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

Do exactly what we did to N to get ΩBN, but this time to Σ:
ΩBΣ
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
S
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