---
name: operad-compose
description: Operad Composition Skill (PLUS +1)
source: local
license: UNLICENSED
---
# Operad Composition Skill (PLUS +1)
> Colored operad composition for structured generation
**Trit**: +1 (PLUS)
**Color**: #D82626 (Red)
**Role**: Generator/Creator
## Core Concept
A colored operad O has:
- **Colors** C (types)
- **Operations** O(c₁,...,cₙ; c) (n-ary operations with input colors cᵢ, output color c)
- **Composition** γ (operadic substitution)
- **Units** 1_c ∈ O(c; c)
```
c₁ c₂ c₃
| /
| /
|/
O(c₁,c₂,c₃; c)
|
c
```
## Operadic Substitution γ
```
γ: O(c₁,...,cₙ; c) × O(d₁,...,dₘ; c₁) → O(d₁,...,dₘ,c₂,...,cₙ; c)
```
Substituting into the first input slot.
### Full Composition
```
γ: O(c₁,...,cₙ; c) × ∏ᵢ O(dᵢ,₁,...,dᵢ,ₖᵢ; cᵢ) → O(d₁,₁,...,dₙ,ₖₙ; c)
```
## Integration with Rubato Composer
```julia
# Musical operad for composition
struct MusicOperad
colors::Set{Symbol} # :melody, :rhythm, :harmony, :texture
operations::Dict{Tuple, Vector{Symbol}} # input colors → output color
end
# Operadic composition for music
function compose_operad(op1, op2, slot::Int)
# op1: (c₁,...,cₙ) → c
# op2: (d₁,...,dₘ) → cₛₗₒₜ
# result: (c₁,...,cₛₗₒₜ₋₁,d₁,...,dₘ,cₛₗₒₜ₊₁,...,cₙ) → c
new_inputs = vcat(
op1.inputs[1:slot-1],
op2.inputs,
op1.inputs[slot+1:end]
)
(inputs=new_inputs, output=op1.output)
end
```
## Gay.jl 3-Color Operad
```julia
# Colored operad with GF(3) colors
const GF3Colors = [:minus, :ergodic, :plus] # -1, 0, +1
struct GF3Operad
# Operations that conserve GF(3)
operations::Vector{NamedTuple}
end
# Valid operations sum to 0 mod 3
function valid_gf3_op(inputs::Vector{Int}, output::Int)
(sum(inputs) + output) % 3 == 0
end
# Generate all valid operations
function gf3_operations(arity::Int)
[(inputs=ins, output=out)
for ins in Iterators.product(fill(-1:1, arity)...)
for out in -1:1
if valid_gf3_op(collect(ins), out)]
end
```
## Little Disks Operad (E₂)
Configuration spaces of n non-overlapping disks:
```
E₂(n) = { (z₁,r₁),...,(zₙ,rₙ) : disks don't overlap } / scaling
```
- **Composition**: Insert small disk configuration into a slot
- **Braiding**: E₂ is braided (operations can pass through each other)
- **Applications**: 2D field theories, loop spaces
## May Operad for Concurrency
```haskell
-- Little intervals operad for A∞ structure
data E1 n = E1 { intervals :: Vector n (Double, Double) }
-- Composition: splice intervals
compose :: E1 n -> Int -> E1 m -> E1 (n + m - 1)
```
## GF(3) Triads
```
persistent-homology (-1) ⊗ open-games (0) ⊗ operad-compose (+1) = 0 ✓
clj-kondo-3color (-1) ⊗ acsets (0) ⊗ operad-compose (+1) = 0 ✓
proofgeneral-narya (-1) ⊗ glass-bead-game (0) ⊗ operad-compose (+1) = 0 ✓
```
## Commands
```bash
# Compose operations
just operad-mult op1 op2 slot
# Generate GF(3) operations of given arity
just operad-gf3 arity
# Visualize operad tree
just operad-tree operation
# Check operad associativity
just operad-assoc op1 op2 op3
```
## Operads in Nature
| Domain | Operad | Colors |
|--------|--------|--------|
| Music | Composition | melody, rhythm, harmony |
| Types | Substitution | types |
| Topology | Little disks | points |
| Logic | Cut-elimination | formulas |
| AI | Skill composition | capabilities |
## References
- May, "The Geometry of Iterated Loop Spaces"
- Loday & Vallette, "Algebraic Operads"
- Spivak, "The Operad of Wiring Diagrams"
