Kan Extensions

Kan Extensions Skill (ERGODIC 0)

Universal schema migration via left/right Kan extensions

Trit: 0 (ERGODIC)
Color: #26D826 (Green)
Role: Coordinator/Transporter

Core Concept

Kan extensions are the "best approximation" to extending a functor along another:

       F
   C ────→ D
   │       ↑
 K │       │ Lan_K F  (left Kan extension)
   ↓       │ Ran_K F  (right Kan extension)
   C'

Adjunction: Lan_K ⊣ Res_K ⊣ Ran_K

Pointwise Formulas

Left Kan Extension (Forward Migration)

(Lan_K F)(d) = colim_{(c,f: K(c)→d)} F(c)

• Colimit over comma category (K ↓ d)
• Extends F forward along K
• Preserves colimits when F does

Right Kan Extension (Backward Migration)

(Ran_K F)(d) = lim_{(c,f: d→K(c))} F(c)

• Limit over comma category (d ↓ K)
• Extends F backward along K
• Preserves limits when F does

Integration with ACSets

using Catlab, DataMigrations

# Schema migration via Kan extension
# K: SchemaOld → SchemaNew
# F: SchemaOld → Set (instance)
# Lan_K F: SchemaNew → Set (migrated instance)

function left_kan_migrate(K::DataMigration, instance::ACSet)
    # Compute colimit for each new object
    return colimit_representables(K, instance)
end

function right_kan_migrate(K::DataMigration, instance::ACSet)
    # Compute limit for each new object
    return limit_representables(K, instance)
end

Schema Transport Patterns

Pattern 1: Forward Schema Evolution

@migration SchemaV1 SchemaV2 begin
    # Lan extends forward
    NewTable => @join begin
        old::OldTable
        # computed from old structure
    end
end

Pattern 2: Backward Compatibility

@migration SchemaV2 SchemaV1 begin
    # Ran projects backward
    OldTable => @join begin
        new::NewTable
        # projected from new structure
    end
end

Pattern 3: Universal Property

For any H: C' → D with natural transformation α: F → H ∘ K
∃! β: Lan_K F → H such that α = β ∘ K ∘ η

GF(3) Triads

sheaf-cohomology (-1) ⊗ kan-extensions (0) ⊗ free-monad-gen (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ kan-extensions (0) ⊗ operad-compose (+1) = 0 ✓
persistent-homology (-1) ⊗ kan-extensions (0) ⊗ topos-generate (+1) = 0 ✓

Commands

# Migrate schema forward (Lan)
just kan-migrate-forward old.json new_schema

# Migrate schema backward (Ran) 
just kan-migrate-backward new.json old_schema

# Check universal property
just kan-universal K F H

All Concepts Are Kan Extensions

References

• Mac Lane, "Categories for the Working Mathematician" Ch. X
• Riehl, "Category Theory in Context" §6
• nLab: https://ncatlab.org/nlab/show/Kan+extension