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

### Ulla and Henriette live in RP∞

Summarizing, the space Σ is a disjoint union of components - one component for each natural number (corresponding to the number of elements there might be in a finite set).

However, each component is in itself a fancy space (the ones corresponding to 0 and 1 are admittedly not that fancy, but from 2 on they become increasingly fascinating).

Take the component where {Ulla, Henriette} lives. There we have non-trivial path

f{Ulla, Henriette} -> {Ulla, Henriette}
given by cofusing Ulla and Henriette. This path is a loop, and it is not contractible (without letting go of the endpoints). On the other hand; if we use f twice we haven't done anything, so we insert a 2-cell so that cofusing twice is homotopic to not confusing anything.

Remember that in the picture, after identification there is just one vertex {Ulla, Henriette} and one edge f: the equality can have length zero.

So, {Ulla,Henriette} live in RP2. However, it does not stop there; we have

fff = f,
which - when you draw it - shows that {Ulla,Henriette} live in RP3 as well... and this continue to infinity:

{Ulla, Henriette} live in RP

RP is just another word for the classifying space2 of the group Σ2 with two elements.

Σ = 0 + BΣ1 + BΣ2 + BΣ3 + BΣ4 + ...

Negative sets
Bjørn Ian Dundas
2017-07-21 14:40:56 UTC