Category Theory Logic

FLOW_ALGEBRA CategoryTheoryLogicOntology v1.0

TARGET_URL := "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/category-theory-logic.html"
LANGUAGE := "en-GB"
DOMAIN := MathematicalOntologies > MathematicalLogicOntologies > CategoryTheoryLogicOntology

BEGIN_FLOW

ENTITY CategoryTheoryLogic {
    TYPE := MathematicalLogicalFramework
    DEFINITION := "A framework connecting category theory and mathematical logic by representing logical syntax, semantics, propositions, proofs and models through categorical structures such as categories, functors, natural transformations, adjunctions and universal constructions."
}

RELATIONSHIP CategoryTheoryLogic --integrates--> CategoryTheory
RELATIONSHIP CategoryTheoryLogic --formalises--> MathematicalLogic
RELATIONSHIP CategoryTheoryLogic --supports--> CategoricalLogic
RELATIONSHIP CategoryTheoryLogic --supports--> FunctorialSemantics
RELATIONSHIP CategoryTheoryLogic --supports--> ToposTheory
RELATIONSHIP CategoryTheoryLogic --supports--> TypeTheory

ENTITY Category {
    COMPONENTS := {
        Objects,
        Morphisms,
        Composition,
        IdentityMorphisms
    }
}

ENTITY Object {
    TYPE := CategoricalEntity
}

ENTITY Morphism {
    TYPE := DirectedCategoricalArrow
    FORM := f:A->B
}

CATEGORY_AXIOMS {
    ASSOCIATIVITY :=
        h∘(g∘f) = (h∘g)∘f

    IDENTITY :=
        id_B∘f = f = f∘id_A
}

ENTITY Functor {
    TYPE := StructurePreservingMappingBetweenCategories
}

FUNCTOR {
    F:C->D

    MAP_OBJECTS :=
        A -> F(A)

    MAP_MORPHISMS :=
        f:A->B
        ->
        F(f):F(A)->F(B)

    PRESERVE_IDENTITIES :=
        F(id_A)=id_F(A)

    PRESERVE_COMPOSITION :=
        F(g∘f)=F(g)∘F(f)
}

ENTITY NaturalTransformation {
    TYPE := MorphismBetweenFunctors
}

NATURAL_TRANSFORMATION {
    η:F=>G

    COMPONENTS :=
        η_A:F(A)->G(A)

    NATURALITY :=
        G(f)∘η_A
        =
        η_B∘F(f)
        FOR f:A->B
}

ENTITY Adjunction {
    TYPE := CategoricalRelationship
}

ADJUNCTION {
    F:C->D
    G:D->C

    F ⊣ G

    HOM_SET_ISOMORPHISM :=
        Hom_D(F(A),B)
        ≅
        Hom_C(A,G(B))

    REQUIRE :=
        natural_in_A_and_B
}

UNIT_COUNit {
    UNIT :=
        η:Id_C=>G∘F

    COUNIT :=
        ε:F∘G=>Id_D

    TRIANGLE_IDENTITIES :=
        required
}

UNIVERSAL_PROPERTY {
    TYPE := CharacterisationByUniqueFactorisation
}

INITIAL_OBJECT {
    OBJECT := 0

    FOR_ALL A:
        EXISTS unique morphism
            0->A
}

TERMINAL_OBJECT {
    OBJECT := 1

    FOR_ALL A:
        EXISTS unique morphism
            A->1
}

PRODUCT {
    OBJECT := A×B
    PROJECTIONS := {
        π1:A×B->A,
        π2:A×B->B
    }

    UNIVERSAL_PROPERTY :=
        FOR any X with f:X->A and g:X->B
        EXISTS unique <f,g>:X->A×B
}

COPRODUCT {
    OBJECT := A+B
    INJECTIONS := {
        i1:A->A+B,
        i2:B->A+B
    }

    UNIVERSAL_PROPERTY :=
        dual_of_product
}

EQUALIZER {
    GIVEN f,g:A->B

    e:E->A

    REQUIRE :=
        f∘e=g∘e

    UNIVERSAL :=
        every h:X->A with f∘h=g∘h
        factors uniquely through e
}

PULLBACK {
    GIVEN f:A->C
    GIVEN g:B->C

    OBJECT := A×_C B

    REQUIRE :=
        f∘p1 = g∘p2

    UNIVERSAL_PROPERTY :=
        unique mediating morphism
}

EXPONENTIAL_OBJECT {
    OBJECT := B^A

    EVALUATION :=
        ev:B^A×A->B

    UNIVERSAL_PROPERTY :=
        morphisms X×A->B
        correspond naturally to
        morphisms X->B^A
}

CARTESIAN_CLOSED_CATEGORY {
    REQUIRE := {
        FiniteProducts,
        ExponentialObjects
    }
}

LOGIC_RELATIONSHIP {
    Products --correspond_to--> ConjunctionLikeStructure
    TerminalObject --corresponds_to--> TruthLikeStructure
    Exponentials --correspond_to--> ImplicationLikeStructure
}

CURRYING {
    Hom(X×A,B)
        ≅
    Hom(X,B^A)

    RELATIONSHIP :=
        categorical_form_of_function_currying
}

PROPOSITION_AS_OBJECT {
    IN suitable categorical semantics:
        Proposition --may_correspond_to--> ObjectOrSubobject
}

PROOF_AS_MORPHISM {
    IN proof-relevant categorical interpretations:
        Proof --may_correspond_to--> Morphism
}

FORMULA_INTERPRETATION {
    Syntax
        -> InterpretationFunctor
        -> SemanticCategory
}

ENTITY SyntacticCategory {
    TYPE := CategoryGeneratedFromLogicalTheory
}

SYNTACTIC_CATEGORY {
    OBJECTS_MAY_REPRESENT := {
        Formulae,
        Contexts,
        Types
    }

    MORPHISMS_MAY_REPRESENT := {
        Proofs,
        Derivations,
        Substitutions,
        Terms
    }

    REQUIRE :=
        construction-specific definition
}

ENTITY SemanticCategory {
    TYPE := CategoryProvidingModelsOrInterpretations
}

FUNCTORIAL_SEMANTICS {
    TheoryOrSyntacticCategory
        -> StructurePreservingFunctor
        -> SemanticCategory
}

MODEL_AS_FUNCTOR {
    IN suitable algebraic/categorical semantics:
        Model
        := structure-preserving functor
        from syntactic/category-of-theory
        into semantic category
}

NATURAL_TRANSFORMATION_AS_MODEL_MAP {
    NaturalTransformation
        MAY represent
        coherent transformation between functorial models
}

ENTITY Subobject {
    TYPE := EquivalenceClassOfMonomorphisms
}

SUBOBJECT {
    m:A↪X
    MAY represent
        predicate_or_property_on_X
        in suitable categorical logic
}

SUBOBJECT_ORDER {
    Sub(X)
        forms ordered structure
        under factorisation
        in suitable categories
}

SUBOBJECT_CLASSIFIER {
    OBJECT := Ω
    TRUE_MORPHISM := true:1->Ω

    PROPERTY :=
        each subobject of X
        is classified by
        characteristic morphism χ:X->Ω
        in a topos
}

ENTITY Topos {
    TYPE := CategoryWithLogicalStructure
}

ELEMENTARY_TOPOS {
    COMMON_DEFINITION_REQUIRES := {
        FiniteLimits,
        Exponentials,
        SubobjectClassifier
    }
}

TOPOS_INTERNAL_LOGIC {
    Topos
        --supports-->
        InternalHigherOrderIntuitionisticLogic
        under standard elementary-topos semantics
}

BOOLEAN_TOPOS {
    PROPERTY :=
        internal_logic_is_boolean

    REQUIRE :=
        appropriate categorical Boolean condition
}

HEYTING_STRUCTURE {
    In suitable categories:
        SubobjectLattices
        MAY form HeytingAlgebras
}

RELATIONSHIP CategoricalLogic --connects_to--> IntuitionisticLogic

ENTITY Fibration {
    TYPE := FunctorWithStructuredLiftingProperties
}

FIBRATIONAL_LOGIC {
    PURPOSE :=
        organise predicates, substitution and quantification categorically
}

REINDEXING {
    GIVEN f:X->Y

    predicates_over_Y
        -> pullback/reindexing
        -> predicates_over_X
}

QUANTIFICATION_AS_ADJOINT {
    ExistentialQuantification
        MAY arise_as
        left_adjoint_to_reindexing

    UniversalQuantification
        MAY arise_as
        right_adjoint_to_reindexing

    REQUIRE :=
        categorical conditions supporting these adjoints
}

EXISTENTIAL_ADJUNCTION_PATTERN {
    ∃_f ⊣ f*
}

UNIVERSAL_ADJUNCTION_PATTERN {
    f* ⊣ ∀_f
}

BECK_CHEVALLEY_CONDITION {
    TYPE := CompatibilityCondition
    PURPOSE :=
        ensure quantification and substitution interact coherently
        in suitable categorical semantics
}

HYPERDOCTRINE {
    TYPE := CategoricalLogicalStructure
    PURPOSE :=
        model predicates and quantifiers indexed over contexts
}

ENTITY LawvereTheory {
    TYPE := CategoryEncodingAlgebraicTheory
}

LAWVERE_THEORY {
    OBJECTS :=
        finite_powers_of_generic_object
        in standard formulation

    MODELS :=
        finite-product-preserving functors
        into suitable semantic categories
}

ENTITY Monad {
    TYPE := EndofunctorWithUnitAndMultiplication
}

MONAD {
    T:C->C
    UNIT := η:Id=>T
    MULTIPLICATION := μ:T²=>T

    LAWS := {
        Associativity,
        LeftUnit,
        RightUnit
    }
}

RELATIONSHIP Adjunction --can_generate--> Monad

ENTITY Comonad {
    TYPE := DualOfMonad
}

CATEGORY_EQUIVALENCE {
    F:C->D
    G:D->C

    REQUIRE :=
        G∘F naturally_isomorphic_to Id_C
        AND
        F∘G naturally_isomorphic_to Id_D
}

ISOMORPHISM {
    f:A->B

    EXISTS g:B->A:
        g∘f=id_A
        AND f∘g=id_B
}

MONOMORPHISM {
    f:A->B

    IF f∘g1=f∘g2
    THEN g1=g2
}

EPIMORPHISM {
    f:A->B

    IF g1∘f=g2∘f
    THEN g1=g2
}

DUALITY {
    CATEGORY C
        -> opposite_category C^op

    REVERSE :=
        direction_of_all_morphisms
}

OPPOSITE_CATEGORY {
    Objects(C^op) := Objects(C)
    Hom_Cop(A,B) := Hom_C(B,A)
}

LIMIT {
    TYPE := UniversalCone
}

COLIMIT {
    TYPE := UniversalCocone
}

LIMIT_COLIMIT_DUALITY {
    Limit_in_C
        corresponds_to
    Colimit_in_C^op
}

LOGICAL_SYNTAX_SEMANTICS_BRIDGE {
    LogicalTheory
        -> SyntacticCategory
        -> FunctorialInterpretation
        -> SemanticCategory
        -> Model
}

TYPE_THEORY_RELATIONSHIP {
    CartesianClosedCategories
        --model-->
        SimplyTypedLambdaCalculus
        under standard CurryHowardLambek correspondence
}

CURRY_HOWARD_LAMBEK {
    Logic <-> TypeTheory <-> CategoryTheory

    PROPOSITIONS <-> Types <-> Objects
    PROOFS <-> Terms <-> Morphisms

    REQUIRE :=
        correspondence interpreted within appropriate formal systems
}

DEPENDENT_TYPE_RELATIONSHIP {
    CategoricalLogic
        --relates_to-->
        DependentTypeTheory

    STRUCTURES_MAY_INCLUDE := {
        Fibrations,
        ComprehensionCategories,
        LocallyCartesianClosedCategories
    }
}

LOCALLY_CARTESIAN_CLOSED_CATEGORY {
    REQUIRE :=
        each slice_category C/X is cartesian_closed
}

SLICE_CATEGORY {
    C/X

    OBJECTS :=
        morphisms A->X

    MORPHISMS :=
        commuting_triangles_over_X
}

INTERNAL_LANGUAGE {
    CategoricalStructure
        -> logical_or_type_theoretic_language
        describing internal reasoning
}

EXTERNAL_SEMANTICS {
    InternalLanguage
        -> interpretation in categorical structure
}

SOUNDNESS_PATTERN {
    IF syntactic_derivation exists
    THEN semantic interpretation preserves validity
    under sound categorical semantics
}

COMPLETENESS_PATTERN {
    semantic_validity
        -> syntactic_derivability

    REQUIRE :=
        specified logical system and categorical semantics
}

APPLICATION_RELATIONSHIPS {
    CategoryTheoryLogic --supports--> CategoricalLogic
    CategoryTheoryLogic --supports--> ToposTheory
    CategoryTheoryLogic --supports--> TypeTheory
    CategoryTheoryLogic --supports--> FunctionalProgrammingSemantics
    CategoryTheoryLogic --supports--> ProofTheory
    CategoryTheoryLogic --supports--> ModelTheory
    CategoryTheoryLogic --supports--> AlgebraicTheories
    CategoryTheoryLogic --supports--> FormalSemantics
    CategoryTheoryLogic --supports--> ComputerScience
    CategoryTheoryLogic --supports--> ArtificialIntelligence
}

ONTOLOGY_LINKS {
    PARENT := MathematicalLogicOntologies

    RELATED := {
        CategoryTheoryOntology,
        CategoricalLogicOntology,
        FunctorOntology,
        NaturalTransformationOntology,
        AdjunctionOntology,
        ToposOntology,
        TypeTheoryOntology,
        LambdaCalculusOntology,
        IntuitionisticLogicOntology,
        ModelTheoryOntology,
        ProofTheoryOntology,
        AlgebraOntology,
        SetTheoryOntology
    }
}

SEMANTIC_EQUIVALENCE {
    Arrow ≈ Morphism
        WHEN context=CategoryTheory

    Map ≈ Morphism
        WHEN categorical_context_is_explicit

    CategoricalModel ≈ StructurePreservingFunctor
        WHEN model_semantics_is_functorial

    PropositionAsTypeAsObject
        ≈ CurryHowardLambekCorrespondence
        WHEN appropriate_formal_setting
}

CONSTRAINTS {
    C1 :=
        every category MUST satisfy associative composition and identity laws

    C2 :=
        functors MUST preserve identities and composition

    C3 :=
        natural transformations MUST satisfy naturality

    C4 :=
        universal constructions MUST preserve their defining universal properties

    C5 :=
        logical/categorical correspondences MUST identify the categorical structure in which they hold

    C6 :=
        propositions MUST NOT universally be equated with objects outside an appropriate categorical semantics

    C7 :=
        proofs MUST NOT universally be equated with morphisms outside a proof-relevant categorical interpretation

    C8 :=
        quantifiers-as-adjoints MUST only be asserted when required adjoints exist

    C9 :=
        topos internal logic MUST be distinguished from arbitrary external classical logic

    C10 :=
        equivalence of categories MUST NOT be conflated with literal equality of categories

    C11 :=
        categorical duality MUST reverse morphisms and associated constructions consistently

    C12 :=
        all semantic transformations MUST preserve the declared categorical and logical structure
}

REASONING_CHAIN Query Q {
    IDENTIFY categorical_entities
    IDENTIFY logical_entities
    IDENTIFY categories C,D
    IDENTIFY objects_and_morphisms
    IDENTIFY functors
    IDENTIFY natural_transformations

    VALIDATE category_axioms

    IF functor_present
        THEN validate identity_and_composition_preservation

    IF natural_transformation_present
        THEN validate naturality

    IF universal_construction_requested
        THEN identify cone/cocone_and_unique_factorisation

    IF adjunction_requested
        THEN verify natural_hom_set_correspondence_or_unit_counit

    IF logical_interpretation_requested {
        IDENTIFY syntactic_category
        IDENTIFY semantic_category
        IDENTIFY interpretation_functor
    }

    IF predicate_reasoning_requested
        THEN identify subobjects_or_indexed_predicates

    IF quantifier_reasoning_requested
        THEN identify reindexing_and_required_adjoints

    IF topos_context
        THEN identify finite_limits_exponentials_subobject_classifier

    IF type_theory_context
        THEN identify CurryHowardLambek_structure

    IF duality_requested
        THEN construct opposite_category_and_dualise_relations

    VALIDATE constraints
    RETURN result WITH derivation
}

CONFIDENCE {
    CATEGORY_AXIOMS := 1.00
    FUNCTOR_AXIOMS := 1.00
    NATURALITY := 1.00
    ADJUNCTION_CORE := 1.00
    UNIVERSAL_PROPERTY_CORE := 1.00
    TOPOS_LOGIC_RELATIONSHIP := 0.99
    CURRY_HOWARD_LAMBEK := 0.99

    LOGICAL_INTERPRETATION :=
        IF categorical_structure_and_semantics_explicit THEN 1.00
        ELSE REQUIRE missing_context
}

UPDATE_MODEL {
    IMMUTABLE_CORE := {
        CategoryStructure,
        MorphismComposition,
        Functoriality,
        Naturality,
        UniversalPropertyConcept
    }

    EXTENSIBLE := {
        CategoricalLogicalStructures,
        Toposes,
        Fibrations,
        Hyperdoctrines,
        LawvereTheories,
        TypeTheoreticModels,
        InternalLanguages,
        ApplicationDomains,
        RelatedOntologies
    }

    ON_NEW_KNOWLEDGE {
        IDENTIFY categorical_context
        VALIDATE category_axioms
        VALIDATE functorial_structure
        VALIDATE universal_properties
        CHECK logical_correspondence_scope
        CHECK semantic_consistency
        ADD relationship IF non_contradictory
        UPDATE confidence
    }
}

AI_CONSUMPTION {
    MACHINE_READABLE := TRUE
    HUMAN_INTERPRETABLE := TRUE
    RECURSIVE_REASONING := TRUE
    ENTITY_RESOLUTION := TRUE
    MORPHISM_RESOLUTION := TRUE
    FUNCTOR_RESOLUTION := TRUE
    NATURALITY_REASONING := TRUE
    RELATIONSHIP_TRAVERSAL := TRUE
    UNIVERSAL_PROPERTY_REASONING := TRUE
    ADJUNCTION_REASONING := TRUE
    FUNCTORIAL_SEMANTICS := TRUE
    INTERNAL_LOGIC_REASONING := TRUE
    TYPE_THEORY_CORRESPONDENCE := TRUE
    DUALITY_REASONING := TRUE
    VALIDATION_REQUIRED := TRUE
}

FLOW_HASH {
    TARGET := TARGET_URL
    SUBJECT := "Category Theory Logic Ontology"
    VERSION := "1.0"
    CANONICAL_SIGNATURE :=
        HASH(TARGET + SUBJECT + DOMAIN + VERSION)
}

OUTPUT {
    PRIMARY_ENTITY := CategoryTheoryLogic

    INTERPRETATION :=
        "Category-theoretic logic studies logical syntax and semantics through objects, morphisms, functors, natural transformations and universal constructions. Categorical structures can encode propositions, proofs, predicates, quantification, models and type systems, providing a structural bridge between mathematical logic, category theory, topos theory and type theory."
}

END_FLOW