Coequalizers

Coequalizers Skill

Quotient redundant skill paths via categorical coequalizers

Version: 1.0.0
Trit: 0 (ERGODIC - coordinates equivalences)
Domain: category-theory, skill-composition, colimits, behavioral-equivalence

Overview

The coequalizers skill provides:

1. Behavioral equivalence checking via bisimulation (from temporal-coalgebra)
2. Parallel morphism quotienting via coequalizers (colimits)
3. Skill overlap detection and gluing (from oapply-colimit pushouts)
4. GF(3) conservation in quotient spaces
5. MCP integration for cross-agent skill synchronization

Core Concept

What Are Coequalizers?

A coequalizer is the colimit of two parallel morphisms:

    X ──f──→ Y
    │  g     │
    └────────→ q
             ↓
             Q  (coequalizer)

Universal property: q ∘ f = q ∘ g

In Sets: Q = Y / ~ where ~ is the smallest equivalence relation such that f(x) ~ g(x) for all x ∈ X.

For skills: If two skill paths produce behaviorally equivalent outputs, the coequalizer gives the canonical quotient.

Key Patterns from asi Repository

1. oapply-colimit: Pushout = Coproduct + Coequalizer

From /skills/oapply-colimit/SKILL.md:

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

Key insight: Pushouts decompose as coproduct + coequalizer. This is how skills with shared interfaces are glued together.

2. Bisimulation-game: Behavioral Equivalence

From /skills/bisimulation-game/SKILL.md:

def bisimilar(skill₁, skill₂, input, depth=10):
    """
    Recursively check if skills produce same observations.
    
    Two skills are bisimilar if:
    - They produce same immediate output
    - Their continuations are pairwise bisimilar
    """
    obs₁ = observe(skill₁, input)
    obs₂ = observe(skill₂, input)
    
    if obs₁ != obs₂:
        return False
    
    # Recursive check on continuations
    if depth > 0:
        for (next₁, inp₁), (next₂, inp₂) in zip(
            skill₁.continuations(input),
            skill₂.continuations(input)
        ):
            if not bisimilar(next₁, next₂, inp₁, depth-1):
                return False
    
    return True

Application: Use bisimulation to establish equivalence relation ~ before applying coequalizer.

3. Adhesive Rewriting: Incremental Query Updating

From /skills/topos-adhesive-rewriting/SKILL.md:

# Decomposition: Q ≅ Q_G +_{Q_L} Q_R
# This IS a coequalizer construction!

function quotient_system(system::SkillSystem, equivalences)
    # Build parallel morphisms from equivalence pairs
    equiv_indices = parts(system, :Equivalence)
    
    # Compute coequalizer (built into Catlab)
    quotient = coequalizer(system, equiv_indices)
    
    # Verify GF(3) conservation
    @assert verify_gf3_conservation(quotient)
    
    return quotient
end

Key: Adhesive categories (like C-Sets) have well-behaved coequalizers for incremental updates.

4. Browser-history-acset: Path Equivalence

From /skills/browser-history-acset/path_equivalence_test.jl:

# Path composition via subpart chains
@assert subpart(acs, subpart(acs, 1, :url_of), :domain_of) == 1

# Multiple visit paths may lead to same outcome
# Coequalizer identifies equivalent paths

Application: Navigation paths through skill graphs that produce same results should be identified.

5. Sheaves on Ordered Locale: Directional Restrictions

From /skills/ordered-locale/sheaves.py:

def gluing(cover: List[FrozenSet], family: Dict[FrozenSet, T]) -> Optional[T]:
    """
    Glue a compatible family over a cover.
    
    This is the sheaf condition - dual to coequalizer.
    """
    # Check compatibility on overlaps
    for U_i, U_j in pairs(cover):
        overlap = U_i & U_j
        if overlap:
            res_i = restrict(U_i, overlap, family[U_i])
            res_j = restrict(U_j, overlap, family[U_j])
            if res_i != res_j:
                return None  # Incompatible family
    
    # Glue: coequalizer of restrictions
    return reduce(lambda a, b: a | b, family.values(), frozenset())

Key: Sheaf gluing IS the dual of coequalizer. Overlapping skill contexts must agree.

6. IrreversibleMorphisms: Lossy Transformations

From /skills/compositional-acset-comparison/IrreversibleMorphisms.jl:

# Irreversible morphisms: information loss
const MORPHISM_CLASSIFICATION = Dict(
    :parent_manifest => :irreversible,     # Append-only chain
    :source_column => :irreversible,       # Lossy embedding
    # ...
)

# Coequalizers preserve irreversibility classification
# If f, g both irreversible, coeq(f,g) is irreversible

Application: Track information loss through quotients. GF(3) trit sum must be preserved.

Implementation Strategy

Schema: Skills with Equivalences

using Catlab.CategoricalAlgebra
using AlgebraicRewriting

@present SchSkillCoequalizer(FreeSchema) begin
    Skill::Ob
    Application::Ob
    Equivalence::Ob
    
    app_src::Hom(Application, Skill)
    app_tgt::Hom(Application, Skill)
    equiv_app1::Hom(Equivalence, Application)
    equiv_app2::Hom(Equivalence, Application)
    
    Trit::AttrType
    Behavior::AttrType
    
    skill_trit::Attr(Skill, Trit)
    app_behavior::Attr(Application, Behavior)
end

@acset_type SkillSystem(SchSkillCoequalizer,
    index=[:app_src, :app_tgt, :equiv_app1, :equiv_app2])

GF(3) Integration

Trit Assignment

• Coequalizers have trit = 0 (ERGODIC)
• They coordinate between validators (-1) and generators (+1)
• Preserve trit sums: if inputs sum to 0 mod 3, output preserves this

Synergistic Triads

bisimulation-game (-1) ⊗ coequalizers (0) ⊗ oapply-colimit (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ coequalizers (0) ⊗ topos-adhesive (+1) = 0 ✓
browser-history-acset (-1) ⊗ coequalizers (0) ⊗ ordered-locale (sheaves) (+1) = 0 ✓

Commands

just coequalizer-find SKILLS...          # Find equivalences among skills
just coequalizer-quotient SYSTEM         # Compute quotient
just coequalizer-verify GF3              # Verify GF(3) conservation
just coequalizer-disperse AGENTS...      # Sync across agents
just coequalizer-pushout SKILL1 SKILL2   # Compose with overlap

References

From asi Repository

• oapply-colimit - Pushout = coproduct + coequalizer
• bisimulation-game - Behavioral equivalence testing
• topos-adhesive-rewriting - Incremental query updating via coequalizers
• browser-history-acset - Path equivalence in ACSets
• ordered-locale (sheaves.py) - Gluing as dual of coequalizer
• compositional-acset-comparison (IrreversibleMorphisms.jl) - Lossy morphisms

Papers

• Lack & Sobociński, "Adhesive and Quasiadhesive Categories" (RAIRO 2005)
• Patterson et al., "Categorical Data Structures for Technical Computing" (Compositionality 2022)
• Brown, "Incremental Query Updating in Adhesive Categories" (Topos Institute 2025)

Web Resources

• nLab: Coequalizer
• Catlab.jl Documentation

Related Skills

• bisimulation-game (-1) - Behavioral equivalence
• oapply-colimit (+1) - Pushout composition
• topos-adhesive-rewriting (+1) - Incremental updates
• temporal-coalgebra (-1) - Coalgebraic bisimulation
• ordered-locale (0) - Sheaf gluing

Skill Name: coequalizers
Type: Category-Theoretic Skill Composition
Trit: 0 (ERGODIC - coordinates equivalences)
GF(3): Conserved via triadic composition