oapply-colimit Skill

Operad algebra evaluation via colimits with bidirectional navigation

Version: 1.1.0 Trit: +1 (Generator - composes systems)

Core Pattern

oapply computes colimit of component diagram over wiring pattern:

using AlgebraicDynamics

# Pattern + components → composite
composite = oapply(wiring_diagram, [machine1, machine2, ...])

Two Composition Modes

Mode	Type	Gluing	Example
Undirected	ResourceSharer	Pushout (shared state)	Lotka-Volterra
Directed	Machine	Wiring (signal flow)	Control systems

Implementation

function oapply(d::UndirectedWiringDiagram, xs::Vector{ResourceSharer})
    # 1. Coproduct of state spaces
    S = coproduct((FinSet ∘ nstates).(xs))
    
    # 2. Pushout identifies shared variables
    S′ = pushout(portmap, junctions)
    
    # 3. Induced dynamics sum at junctions
    return ResourceSharer(induced_interface, induced_dynamics)
end

Specter-Style Navigation for Wiring Diagrams

Navigate wiring diagrams with bidirectional paths:

using SpecterACSet

# Navigate to all boxes
select([wd_boxes, ALL], diagram)

# Navigate to all wires from a specific box
select([wd_wires, pred(w -> source_box(w) == 1)], diagram)

# Transform: rename all boxes
transform([wd_boxes, ALL, box_name], uppercase, diagram)

Wiring Diagram Navigators

Navigator	Select	Transform
wd_boxes	All boxes	Update boxes
wd_wires	All wires	Update wires
wd_ports(box_id)	Ports of box	Update ports
wd_outer_ports	Outer interface	Update interface

Connection to Specter's comp-navs

Specter's comp-navs pattern mirrors oapply:

# Specter: compose navigators (fast - just allocation)
comp_navs(nav1, nav2, nav3)

# oapply: compose machines (colimit of diagram)
oapply(wiring, [machine1, machine2, machine3])

Both use composition as colimit - Specter over paths, oapply over state spaces.

Compositional Dynamics Example

# Lotka-Volterra as composed resource sharers
rabbit = ResourceSharer{Float64}([:pop], [:pop]) do u, p, t
    [p.α * u[1]]  # growth
end

fox = ResourceSharer{Float64}([:pop], [:pop]) do u, p, t
    [-p.δ * u[1]]  # decay
end

# Compose via shared population interface
predation = oapply(predation_diagram, [rabbit, fox])

Sexp Serialization for Wiring Diagrams

# Wiring diagram → Sexp
sexp = sexp_of_wiring_diagram(diagram)

# Navigate: find all box names
box_names = select([SEXP_CHILDREN, pred(is_box), SEXP_HEAD, ATOM_VALUE], sexp)

# Roundtrip
diagram2 = wiring_diagram_of_sexp(sexp)

GF(3) Triads

schema-validation (-1) ⊗ acsets (0) ⊗ oapply-colimit (+1) = 0 ✓
interval-presheaf (-1) ⊗ algebraic-dynamics (0) ⊗ oapply-colimit (+1) = 0 ✓
bisimulation-game (-1) ⊗ coequalizers (0) ⊗ oapply-colimit (+1) = 0 ✓

Related Skills

• coequalizers (0) - Uses pushout = coproduct + coequalizer decomposition
• bisimulation-game (-1) - Behavioral equivalence for skill paths
• topos-adhesive-rewriting (+1) - Incremental updates via coequalizers

Koopman Integration

For time-varying systems, oapply composes observable functors:

# Koopman operator: lift nonlinear → infinite-dim linear
# oapply: compose lifted systems via colimit

composed_koopman = oapply(
    dynamics_diagram,
    [koopman_lift(system1), koopman_lift(system2)]
)

References

• Libkind "An Algebra of Resource Sharers" arXiv:2007.14442
• AlgebraicJulia/AlgebraicDynamics.jl
• Nathan Marz: Specter composition patterns

Scientific Skill Interleaving

This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem:

Graph Theory

• networkx [○] via bicomodule
  • Universal graph hub

Bibliography References

• general: 734 citations in bib.duckdb

Cat# Integration

This skill maps to Cat# = Comod(P) as a bicomodule in the equipment structure:

Trit: -1 (MINUS)
Home: Prof
Poly Op: ⊗
Kan Role: Ran_K
Color: #FF6B6B

GF(3) Naturality

The skill participates in triads satisfying:

(-1) + (0) + (+1) ≡ 0 (mod 3)

This ensures compositional coherence in the Cat# equipment structure.