Internal Logic

Origin. Lawvere, Kock, and others (1960s-1970s). Logic interpreted inside a category. Subobject classifier replaces truth values. Intuitionistic in general topoi. Foundation for categorical proof theory.

Models. Categories as universes of discourse. Objects as types, morphisms as functions. Subobjects as predicates. Power object for quantification. Ω (subobject classifier) for truth.

Formalism.

Subobject classifier: Ω: object of truth values. true: 1 → Ω (the "true" element). χ_m: A → Ω classifies subobject m: S ↣ A.

Propositions: Subobjects of 1 (global sections of Ω). In Set: Ω = {0,1}, two propositions. In general topos: richer Ω structure.

Connectives as operations on Ω: ∧: Ω × Ω → Ω (meet in Ω) ∨: Ω × Ω → Ω (join in Ω) ⇒: Ω × Ω → Ω (Heyting implication) ¬: Ω → Ω (¬p = p ⇒ ⊥)

Quantifiers: ∀_f: Ω^A → Ω^B (right adjoint to f*) ∃_f: Ω^A → Ω^B (left adjoint to f*) Along f: A → B.

Internal language: Mitchell-Bénabou language. Variables range over objects. Formulas interpreted as subobjects.

Topos validity: ⊨ φ iff φ interpreted as 1 ≅ {x | φ(x)}. Intuitionistic in general topoi.

Symbols.

Symbol	Unicode	Name	Meaning
Ω	U+03A9	Omega	Subobject classifier
↣	U+21A3	Monomorphism	Subobject
true	—	True	1 → Ω
χ	U+03C7	Chi	Classifying map
⊨	U+22A8	Satisfies	Internal validity

Metatheory. Every topos has internal logic. Intuitionistic unless Boolean. Higher-order via power objects. Kripke-Joyal semantics. Soundness and completeness internally.

Applies to. Topos theory. Synthetic differential geometry. Sheaf models. Type theory semantics. Categorical foundations.

Limitations. Requires category theory background. Intuitionistic by default. External vs internal subtle. Constructive mathematics assumptions.