      E^op
        ↓ yoneda
    [E^op, Set]
        ↓ sheafification
     Sh(E, J)  ← topos!

Subobject Classifier Ω

In Set: Ω = {0, 1} = Bool In Sh(X): Ω = {open subsets of X} In Sh(C,J): Ω = sieves

For any mono m: A ↣ B
∃! χ_m: B → Ω such that:

    A ───→ 1
    ↓      ↓ true
    B ──→ Ω
       χ_m

Internal Language (Mitchell-Bénabou)

Every topos has an internal type theory:

Types      ↔ Objects
Terms      ↔ Morphisms  
Predicates ↔ Subobjects
∧, ∨, →    ↔ Ω operations
∀, ∃       ↔ Quantifiers via adjoints

Internal Logic

⟦A ∧ B⟧ = ⟦A⟧ ×_Ω ⟦B⟧
⟦A → B⟧ = Ω^{⟦A⟧ → ⟦B⟧}
⟦∀x:X. φ(x)⟧ = ∏_{x:X} ⟦φ(x)⟧

Kripke-Joyal Forcing

Stage-wise truth at object U:

U ⊩ φ ∧ ψ  ⟺  U ⊩ φ and U ⊩ ψ
U ⊩ φ → ψ  ⟺  ∀f:V→U. V ⊩ φ ⟹ V ⊩ ψ  
U ⊩ ∀x:X.φ ⟺  ∀f:V→U, ∀a:X(V). V ⊩ φ[a/x]
U ⊩ ∃x:X.φ ⟺  ∃ cover {Uᵢ→U}, ∃aᵢ:X(Uᵢ). Uᵢ ⊩ φ[aᵢ/x]

Integration with Cider/Clojure

(ns topos.generate
  (:require [acsets.core :as acs]))

;; Subobject classifier for finite topos
(defn omega [topos]
  (let [sieves (all-sieves (:site topos))]
    {:object sieves
     :true (maximal-sieve topos)
     :false (empty-sieve)}))

;; Force a proposition at stage
(defn force [stage prop env]
  (case (:type prop)
    :and (and (force stage (:left prop) env)
              (force stage (:right prop) env))
    :implies (every? (fn [morphism]
                       (let [stage' (compose stage morphism)]
                         (if (force stage' (:antecedent prop) env)
                           (force stage' (:consequent prop) env)
                           true)))
                     (covers stage))
    :forall (every? (fn [[morphism witness]]
                      (force (compose stage morphism) 
                             (:body prop) 
                             (assoc env (:var prop) witness)))
                    (all-witnesses stage (:type prop)))
    :exists (some (fn [[cover witnesses]]
                    (every? (fn [[morph wit]]
                              (force (compose stage morph)
                                     (:body prop)
                                     (assoc env (:var prop) wit)))
                            (zip cover witnesses)))
                  (all-covers-with-witnesses stage (:type prop)))))

;; Generate model satisfying formula
(defn generate-model [topos formula]
  (let [stages (objects topos)]
    (for [stage stages
          :when (force stage formula {})]
      {:stage stage
       :witnesses (collect-witnesses stage formula)})))

Forcing for Set Theory

Cohen forcing generates new sets:

P = partial functions ω → 2 (finite approximations)
G = generic filter (added by forcing)

M[G] = { interpretation of names under G }

GF(3) Triads

sheaf-cohomology (-1) ⊗ dialectica (0) ⊗ topos-generate (+1) = 0 ✓
temporal-coalgebra (-1) ⊗ open-games (0) ⊗ topos-generate (+1) = 0 ✓
three-match (-1) ⊗ kan-extensions (0) ⊗ topos-generate (+1) = 0 ✓

Commands

# Generate subobject classifier
just topos-omega site

# Force proposition at stage
just topos-force stage "∀x. φ(x)"

# Generate satisfying model
just topos-model formula

# Internal language translation
just topos-internal formula

Topos Models in Practice

Topos	Generates	Application
Set	Classical sets	Standard math
Sh(X)	Varying sets over X	Geometry
Sh(G)	G-sets	Symmetry
Eff	Computable functions	Computability
Dialectica	Proof-relevant math	Type theory

References

Mac Lane & Moerdijk, "Sheaves in Geometry and Logic"
Johnstone, "Sketches of an Elephant"
Awodey, "Category Theory" §8
nLab: https://ncatlab.org/nlab/show/forcing

topos-generate

Topos Generation Skill (PLUS +1)

Core Concept