# Categorifications of the real numbers

For the purposes of this question, a categorification of the real numbers is a pair $(\mathcal{C},r)$ consisting of:

• a symmetric monoidal category $\mathcal{C}$
• a function $r\colon \mathrm{ob}(\mathcal{C})\to\mathbb{R}$

such that:

• $r(X\otimes Y) = r(X) r(Y)$ for all objects $X$ and $Y$ of $\mathcal{C}$
• $r(\mathbb{1}) = 1$, where $\mathbb{1}$ is the monoidal unit
• $X\cong X'\implies r(X)=r(X')$

Some examples of categorifications of $\mathbb{R}$ are: finite sets and cardinality; finite-dimensional vector spaces and dimension; topological spaces (with the homotopy type of a CW-complex, say) and Euler characteristic. However, in these examples the map $r$ factors through $\mathbb{N}$ or $\mathbb{Z}$. I am interested in examples where the values of $r$ are not so restricted.

Question: What categorifications of $\mathbb{R}$ are there where $r$ can take all values in $\mathbb{R}$, or perhaps all values in $(0,\infty)$ or $(1,\infty)$?

I am especially interested in examples that already appear somewhere in the mathematical literature. I am also especially interested in examples where $\mathcal{C}$ is symmetric monoidal abelian and r(A)+r(C) = r(B) for every short exact sequence $0\to A\to B\to C\to 0$.

• This page has a number of examples: http://math.ucr.edu/home/baez/counting/ – MTyson Sep 6 '18 at 18:49
• en.wikipedia.org/wiki/Continuous_geometry – Qiaochu Yuan Sep 6 '18 at 20:56
• By an underappreciated result of Shannon, this is true (for nonnegative reals) for the capacity of discrete memoryless channels with the usual Kronecker product of channel matrices. – Steve Huntsman Sep 7 '18 at 1:25
• Here is a cheeky answer (sufficiently cheeky I would rather write it as a comment). The Godbillon-Vey invariant of a connected oriented closed 3-manifold equipped with a co-oriented foliation is a top cohomology class in $H^3(M;\Bbb R) \cong \Bbb R$. One may extend this additively to a $\Bbb R$-valued invariant for disconnected such manifolds (or, equivalently, say "evaluate the GV class against the fundamental class"). Then the above category (with morphisms the isomorphisms) is symmetric monoidal with disjoint union, and one may take $\text{exp}(\text{gv}(M, \mathcal F))$. But this is silly. – Mike Miller Sep 7 '18 at 15:42
• Doesnt the Category of finitely-generated Hilbert $N(G)$-modules for $G$ some fixed infinite group ans $r$ the von Neumann dimension work ? – Berni Waterman Sep 7 '18 at 17:47

Janelidze and Street have described such a symmetric monoidal category over the nonnegative real numbers in

• George Janelidze and Ross Street, Real sets, Tbilisi Math. J., Volume 10, Issue 3 (2017), 23-49. (link)

The arXiv version is here.

• Any explanation for the down-vote? – Todd Trimble Sep 6 '18 at 21:18
• A coulpe of pointers to the relevant definitions in this paper would be useful. In particular, what is the symmetric monoidal category you are referring to (I presume it's the "series monoids")? And what is the function $r:Ob(\mathit C)\to\mathbb R$, or where is it discussed in the article? – André Henriques Sep 6 '18 at 23:14
• @AndréHenriques As far as I can tell, the category is $\mathrm{RSet_g}$ (Definition 6.8) and the functor is $\# : \mathrm{RSet_g} \rightarrow [0,\infty]$ (so not to $\mathbb{R}$ but to the extended reals). It's a "categorification" of Higgs's characterization of $[0,\infty]$, as described in Theorem 4.6. – Robert Furber Sep 7 '18 at 20:29

If you drop the condition that your category is symmetric monoidal, and content yourself with a category that's merely monoidal, then the tensor category of $R$-$R$-bimodules for $R$ a factor (factor = von Neumann algebra with trivial center) is an example .

The "quantum dimension" or "statistical dimension" of an $R$-$R$-bimodule is an $\mathbb R_{\ge 0}\cup\{\infty\}$-valued invariant with all the properties that you want.

See, e.g., my paper: https://arxiv.org/abs/1110.5671, specifically Proposition 5.2.

An example might be provided by various genera. A genus is not only multiplicative with respect to the Cartesian product of manifolds but also additive with respect to their disjoint sum, and is not only isomorphism but also cobordism invariant. Some genera do not factor through $\mathbb Z$ -- for example the $\hat A$-genus takes non-integer values on some non-spin manifolds.

I don't know though whether there is a genus which attains all real values. The $\Gamma$-genus of Morava (arXiv:1101.1647) seems to have lots of non-rational values but I am too ignorant to say anything trustworthy about it.

In a somewhat similar spirit to მამუკა ჯიბლაძე's answer: generalisations of Euler characteristic. As mentioned by the OP, classical Euler characteristic associates an integer to a finite complex in a way that is additive with respect to unions and multiplicative with respect to products. There are variations that apply not just to finite complexes (some are described in The Euler characteristic of a category (arXiv:math/0610260), by Tom Leinster). The extended Euler characteristic may have values that are not integer or even rational (for example, the Euler characteristic of the symmetric groupoid is $e$). But it is still a "ring map".

Can we do an example along these lines:
An appropriate collection of metric spaces, with $\otimes$ the Cartesian product, and $r$ a fractal dimension? Or more precisely, $r(X) = \exp(\dim(X))$.

Perhaps the arrows are weakly contracting $d(f(x),f(y) \le d(x,y)$. And perhaps we want the finite-dimenaional (so that $\dim(X) = \infty$ is disallowed) fractals in the sense of Taylor (so that $\dim(X \times Y) = \dim(X)+\dim(Y)$).

Via the notion of groupoid cardinality (generalizing ordinary set-theoretic cardinality), finite non-empty groupoids can be seen as a categorification of the positive rationals, and more general groupoids as a categorification of the non-negative reals:

• John Baez and James Dolan, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited—2001 and Beyond, Springer, 2001. (arXiv link)

An example that Baez and Dolan discuss is the groupoid of finite sets and bijections, which has cardinality $e$. (The notion of Euler characteristic of a category, mentioned in Gregory Arone's answer, can be seen as a further generalization of groupoid cardinality, see Example 2.7 of Leinster's article.)

• I am sure Richard knows this example/phenomenon and it doesn't seem to address the actual question – Yemon Choi Sep 7 '18 at 1:02
• Why doesn't it address the question? Here we have $\mathcal{C} = Gpd$ a symmetric monoidal (2-)category and $r : ob(Gpd) \to [0,\infty]$ given by groupoid cardinality, satisfying the three criteria. – Noam Zeilberger Sep 7 '18 at 1:10
• moreover, this specifically is an example where $r$ can take all non-negative real values. – Noam Zeilberger Sep 7 '18 at 1:16
• @YemonChoi You can write any nonnegative real as a sum of reciprocals of powers of $2$ via its binary representation. For each $1/2^n$, adjoin $B(\mathbb{Z}/2^n\mathbb{Z})$ to the groupoid. – MTyson Sep 7 '18 at 3:51
• @მამუკაჯიბლაძე See Example 2.7 in Leinster's article – Noam Zeilberger Sep 7 '18 at 7:07