catcolab-decapodes

# 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 ```rust // 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 ```typescript { "type": "ObDecl", "name": "Temperature", "theory_type": "Form0", "description": "scalar temperature field on vertices" } ``` ### Operator Declaration ```typescript { "type": "MorDecl", "name": "gradient_T", "dom": "Temperature", "cod": "HeatFlux", "theory_type": "d0", "description": "gradient of temperature" } ``` ### PDE Declaration ```typescript { "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: ```typescript // 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 ```bash # 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 ```julia 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](https://algebraicjulia.github.io/Decapodes.jl/) - [CatColab DEC Help](https://catcolab.org/help/logics/decapodes) --- **Skill Name**: catcolab-decapodes **Type**: Discrete Exterior Calculus / PDE Modeling **Trit**: -1 (MINUS) **GF(3)**: Conserved via triadic composition