Elements of ∞-Categories Skill: Model-Independent Foundations

Status: ✅ Production Ready Trit: 0 (ERGODIC - coordinator) Color: #26D826 (Green) Principle: ∞-categories via model-independent axioms Frame: Riehl-Verity ∞-cosmos formalism

---

Overview

Elements of ∞-Category Theory provides model-independent foundations for ∞-categories. Rather than committing to quasi-categories, complete Segal spaces, or another model, the ∞-cosmos framework captures the common structure.

1. ∞-cosmos: Enriched category of ∞-categories
2. Isofibrations: Right class of factorization system
3. Comma ∞-categories: Slice constructions
4. Adjunctions/equivalences: Model-independent definitions

Core Framework

∞-cosmos K has:
  - Objects: ∞-categories
  - Mapping spaces: Kan complexes Map_K(A, B)
  - Isofibrations: p : E ↠ B with lift property
  - Comma objects: A/f for f : A → B

class InfinityCosmos k where
  type Ob k :: Type
  mapping :: Ob k → Ob k → KanComplex
  isofibration :: (e : Ob k) → (b : Ob k) → Prop
  comma :: {a b : Ob k} → (f : Map a b) → Ob k

Key Concepts

1. ∞-Cosmos Structure

-- Core axioms of an ∞-cosmos
record ∞-Cosmos : Type₁ where
  field
    Ob : Type
    Hom : Ob → Ob → KanComplex
    id : (A : Ob) → Hom A A
    _∘_ : Hom B C → Hom A B → Hom A C
    
    -- Limits
    terminal : Ob
    product : Ob → Ob → Ob
    pullback : {A B C : Ob} → Hom A C → Hom B C → Ob
    
    -- Isofibrations
    isofib : {E B : Ob} → Hom E B → Prop
    factorization : (f : Hom A B) → 
      Σ E, Σ (p : Hom E B), isofib p × trivial-cofib(A → E)

2. Comma ∞-Categories

-- Comma construction
comma : {K : ∞-Cosmos} {A B C : K.Ob} 
      → K.Hom A C → K.Hom B C → K.Ob
comma f g = pullback (mapping-isofib A C f) (ev₀ : C^𝟚 → C) 
            ×_{C} pullback (mapping-isofib B C g) (ev₁ : C^𝟚 → C)

-- Slice as comma
slice : {K : ∞-Cosmos} (B : K.Ob) (b : pt → B) → K.Ob  
slice B b = comma (id B) b

3. Adjunctions

-- Model-independent adjunction
record Adjunction (L : Hom A B) (R : Hom B A) : Type where
  field
    unit : id A ⇒ R ∘ L
    counit : L ∘ R ⇒ id B
    triangle-L : (counit ∘ L) ∘ (L ∘ unit) ≡ id L
    triangle-R : (R ∘ counit) ∘ (unit ∘ R) ≡ id R

Commands

# Verify ∞-cosmos axioms
just infinity-cosmos-check structure.rzk

# Compute comma construction
just comma-category f.rzk g.rzk

# Check adjunction conditions
just adjunction-verify L.rzk R.rzk

Integration with GF(3) Triads

yoneda-directed (-1) ⊗ elements-infinity-cats (0) ⊗ synthetic-adjunctions (+1) = 0 ✓  [Yoneda-Adjunction]
covariant-fibrations (-1) ⊗ elements-infinity-cats (0) ⊗ rezk-types (+1) = 0 ✓  [Model Transport]

Related Skills

• synthetic-adjunctions (+1): Generate adjunction data
• covariant-fibrations (-1): Validate fibration conditions
• segal-types (-1): Concrete Segal space model

---

Skill Name: elements-infinity-cats Type: ∞-Cosmos Coordinator Trit: 0 (ERGODIC) Color: #26D826 (Green)