Good news! Janelidze and Street have tackled some puzzles that are perennial favorites here on the n n -Café:

The category of these extended positive real sets is equipped with a countable tensor product. We develop somewhat the theory of categories with countable tensors; we call the commutative such categories series monoidal and conclude by only briefly mentioning the non-commutative possibility called ω-monoidal . We include some remarks on sets having cardinalities in [ − ∞ , ∞ ] [-\infty,\infty] .

Abstract. After reviewing a universal characterization of the extended positive real numbers published by Denis Higgs in 1978, we define a category which provides an answer to the questions:

First they define a series magma, which is a set A A equipped with an element 0 0 and a summation function

∑ : A ℕ → A \sum \colon A^{\mathbb{N}} \to A

obeying a nice generalization of the law a + 0 = 0 + a = a a + 0 = 0 + a = a . Then they define a series monoid in which this summation function obeys a version of the commutative law.

(Yeah, the terminology here seems a bit weird: their summation function already has associativity built in, so their ‘series magma’ is associative and their ‘series monoid’ is also commutative!)

The forgetful functor from series monoids to sets has a left adjoint, and as you’d expect, the free series monoid on the one-element set is ℕ ∪ { ∞ } \mathbb{N} \cup \{\infty\} . A more interesting series monoid is [ 0 , ∞ ] [0,\infty] , and one early goal of the paper is to recall Higgs’ categorical description of this. That’s Denis Higgs. Peter Higgs has a boson, but Denis Higgs has a nice theorem.

First, some preliminaries:

Countable products of series monoids coincide with countable coproducts, just as finite products of commutative monoids coincide with finite coproducts.

There is a tensor product of series monoids, which is very similar to the tensor product of commutative monoids —- or, to a lesser extent, the more familiar tensor product of abelian groups. Monoids with respect to this tensor product are called series rigs. For abstract nonsense reasons, because ℕ ∪ { ∞ } \mathbb{N} \cup \{\infty\} is the free series monoid on one elements, it also becomes a series rig… with the usual multiplication and addition. (Well, more or less usual: if you’re not familiar with this stuff, a good exercise is to figure out what 0 0 times ∞ \infty must be.)

Now for the characterization of [ 0 , ∞ ] [0,\infty] . Given an endomorphism f : A → A f \colon A \to A of a series monoid A A you can define a new endomorphism f ¯ : A → A \overline{f} \colon A \to A by

f ¯ = f + f ∘ f + f ∘ f ∘ f + ⋯ \overline{f} = f + f\circ f + f \circ f \circ f + \cdots

where the infinite sum is defined using the series monoid structure on A A . Following Higgs, Janelidze and Street define a Zeno morphism to be an endomorphism h maps A → A h \maps A \to A such that

h ¯ = 1 A \overline{h} = 1_A

The reason for this name is that in [ 0 , ∞ ] [0,\infty] we have

1 = 1 2 + ( 1 2 ) 2 + ( 1 2 ) 3 + ⋯ 1 = \frac{1}{2} + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 + \cdots

putting us in mind of Zeno’s paradox:

That which is in locomotion must arrive at the half-way stage before it arrives at the goal. — Aristotle, Physics VI:9, 239b10.

So, it makes lots of sense to think of any Zeno morphism h : A → A h \colon A \to A as a ‘halving’ operation. Hence the name h h .

In particular, one can show any Zeno morphism obeys

h + h = 1 A h + h = 1_A

Higgs called a series monoid equipped with a Zeno morphism a magnitude module, and he showed that the free magnitude module on one element is [ 0 , ∞ ] [0,\infty] . By the same flavor of abstract nonsense as before, this implies that [ 0 , ∞ ] [0,\infty] is a series rig…. with the usual addition and multiplication.

Categorification

Next, Janelidze and Street categorify the entire discussion so far! They define a ‘series monoidal category’ to be a category A A with an object 0 ∈ A 0 \in A and summation functor

∑ : A ℕ → A \sum \colon A^{\mathbb{N}} \to A

obeying some reasonable properties… up to natural isomorphisms that themselves obey some reasonable properties. So, it’s a category where we can add infinite sequences of objects. For example, every series monoid gives a series monoidal category with only identity morphisms. The maps between series monoidal categories are called ‘series monoidal functors’.

They define a ‘Zeno functor’ to be a series monoidal functor h : A → A h \colon A \to A obeying a categorified version of the definition of Zeno morphism. A series monoidal category with a Zeno functor is called a ‘magnitude category’.

As you’d guess, there are also ‘magnitude functors’ and ‘magnitude natural transformations’, giving a 2-category MgnCat MgnCat . There’s a forgetful 2-functor

U : MgnCat → Cat U \colon MgnCat \to Cat

and it has a left adjoint (or, as Janelidze and Street say, a left ‘biadjoint’)

F : Cat → MgnCat F \colon Cat \to MgnCat

Applying F F to the terminal category 1 1 , they get a magnitude category RSet g RSet_g of positive real sets. These are like sets, but their cardinality can be anything in [ 0 , ∞ ] [0,\infty] !

For example, Janelidze and Street construct a positive real set of cardinality π \pi . Unfortunately they do it starting from the binary expansion of π \pi , so it doesn’t connect in a very interesting way with anything I know about the number π \pi .

What’s that little subscript g g ? Well, unfortunately RSet g RSet_g is a groupoid: the only morphisms between positive real sets we get from this construction are the isomorphisms.

So, there’s a lot of great stuff here, but apparently a lot left to do.

Digressive Postlude

There is more to say, but I need to get going — I have to walk 45 minutes to Paris 7 to talk to Mathieu Anel about symplectic geometry, and then have lunch with him and Paul-André Melliès. Paul-André kindly invited me to participate in his habilitation defense on Monday, along with Gordon Plotkin, André Joyal, Jean-Yves Girard, Thierry Coquand, Pierre-Louis Curien, George Gonthier, and my friend Karine Chemla (an expert on the history of Chinese mathematics). Paul-André has some wonderful ideas on linear logic, Frobenius pseudomonads, game semantics and the like, and we want to figure out more precisely how all this stuff is connected to topological quantum field theory. I think nobody has gotten to the bottom of this! So, I hope to spend more time here, figuring it out with Paul-André.