Code Example

This is an example of adding a code example compiled with Literate.jl in the docs.

First we want to load our package with using

using CellularSheaves
using LinearAlgebra

Using the Package

Here is a simple example of creating a sheaf using the @cellular_sheaf macro.

A = [1.0 0.0 1.0 0.0]
B = [1.0 0.0 0.0 1.0]
C = [1.0 0.0 0.0 0.0]

sheaf = @cellular_sheaf A, B, C begin
    x::Stalk{4}, y::Stalk{4}, z::Stalk{4}

    A(x) == B(y)
    A(x) == C(z)
    B(y) == C(z)

end

A network sheaf with 3 vertex stalks and 3 edge stalks.

Let's compute a global section of this sheaf. We start with a random 0-cochain:

x0 = rand(sum(vertex_stalks(sheaf)))

global_section = nearest_global_section(sheaf, x0)

3-blocked 12-element BlockVector{Float64}:
 0.44580000522136615
 0.4097909504750348 
 0.14401925806699073
 0.8306461649657294 
 ───────────────────
 0.362111333313761  
 0.33024671280308004
 0.947482906794697  
 0.22770792997459566
 ───────────────────
 0.5898192632883568 
 0.2940774421495559 
 0.6314378706434358 
 0.2936964101982473 

Now we can check that this is indeed a global section by verifying that the coboundary map applied to it is zero.

d = coboundary_map(sheaf)

norm(d * global_section)

2.7194799110210365e-16