suppress writing the nerve operation.

OK, thanks. I think this identifies something that I don’t understand about where you’re coming from: in topology, I’m accustomed to the left adjoint of $\Omega$ being called $\Sigma$ (or $S$), not $B$. I thought that was universal. Why not do so here?

I took care over pointed categories because of what Mike wrote at the end of this comment.

I took care over pointed categories because of what Mike wrote at the end of this comment.

Yes, in order to have $(\Omega, B)$ restrict to an equivalence it is important that we use pointed spaces. But for distinguishing between $B$ and $\Sigma$ that’s not the issue.

The main principles I want to adhere to are that $B$ always denotes an upwards level-shift, aka delooping, and level-shifts are never nameless. I also prefer that operations which turn directed things into undirected things, such as the geometric realization of the nerve of a category, are not nameless. (Whether it’s the “geometric realization” part or the “nerve” part which makes a category undirected depends on whether today we are regarding simplicial sets as directed gadgets or undirected ones.)

1. $Monoid \;\underoverset{B}{\Omega}{\rightleftarrows}\; Cat_\ast \xrightarrow{forget} Cat$. As Urs said, the forgetful functor can be nameless.

2. As Urs said, if $X$ is only a plain category, then I wouldn’t write $B X$ to mean anything, since a plain category cannot be upwards level-shifted or delooped. $N X$ means the nerve of $X$, which is a simplicial set and makes sense for any category, but if we are thinking up to homotopy and identifying simplicial sets with spaces, then we might also leave geometric realization nameless and write $N X$ for $|N X|$. Plain ${|X|}$ doesn’t “really” have a meaning, since as you say we generally geometrically realize presheaves, not categories, but since there’s a canonical presheaf associated to $X$, namely $N X$, I don’t have a problem with writing ${|X|}$ to mean ${|N X|}$—though it should probably be explained on first usage.

   The same answers apply when $X$ is pointed, except that $N X$ will be a pointed simplicial set and ${|X|}$ will be a pointed space.

   If $X$ is a monoid, then I would write $B X$ to mean (1) the category with one object whose endomorphisms are $X$. If $X$ is a group, then I would also write $B X$ to mean (2) the nerve of that category or (3) the geometric realization of that category. I would, I guess, also grudgingly accept (2) and (3) when $X$ is a mere monoid, since they are standard usages in algebraic topology, although I’d prefer to separate the level-shifting operation from the making-things-invertible operation notationally.

   For a monoid $X$, I might also grudgingly accept $N X$ for the nerve of the category $B X$, since at least the level-shift is being notated somehow, and the category $B X$ is the only thing canonically associated to $X$ which has a nerve. Writing ${|X|}$ for the geometric realization of that nerve is stretching it a bit more, but if it were explained on first usage, it might be okay.

3. For a monoid $M$, yes, I would write the geometric realization of the nerve of its associated 1-object category as $|N B M|$. By the answer to ${|X|}$ when $X$ is a category, I might abbreviate this as $|B M|$. By the answer to $N X$ when $X$ is a category, if thinking of it only as a homotopy type, I might abbreviate it $N B M$. By answer (3) to $B X$ when $X$ is a monoid, I might also (grudgingly, unless $M$ is a group) abbreviate it as $B M$. By the answers to $N X$ and $|X|$ when $X$ is a monoid, I might (grudgingly) abbreviate it to $|N M|$ or $|M|$. Finally, combining the justifications for $N B M$ and $|N M|$, I could even accept just $N M$, if thinking of it only as a homotopy type. But I don’t think just plain “$M$” would ever be okay, since something is being done to it.

Let me mention two last examples, which really contributed to convincing me to take a hard line that $B$ should always denote a level-shift. Firstly, if $C$ is a monoidal category, then $|N C|$ is an $A_\infty$-space, and as such it has a delooping $B(|N C|)$. If $|N C|$ is denoted by $B C$, then $B(|N C|)$ would be denoted by $B(B C)$. I have actually seen this notation used! But notice that it cannot be rewritten as $B^2 C$, since the two “$B$“s have very different meanings. I find this an unacceptable state of affairs.

Secondly, if we accept that the functor $Monoid\to Cat$ needs some name, since as I argued above, it is not an inclusion, then $B$ seems like the natural name for it. But this in incompatible with notating the functor “geometric realization of the nerve” by $B$, since then in the commutative square $$\array{ Groups & \overset{B}{\to} & Groupoids\\ \downarrow & & \downarrow^B\\ Topological Groups & \underset{B}{\to} & Spaces }$$ all the functors would be denoted by $B$, except the left-hand vertical one which is simply an inclusion and usually nameless. Thus we would have $B\circ B = B$, also an unacceptable state of affairs.

Maybe I’m overoptimistic, but I guess most mathematicians worthy of the name have at least heard of ‘Möbius inversion’ and are thus vaguely curious about what it means, or at least annoyed at the existence of yet another buzzword they don’t understand. So if you can explain it in a way they remember, they’ll thank you for it.

So in fact I guess the first few slides, review for me, may be extremely useful for people who have never understood any version of ‘Möbius inversion’ or the ‘Möbius function’.

If at the bottom of page 32 (that’s what it’s called, anyway) you say the magic phrase Riemann zeta function, they’ll know this is important stuff. And if you wanted to really pull out all the stops, you could note that the left-hand side of this equation blows up at the famous zeroes of the Riemann zeta function.

(Which, however, occur at points where the sum doesn’t converge.)

Yes, I’d assume most people in the audience would have seen the definition of the classical Möbius function at some point, even if they’d forgotten it. The first few slides, reviewing this, were mostly by way of scratching an itch, so that people didn’t sit there thinking “but what was that thing called the Möbius function from my undergraduate days?”

(When I wrote that most people wouldn’t have heard of or care about either kind of Möbius inversion, I was referring to Möbius inversion for categories.)

you could note that the left-hand side of this equation blows up at the famous zeroes of the Riemann zeta function.

Good idea. I could also be more clear than I was in Belgium about the reason for the notation “$\zeta$” for the function with constant value $1$.

Hmm. I understand what you just wrote, but I don’t have any explanation.

Can you say a bit more about what kind of answer to “What’s going on?” you’re hoping for?

The logarithmic relationship between the first two is not that surprising to me, since for a finite-dimensional vector space over a field with $q$ elements, its cardinality $s$ and rank $r$ are related by $s = q^r$. Therefore, an alternating product of cardinalities corresponds to an alternating sum of ranks. If we regard homology groups as a sort of approximation to homotopy groups, then morally it seems clear that the ordinary Euler characteristic is a sort of logarithm of the groupoid cardinality. Of course the Euler characteristic is usually defined using integral or rational homology, and many of the spaces we apply it to have infinite homotopy groups, so it seems like usually “the base of the logarithm is infinity” or some such nonsense.

Is there anything more precise we can say in that direction? And is there some reason why a finite poset presentation is a “logarithm” of a groupoid presentation?

Urs wrote:

What’s going on?

I think this is a big mystery, which is why I gave a talk on this subject called The Mysteries of Counting: Euler Characteristic versus Homotopy Cardinality. This was before Tom provided a new clue to the puzzle. Unfortunately, Tom’s clue leaves me still mystified about the way cardinality has an ‘additive’ and ‘multiplicative’ face, which like Jekyll and Hyde are secretly the same yet hardly ever seen in the same place at the same time.

His work brings them closer together. But I’d still like to see a nice category of topological spaces (or homotopy types) that includes both the “cohomologically finite” and “homotopically finite” spaces mentioned in my talk, and comes equipped with a notion of cardinality generalizing both Euler characteristic and homotopy cardinality, that is both additive under disjoint union and multiplicative under twisted products.

The page for my talk includes lots of references for anyone interested in Euler cardinality. Some perhaps lesser-known papers that I find really exciting are:

• William J. Floyd and Steven P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), 1-29.

• R. I. Grigorchuk, Growth functions, rewriting systems and Euler characteristic, Mat. Zametki 58 (1995), 653-668, 798.

I think this is a big mystery, […]

Thanks for reminding me of that page of yours. I have now incorporated a little bit of its content and many of its references into the $n$Lab page.

I think Mike is correctly aiming at getting to the bottom of the nature of Euler char with his emphasis of the fact that Euler characteristic is just the categorical dimension in the correct ambient symmetric monoidal category/$\infty$-category.

So I agree that his query box-question in the entry looks like the one to concentrate on:

in which ambient symmetric monoidal category/$\infty$-category is the Euler characteristic of a category/enriched category/higher category nothing but the categorical dimension (trace of the identity morphism)?

in which ambient symmetric monoidal category/∞-category is the Euler characteristic of a category/enriched category/higher category nothing but the categorical dimension (trace of the identity morphism)?

Here’s a specific question I’d like to know the answer to: the category of Z-categories should be symmetric monoidal, with a smash product analogous to that for spectra. And there should be a “suspension Z-category” functor $\Sigma^\infty_+$ from ω-categories to Z-categories. For which ω-categories X is the Z-category $\Sigma^\infty_+ X$ dualizable, and what is the resulting Euler characteristic?

the category of Z-categories should be symmetric monoidal

Can we just briefly see if we agree on what this is?

I suppose you are thinking of globular sets, unbounded in both directions, equipped with a composition operation?

More systematically it could be defined to be: the stabilization of $(\infty,n)Cat$. Maybe here we want $n= 1$ or somehow imagine sending $n \to \infty$. Maybe better the former.

Or is it? This definition would stabilize with respect to looping whose loops are invertible. Maybe $Z Cat$ should instead be stabilized with respect to “lax looping” (or “comma looping”, I guess)?

The former version has the advantage that we know in some detail how to make sense of it and we also know that it indeed is a sensible directed analog of spectra. And for comparison with Tom’s 1-categorical Euler characteristic it should be sufficient.

Another question I have: if we want ($n$-)categorical Euler characteristic to be a categorical trace, the ambient symmetric monoidal context has to be such that $End(1)$ “contains rational numbers” in some way. That seems unlikely for any category of (higher) categories. Or is it? We might need to abelianize in some other way, too, maybe.

There are two things that I would consider calling Z-categories:

1. globular sets (or some other kind of shapes) unbounded in both directions, with compositions

2. a sequence $\{X_n\}_{n\in \mathbb{N}}$ of basepointed ω-categories with equivalences $$X_n \simeq \Omega_{dir} X_{n+1} \coloneqq Hom_{X_{n+1}}(\ast,\ast)$$

The second is, I think, what you mean by stabilizing with respect to lax or comma looping. We could also, I guess, consider the usual $(\infty,1)$-categorical stabilization of ω-Cat (or, I guess, $(\infty,n)$-Cat), but I wouldn’t expect it to be very interesting. In particular, I think if $X$ is a pointed ω-category, then the homotopy pullback

$$\array{ \Omega_{inv} X & \to & \ast \\ \downarrow && \downarrow \\ \ast &\to&X}$$

in the $(\infty,1)$-category of ω-categories, consists only of automorphisms of $\ast$, automorphisms of those, etc.—only invertible things. Even the comma object in the $(\infty,2)$-category of ω-categories would only have invertible higher cells. So it seems that there isn’t really any directedness appearing at all in that construction: the spectrification condition will force the objects to just be ordinary ∞-groupoids and the resulting “Z-category” to just be an ordinary undirected spectrum.

Whether (1) and (2) above are equivalent is also an interesting question. My guess would be that the objects (2) are equivalent to the objects of (1) with some additional property like that for every cell $x$, there exists an $n$ such that $s^n x = t^n x$ is an identity cell.

if we want (n-)categorical Euler characteristic to be a categorical trace, the ambient symmetric monoidal context has to be such that End(1) “contains rational numbers” in some way.

Yes, I guess so. That is also puzzling to me. Is there any known symmetric monoidal category containing a dualizable object whose Euler characteristic is fractional?

By the way, let me take this opportunity to shamelessly advertise the expository paper Traces in symmetric monoidal categories, which Kate and I have just posted on our websites. It has a lot of toy examples (most of which were in the first version of “Shadows and traces in bicategories”), and some informal discussion about what traces have to do with fixed points, which is something that puzzled me for a long time.