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)))

12-element Vector{Float64}:
 0.69409996154618
 0.9840756958533826
 0.7176294059540133
 0.16639286962869604
 0.5179366841326856
 0.4478268691593501
 0.5881355711002596
 0.4047067850401903
 0.7100517951156654
 0.7992347891131538
 0.9019572877814998
 0.5938009515950424

We use the deterministic direct LDLt backend to guarantee a result.

global_section = nearest_global_section(sheaf, x0; method=:ldl)

3-blocked 12-element BlockVector{Float64}:
 0.4575448311591334 
 0.9840756958533826 
 0.48107427556696664
 0.16639286962869604
 ───────────────────
 0.5259245029092976 
 0.4478268691593501 
 0.5881355711002596 
 0.41269460381680234
 ───────────────────
 0.9386191067261    
 0.7992347891131538 
 0.9019572877814998 
 0.5938009515950424 

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)

1.5700924586837752e-16