plurigrid CatColab Decapodes: Discrete Exterior Calculus
Trit: -1 (MINUS - validator/verifier) Color: Purple (#8A2BE2)
Overview
Decapodes in CatColab enable Discrete Exterior Calculus (DEC) for modeling PDEs:
- Differential forms: 0-forms (scalars), 1-forms (vectors), 2-forms (flux)
- Operators: d (exterior derivative), ★ (Hodge star), Δ (Laplacian)
- Multiphysics: Compose PDEs from different domains
- Automatic code generation: Export to AlgebraicJulia/Decapodes.jl
This is CatColab's most advanced logic, connecting category theory to numerical PDE simulation.
Mathematical Foundation
Discrete Exterior Calculus discretizes differential geometry on meshes:
┌─────────────────────────────────────────────────────┐
│ DISCRETE EXTERIOR CALCULUS │
├─────────────────────────────────────────────────────┤
│ Spaces (Differential Forms): │
│ Ω⁰ (0-forms): Scalars on vertices (temperature) │
│ Ω¹ (1-forms): Vectors on edges (velocity) │
│ Ω² (2-forms): Flux through faces (flow rate) │
│ │
│ Operators: │
│ d: Ωᵏ → Ωᵏ⁺¹ (exterior derivative) │
│ ★: Ωᵏ → Ωⁿ⁻ᵏ (Hodge star) │
│ δ = ★d★: Ωᵏ → Ωᵏ⁻¹ (codifferential) │
│ Δ = dδ + δd: Laplacian │
│ │
│ De Rham Complex: │
│ Ω⁰ ──d──► Ω¹ ──d──► Ω² ──d──► Ω³ │
│ │ │ │ │ │
│ ★ ★ ★ ★ │
│ ▼ ▼ ▼ ▼ │
│ Ω³ ◄──d── Ω² ◄──d── Ω¹ ◄──d── Ω⁰ │
└─────────────────────────────────────────────────────┘
Double Theory
// DEC double theory (simplified)
pub fn th_decapodes() -> DiscreteDblTheory {
let mut cat = FpCategory::new();
// Form spaces
cat.add_ob_generator(name("Form0")); // Scalars
cat.add_ob_generator(name("Form1")); // 1-forms
cat.add_ob_generator(name("Form2")); // 2-forms
// Differential operators
cat.add_mor_generator(name("d0"), name("Form0"), name("Form1"));
cat.add_mor_generator(name("d1"), name("Form1"), name("Form2"));
// Hodge star
cat.add_mor_generator(name("star0"), name("Form0"), name("Form2"));
cat.add_mor_generator(name("star1"), name("Form1"), name("Form1"));
cat.add_mor_generator(name("star2"), name("Form2"), name("Form0"));
// Constraint: d ∘ d = 0
cat.add_equation(
compose(name("d0"), name("d1")),
zero_morphism(name("Form0"), name("Form2"))
);
cat.into()
}
CatColab Implementation
Form Space Declaration
{
"type": "ObDecl",
"name": "Temperature",
"theory_type": "Form0",
"description": "scalar temperature field on vertices"
}
Operator Declaration
{
"type": "MorDecl",
"name": "gradient_T",
"dom": "Temperature",
"cod": "HeatFlux",
"theory_type": "d0",
"description": "gradient of temperature"
}
PDE Declaration
{
"type": "EqDecl",
"name": "heat_equation",
"equation": "∂T/∂t = κ·Δ(T)",
"description": "heat diffusion equation"
}
Physics Examples
Example 1: Heat Equation
∂T/∂t = κ·ΔT
Where:
T: Form0 (temperature)
κ: diffusivity constant
Δ = ★d★d (Laplace-Beltrami)
CatColab composition:
T ──d──► dT ──★──► ★dT ──d──► d★dT ──★──► ★d★dT = ΔT
Example 2: Navier-Stokes (2D)
∂ω/∂t + (v·∇)ω = ν·Δω
Where:
ω: Form2 (vorticity)
v: Form1 (velocity)
ν: viscosity
Composition:
Advection: v ──∧──► v∧ω (wedge product)
Diffusion: ω ──Δ──► Δω
Example 3: Maxwell's Equations
dE = -∂B/∂t (Faraday)
dB = 0 (no monopoles)
δE = ρ/ε₀ (Gauss)
δB = μ₀J + μ₀ε₀∂E/∂t (Ampère-Maxwell)
Where:
E: Form1 (electric field)
B: Form2 (magnetic field)
J: Form1 (current density)
Example 4: Shallow Water Equations
∂h/∂t + ∇·(hv) = 0 (continuity)
∂v/∂t + (v·∇)v = -g∇h (momentum)
Where:
h: Form0 (water height)
v: Form1 (velocity)
Multiphysics Composition
Decapodes compose via operad algebras:
┌─────────────────────────────────────────────────────┐
│ MULTIPHYSICS COMPOSITION │
├─────────────────────────────────────────────────────┤
│ Physics 1: Heat Transfer │
│ ∂T/∂t = κ·ΔT │
│ │
│ Physics 2: Advection │
│ ∂c/∂t = -v·∇c │
│ │
│ Composed: Advection-Diffusion │
│ ∂c/∂t = κ·Δc - v·∇c │
│ │
│ Interface: Shared Form1 velocity v │
└─────────────────────────────────────────────────────┘
CatColab 0.2 Integration
CatColab 0.2 (Wren) introduced Decapodes integration:
// Export to Decapodes.jl
const analysis = await model.analyze({
type: "decapodes-export",
mesh: "sphere_mesh.obj",
time_span: [0, 100],
parameters: { κ: 0.1 }
});
// Returns animated solution visualization
GF(3) Triads
catcolab-decapodes (-1) ⊗ topos-catcolab (0) ⊗ catcolab-stock-flow (+1) = 0 ✓
fokker-planck-analyzer (-1) ⊗ catcolab-decapodes (0) ⊗ langevin-dynamics (+1) = 0 ✓
Commands
# Create Decapodes model
just catcolab-new decapodes "heat-transfer"
# Generate Julia code
just catcolab-export heat-transfer --format=decapodes
# Simulate on mesh
just catcolab-simulate heat-transfer --mesh sphere.obj --time 100
# Compose physics models
just catcolab-compose heat-transfer advection --interface velocity
Integration with Decapodes.jl
using Decapodes
using CombinatorialSpaces
# Load CatColab model
decapode = load_decapode("heat-transfer.json")
# Create mesh
mesh = loadmesh(Icosphere(3))
# Generate simulation code
sim = evalsim(decapode)
# Run simulation
u0 = initial_conditions(mesh)
prob = ODEProblem(sim, u0, (0.0, 10.0))
sol = solve(prob, Tsit5())
# Visualize
animate(sol, mesh, "heat_animation.mp4")
References
- Hirani (2003) "Discrete Exterior Calculus" (PhD thesis)
- Desbrun et al. (2005) "Discrete Differential Forms for Computational Modeling"
- Patterson et al. (2023) "Decapodes: A diagrammatic framework for multiphysics"
- Decapodes.jl
- CatColab DEC Help
Skill Name: catcolab-decapodes Type: Discrete Exterior Calculus / PDE Modeling Trit: -1 (MINUS) GF(3): Conserved via triadic composition