Grothendieck’s Sheaf (束)

Natural Numbers (N) = {1,2,3, 4…}
1-dimension: a Line
2-dimension: a plane
n-dimensional flat space: a Vector Space

Now imagine in a world where we replace every natural number by vector space:
1 by a Line
2 by a Plane
n by a flat space Vector Space

Sum of numbers = Direct sum of vector space.
E.g. Add a 1-D Line to a 2-D Plane = 3-D Space

Product of numbers = Tensor Product (of two vector spaces of respective dimension m & n) with dimension m.n

This new world would be much interesting and richer than the Natural Number world: vector spaces have symmetries, whereas numbers are just numbers with no symmetries.
(Interesting): we can add 1 to 2 in only ONE way (1+2), but there are many ways to embed (add) a Line in a Plane (perpendicular, slanting in any angle, etc).
(Richer): the Lie Group SO(3)…