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}:
 0.7821314973723056
 0.039485372628759285
 0.21363989632016633
 0.3695763272169056
 0.24971446370614858
 0.28225303763984655
 0.2393441621662179
 0.5631213469737266
 0.5025958776169936
 0.3347573625738054
 0.44857527268125164
 0.548624124453838

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}:
 0.6359706704768614  
 0.039485372628759285
 0.06747906942472215 
 0.3695763272169056  
 ────────────────────
 0.19502142831700275 
 0.28225303763984655 
 0.2393441621662179  
 0.5084283115845808  
 ────────────────────
 0.7034497399015837  
 0.3347573625738054  
 0.44857527268125164 
 0.548624124453838   

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