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

### The sphere spectrum

The result ΩB**Σ** 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
*S*^{n}, and let
Ω^{n}*S*^{n} 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 ΩB**Σ**.

[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