Core Sheaf Workflows

This page highlights the main externally facing APIs for constructing sheaves, assembling operators, and solving section problems.

When to use this page

Use this guide when you want to:

• Build a sheaf model on a fixed graph.
• Form the sheaf Laplacian and evaluate consistency energy.
• Compute or approximate globally consistent sections.

Constructing and Inspecting Sheaves

• Create sheaves using EuclideanSheaf, sheaf_from_graph, and @cellular_sheaf.
• Inspect structure with vertex_stalks, edge_stalks, and underlying_graph.
• See: Euclidean Sheaves, Sheaf Interface, and Cellular Sheaf DSL.

using CellularSheaves, Graphs

g = path_graph(4)
s = sheaf_from_graph(g, 2, n -> Matrix{Float64}(I, n, n))

vertex_stalks(s)
edge_stalks(s)
underlying_graph(s)

Laplacians and Section Computations

• Build operators with coboundary_map and sheaf_laplacian_matrix.
• Optimize and project with energy_function, nearest_global_section, nullspace_ldlt, and harmonic_extension.
• See: Euclidean Sheaves and Sheaf Interface.

L = sheaf_laplacian_matrix(s)
b = rand(sum(vertex_stalks(s)))

x_star = nearest_global_section(s, b)
E = energy_function(s)(x_star)

Typical workflow:

1. Construct s from graph structure and restriction maps.
2. Evaluate sheaf_laplacian_matrix(s) to inspect conditioning and size.
3. Project data with nearest_global_section or solve constrained values with harmonic_extension.