# 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) is the rotation of a 3-dimensional space. The number 3 is just an attribute of its dimensionality.

Numbers forms a Set
Vector Space forms a Category

Category not only contains ‘objects’ the vector spaces, also ‘morphisms’ from any object to any other object.

Note 1: Morphism of an object to itself = symmetries

Note 2: The Functional Language: Haskel is the next generation of computer languages based on Category, rather than on Set theory.

Categorification: The paradigm shift in modern math, by creating a new world in which the familiar concepts are elevated to a higher level — 欲穷千里路, 更上一层楼。( 6th CE Tang Dynasty Poem: “To see further thousand miles away, climb up one level higher”)

Examples: from sets to categories:

$\boxed { \text {Set} \rightarrow \text {Category} }$ Categorification (1)

Function (sine) to 30 degrees: eg.
$\sin (30 ^\circ) = \frac {1}{2}$
The function ‘sine’ assigns to a point of the circle (manifold) a number 1/2.

A Sheaf assigns to a point in the manifold a vector space.

$\boxed { \text {Number} \rightarrow \text {Vector Space} }$ Categorification (2)

$\boxed { \text {Function} \rightarrow \text {Sheaf} }$ Categorification (3)

Functions were the concepts of archaic math, and sheaves are the concepts of modern math. Sheaf can have symmetries. By elevating a function to a sheaf, we can explore these symmetries to learn more than we could ever learn using functions.

Sheaves applied in the right column of Langrands Program:

Number Theory | Curves over Finite Fields | Riemann Surfaces

$\begin{array}{ |l|l|l|l|} \hline : & Number \: Theory & Curves \: over \: Finite \: Fields & Riemann \: Surfaces \\ \hline L: & Galois \: Group & Galois \: Group & Fundamental \: Group \\ \hline R: & Automorphic \: functions & Automorphic \: functions \: (sheaves) & Automorphic \: sheaves\\ \hline \end{array}$
The 2 rows (L & R) below each column contains the names of the objects on the 2 sides of the Langrands relation specific to that column.

Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY