The sphere spectrum
BID, Lisboa 2017
1. Extensions of number systems
2. Negative numbers
3. Briefly on sheep
4. A trade deficit
5. Out in space
6. Ulla and Henriette live in RP∞
7. Negative sets
8. The sphere spectrum
9. Brave New World
There is a saying:
"use algebra to add and topology to subtract"
which is implemented as follows, eventually giving us "negative" sets.
The idea has some similarities with string theory: There particles are viewed as loops or strings to fatten up their symmetry groups. Timelines become surfaces (a loop floating in time, making particle interactions nice and smooth, as opposed to the branchings we're accustomed to). Contrary to string theory, the below are not speculations, but we need to adjoust the setup a wee bit to make it work.
In string theory, a collision between two particles look like the "pair of pants" picture. The annihilation of a particle and its antiparticle looks like an old fashioned telephone handset. Note that the loop of the particle and the antiparticle are oppositely oriented. Negative sets are sets that spin the "wrong" way.
As a warm up, let's make a "string theory" model for the integers Z":
To the left you see 3+5=8 in this model, and to the right you see 1+(-1)=0.
- For each natural number a, consider a loop with arc length a (so, zero is just a point).
- Between the loops of a, b and a+b, inset a surface (a,b).
The shape of the surface is immaterial, and in this way we get associativity, e.g. (3+2)+4=3+(2+4): the illustration to the left shows how you form (3+2)+4, and the figure to the right shows how this is deformed via a symmetric version of "3+2+4" to a copy of 3+(2+4).
This is a visualization only: in reality you create the classifying space BN (wrt. addition) and look at the space
The loopspace Ω BN is a fat and "flabby" version of the integers: it has exactly one path component for each integer, but each component is contractible: up to homotopy it contains no extra information.