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

Le Voila Do exactly what we did to N to get ΩBN, but this time to Σ:
one loop for every set!

The sphere spectrum

The result ΩBΣ is more commonly known as the (infinite loop space assocated with) the sphere spectrum
The reason for this name comes from the fact that we can get this space in a different way: consider the n-sphere Sn, and let ΩnSn be what you get if you take the loop space of the loop space of the... (n times) of the n sphere. If we let n go to infinity you get another model for ΩBΣ.


[Digression: what we did here for finite sets, could of course have been done in other similar situations. If you for instance do it to finite dimensional vector spaces, you get a version of linear algebra where you can deal with negative dimensional vector spaces. This is called topological K-theory and is useful in many situations (bundles...). If you do it to finitely generated projective modules over rings, you get algebraic K-theory which contains a wealth of arithmetic information.]

Just as you have an inclusion of N in Z, you get an inclusion of Σ in S.

S has one path component for each integer. However, now all path components are equivalent -- every component contains the information of all of Σ (you may for instance find the loop of {Ulla, Henriette} in every component of S and strangely also reappears in algebraic K-theory as the fact that the multiplicative subgroup of the integers is {-1,1} - where (-1)2=1).

Brave New World
Bjørn Ian Dundas
2017-07-21 14:40:56 UTC