455 lines
13 KiB
Rust
455 lines
13 KiB
Rust
//! # Topos Theory
|
||
//!
|
||
//! A topos is a category with additional structure that makes it behave
|
||
//! like a generalized universe of sets. It provides an internal logic
|
||
//! for reasoning about mathematical structures.
|
||
//!
|
||
//! ## Key Features
|
||
//!
|
||
//! - **Subobject classifier**: An object Ω with a universal property
|
||
//! for classifying subobjects (generalizes {true, false} in Set)
|
||
//! - **Internal logic**: Intuitionistic logic derived from the topos structure
|
||
//! - **Exponentials**: All function spaces exist
|
||
//! - **Limits and colimits**: All finite limits and colimits exist
|
||
|
||
use crate::category::{
|
||
Category, CategoryWithMono, CategoryWithProducts, CartesianClosedCategory,
|
||
Object, ObjectData, Morphism, MorphismData,
|
||
};
|
||
use crate::{CategoryError, MorphismId, ObjectId, Result};
|
||
use dashmap::DashMap;
|
||
use serde::{Deserialize, Serialize};
|
||
use std::collections::HashMap;
|
||
use std::sync::Arc;
|
||
|
||
/// The subobject classifier Ω
|
||
///
|
||
/// In a topos, the subobject classifier has a characteristic morphism
|
||
/// true: 1 -> Ω such that for any monomorphism m: A >-> B, there exists
|
||
/// a unique χ_m: B -> Ω making the pullback square commute.
|
||
#[derive(Debug, Clone, Serialize, Deserialize)]
|
||
pub struct SubobjectClassifier<T: Clone + std::fmt::Debug> {
|
||
/// The classifier object Ω
|
||
pub omega: Object<T>,
|
||
/// The terminal object 1
|
||
pub terminal: Object<T>,
|
||
/// The truth morphism: true: 1 -> Ω
|
||
pub truth: MorphismId,
|
||
/// Cached characteristic morphisms
|
||
pub characteristics: HashMap<MorphismId, MorphismId>,
|
||
}
|
||
|
||
impl<T: Clone + std::fmt::Debug + PartialEq> SubobjectClassifier<T> {
|
||
/// Creates a new subobject classifier
|
||
pub fn new(omega: Object<T>, terminal: Object<T>, truth: MorphismId) -> Self {
|
||
Self {
|
||
omega,
|
||
terminal,
|
||
truth,
|
||
characteristics: HashMap::new(),
|
||
}
|
||
}
|
||
|
||
/// Registers a characteristic morphism for a monomorphism
|
||
pub fn register_characteristic(&mut self, mono: MorphismId, chi: MorphismId) {
|
||
self.characteristics.insert(mono, chi);
|
||
}
|
||
|
||
/// Gets the characteristic morphism for a monomorphism
|
||
pub fn characteristic_of(&self, mono: &MorphismId) -> Option<MorphismId> {
|
||
self.characteristics.get(mono).copied()
|
||
}
|
||
}
|
||
|
||
/// A topos is a category with special structure
|
||
///
|
||
/// Key properties:
|
||
/// 1. Has all finite limits
|
||
/// 2. Has all finite colimits
|
||
/// 3. Is cartesian closed (has exponentials)
|
||
/// 4. Has a subobject classifier
|
||
#[derive(Debug)]
|
||
pub struct Topos<C: Category> {
|
||
/// The underlying category
|
||
pub category: C,
|
||
/// The subobject classifier
|
||
subobject_classifier: Option<SubobjectClassifier<ObjectData>>,
|
||
/// Truth values in the internal logic
|
||
truth_values: Vec<MorphismId>,
|
||
/// Cached exponential objects
|
||
exponentials: Arc<DashMap<(ObjectId, ObjectId), ObjectId>>,
|
||
/// Cached pullbacks
|
||
pullbacks: Arc<DashMap<(MorphismId, MorphismId), PullbackData>>,
|
||
}
|
||
|
||
/// Data for a pullback square
|
||
#[derive(Debug, Clone, Serialize, Deserialize)]
|
||
pub struct PullbackData {
|
||
/// The pullback object P
|
||
pub pullback: ObjectId,
|
||
/// First projection P -> A
|
||
pub proj1: MorphismId,
|
||
/// Second projection P -> B
|
||
pub proj2: MorphismId,
|
||
}
|
||
|
||
impl<C: Category> Topos<C> {
|
||
/// Creates a new topos from a category
|
||
///
|
||
/// Note: This does not verify that the category actually forms a topos.
|
||
/// Use `verify_topos_axioms` to check.
|
||
pub fn new(category: C) -> Self {
|
||
Self {
|
||
category,
|
||
subobject_classifier: None,
|
||
truth_values: Vec::new(),
|
||
exponentials: Arc::new(DashMap::new()),
|
||
pullbacks: Arc::new(DashMap::new()),
|
||
}
|
||
}
|
||
|
||
/// Sets the subobject classifier
|
||
pub fn with_subobject_classifier(
|
||
mut self,
|
||
classifier: SubobjectClassifier<ObjectData>,
|
||
) -> Self {
|
||
self.subobject_classifier = Some(classifier);
|
||
self
|
||
}
|
||
|
||
/// Gets the subobject classifier if it exists
|
||
pub fn subobject_classifier(&self) -> Option<&SubobjectClassifier<ObjectData>> {
|
||
self.subobject_classifier.as_ref()
|
||
}
|
||
|
||
/// Adds a truth value
|
||
pub fn add_truth_value(&mut self, morphism: MorphismId) {
|
||
self.truth_values.push(morphism);
|
||
}
|
||
|
||
/// Gets all truth values
|
||
pub fn truth_values(&self) -> &[MorphismId] {
|
||
&self.truth_values
|
||
}
|
||
|
||
/// Gets the underlying category
|
||
pub fn category(&self) -> &C {
|
||
&self.category
|
||
}
|
||
}
|
||
|
||
impl<C: Category + CategoryWithProducts> Topos<C> {
|
||
/// Computes a pullback of f: A -> C and g: B -> C
|
||
///
|
||
/// Returns the pullback object P with projections
|
||
/// such that the square commutes.
|
||
pub fn pullback(
|
||
&self,
|
||
f: &C::Morphism,
|
||
g: &C::Morphism,
|
||
) -> Option<(C::Object, C::Morphism, C::Morphism)> {
|
||
// Check that f and g have the same codomain
|
||
if self.category.codomain(f) != self.category.codomain(g) {
|
||
return None;
|
||
}
|
||
|
||
// For a concrete implementation, we would compute the actual pullback
|
||
// This is a simplified version using products as an approximation
|
||
let a = self.category.domain(f);
|
||
let b = self.category.domain(g);
|
||
|
||
// P is a subobject of A x B
|
||
let product = self.category.product(&a, &b)?;
|
||
let p1 = self.category.proj1(&product)?;
|
||
let p2 = self.category.proj2(&product)?;
|
||
|
||
Some((product, p1, p2))
|
||
}
|
||
|
||
/// Computes the equalizer of f, g: A -> B
|
||
///
|
||
/// The equalizer E is the largest subobject of A where f = g
|
||
pub fn equalizer(
|
||
&self,
|
||
f: &C::Morphism,
|
||
g: &C::Morphism,
|
||
) -> Option<(C::Object, C::Morphism)> {
|
||
// f and g must have the same domain and codomain
|
||
if self.category.domain(f) != self.category.domain(g) {
|
||
return None;
|
||
}
|
||
if self.category.codomain(f) != self.category.codomain(g) {
|
||
return None;
|
||
}
|
||
|
||
// Simplified: return domain with identity if f = g
|
||
// A real implementation would compute the actual equalizer
|
||
let a = self.category.domain(f);
|
||
let id = self.category.identity(&a)?;
|
||
|
||
Some((a, id))
|
||
}
|
||
}
|
||
|
||
impl<C: Category + CategoryWithProducts + CategoryWithMono> Topos<C> {
|
||
/// Verifies that this is a valid topos
|
||
///
|
||
/// Checks:
|
||
/// 1. Finite limits exist (simplified: products and equalizers)
|
||
/// 2. Has subobject classifier
|
||
/// 3. Is cartesian closed (simplified check)
|
||
pub fn verify_topos_axioms(&self) -> ToposVerification {
|
||
let mut verification = ToposVerification::new();
|
||
|
||
// Check subobject classifier
|
||
if self.subobject_classifier.is_some() {
|
||
verification.has_subobject_classifier = true;
|
||
}
|
||
|
||
// Check products (simplified)
|
||
let objects = self.category.objects();
|
||
if objects.len() >= 2 {
|
||
let a = &objects[0];
|
||
let b = &objects[1];
|
||
verification.has_finite_products = self.category.product(a, b).is_some();
|
||
}
|
||
|
||
// Check for terminal object (simplified)
|
||
verification.has_terminal = !objects.is_empty();
|
||
|
||
verification
|
||
}
|
||
}
|
||
|
||
/// Result of topos axiom verification
|
||
#[derive(Debug, Clone, Serialize, Deserialize)]
|
||
pub struct ToposVerification {
|
||
pub has_subobject_classifier: bool,
|
||
pub has_finite_products: bool,
|
||
pub has_finite_coproducts: bool,
|
||
pub has_equalizers: bool,
|
||
pub has_coequalizers: bool,
|
||
pub has_terminal: bool,
|
||
pub has_initial: bool,
|
||
pub is_cartesian_closed: bool,
|
||
}
|
||
|
||
impl ToposVerification {
|
||
pub fn new() -> Self {
|
||
Self {
|
||
has_subobject_classifier: false,
|
||
has_finite_products: false,
|
||
has_finite_coproducts: false,
|
||
has_equalizers: false,
|
||
has_coequalizers: false,
|
||
has_terminal: false,
|
||
has_initial: false,
|
||
is_cartesian_closed: false,
|
||
}
|
||
}
|
||
|
||
pub fn is_topos(&self) -> bool {
|
||
self.has_subobject_classifier
|
||
&& self.has_finite_products
|
||
&& self.has_terminal
|
||
&& self.is_cartesian_closed
|
||
}
|
||
}
|
||
|
||
impl Default for ToposVerification {
|
||
fn default() -> Self {
|
||
Self::new()
|
||
}
|
||
}
|
||
|
||
/// Internal logic operations in a topos
|
||
///
|
||
/// The subobject classifier Ω supports logical operations
|
||
/// that form an internal Heyting algebra.
|
||
#[derive(Debug)]
|
||
pub struct InternalLogic {
|
||
/// Conjunction: ∧: Ω x Ω -> Ω
|
||
pub conjunction: Option<MorphismId>,
|
||
/// Disjunction: ∨: Ω x Ω -> Ω
|
||
pub disjunction: Option<MorphismId>,
|
||
/// Implication: →: Ω x Ω -> Ω
|
||
pub implication: Option<MorphismId>,
|
||
/// Negation: ¬: Ω -> Ω
|
||
pub negation: Option<MorphismId>,
|
||
/// Universal quantifier for each object
|
||
pub universal: HashMap<ObjectId, MorphismId>,
|
||
/// Existential quantifier for each object
|
||
pub existential: HashMap<ObjectId, MorphismId>,
|
||
}
|
||
|
||
impl InternalLogic {
|
||
pub fn new() -> Self {
|
||
Self {
|
||
conjunction: None,
|
||
disjunction: None,
|
||
implication: None,
|
||
negation: None,
|
||
universal: HashMap::new(),
|
||
existential: HashMap::new(),
|
||
}
|
||
}
|
||
|
||
/// Checks if the logic is complete (all operations defined)
|
||
pub fn is_complete(&self) -> bool {
|
||
self.conjunction.is_some()
|
||
&& self.disjunction.is_some()
|
||
&& self.implication.is_some()
|
||
&& self.negation.is_some()
|
||
}
|
||
|
||
/// Checks if the logic is classical (excluded middle holds)
|
||
/// In general, topos logic is intuitionistic
|
||
pub fn is_classical(&self) -> bool {
|
||
// Would need to verify ¬¬p = p for all p
|
||
// By default, topos logic is intuitionistic
|
||
false
|
||
}
|
||
}
|
||
|
||
impl Default for InternalLogic {
|
||
fn default() -> Self {
|
||
Self::new()
|
||
}
|
||
}
|
||
|
||
/// A subobject in a topos
|
||
///
|
||
/// Subobjects are equivalence classes of monomorphisms into an object
|
||
#[derive(Debug, Clone, Serialize, Deserialize)]
|
||
pub struct Subobject {
|
||
/// The source object of the monomorphism
|
||
pub source: ObjectId,
|
||
/// The target object (what we're a subobject of)
|
||
pub target: ObjectId,
|
||
/// The monomorphism
|
||
pub mono: MorphismId,
|
||
/// The characteristic morphism χ: target -> Ω
|
||
pub characteristic: Option<MorphismId>,
|
||
}
|
||
|
||
impl Subobject {
|
||
pub fn new(source: ObjectId, target: ObjectId, mono: MorphismId) -> Self {
|
||
Self {
|
||
source,
|
||
target,
|
||
mono,
|
||
characteristic: None,
|
||
}
|
||
}
|
||
|
||
pub fn with_characteristic(mut self, chi: MorphismId) -> Self {
|
||
self.characteristic = Some(chi);
|
||
self
|
||
}
|
||
}
|
||
|
||
/// Lattice of subobjects for an object in a topos
|
||
///
|
||
/// In a topos, the subobjects of any object form a Heyting algebra
|
||
#[derive(Debug)]
|
||
pub struct SubobjectLattice {
|
||
/// The object whose subobjects we're tracking
|
||
pub object: ObjectId,
|
||
/// All subobjects (ordered by inclusion)
|
||
pub subobjects: Vec<Subobject>,
|
||
/// Meet (intersection) results
|
||
meets: HashMap<(usize, usize), usize>,
|
||
/// Join (union) results
|
||
joins: HashMap<(usize, usize), usize>,
|
||
}
|
||
|
||
impl SubobjectLattice {
|
||
pub fn new(object: ObjectId) -> Self {
|
||
Self {
|
||
object,
|
||
subobjects: Vec::new(),
|
||
meets: HashMap::new(),
|
||
joins: HashMap::new(),
|
||
}
|
||
}
|
||
|
||
/// Adds a subobject to the lattice
|
||
pub fn add(&mut self, subobject: Subobject) -> usize {
|
||
let index = self.subobjects.len();
|
||
self.subobjects.push(subobject);
|
||
index
|
||
}
|
||
|
||
/// Computes the meet (intersection) of two subobjects
|
||
pub fn meet(&self, a: usize, b: usize) -> Option<usize> {
|
||
self.meets.get(&(a.min(b), a.max(b))).copied()
|
||
}
|
||
|
||
/// Computes the join (union) of two subobjects
|
||
pub fn join(&self, a: usize, b: usize) -> Option<usize> {
|
||
self.joins.get(&(a.min(b), a.max(b))).copied()
|
||
}
|
||
|
||
/// Records a meet computation
|
||
pub fn record_meet(&mut self, a: usize, b: usize, result: usize) {
|
||
self.meets.insert((a.min(b), a.max(b)), result);
|
||
}
|
||
|
||
/// Records a join computation
|
||
pub fn record_join(&mut self, a: usize, b: usize, result: usize) {
|
||
self.joins.insert((a.min(b), a.max(b)), result);
|
||
}
|
||
}
|
||
|
||
#[cfg(test)]
|
||
mod tests {
|
||
use super::*;
|
||
use crate::category::SetCategory;
|
||
|
||
#[test]
|
||
fn test_topos_creation() {
|
||
let cat = SetCategory::new();
|
||
let topos = Topos::new(cat);
|
||
|
||
assert!(topos.subobject_classifier().is_none());
|
||
}
|
||
|
||
#[test]
|
||
fn test_subobject_classifier() {
|
||
let omega = Object::new(ObjectData::FiniteSet(2)); // {false, true}
|
||
let terminal = Object::new(ObjectData::Terminal);
|
||
let truth = MorphismId::new();
|
||
|
||
let classifier = SubobjectClassifier::new(omega, terminal, truth);
|
||
|
||
assert_eq!(classifier.omega.data, ObjectData::FiniteSet(2));
|
||
}
|
||
|
||
#[test]
|
||
fn test_internal_logic() {
|
||
let logic = InternalLogic::new();
|
||
|
||
assert!(!logic.is_complete());
|
||
assert!(!logic.is_classical());
|
||
}
|
||
|
||
#[test]
|
||
fn test_subobject() {
|
||
let source = ObjectId::new();
|
||
let target = ObjectId::new();
|
||
let mono = MorphismId::new();
|
||
|
||
let sub = Subobject::new(source, target, mono);
|
||
|
||
assert_eq!(sub.source, source);
|
||
assert!(sub.characteristic.is_none());
|
||
}
|
||
|
||
#[test]
|
||
fn test_topos_verification() {
|
||
let verification = ToposVerification::new();
|
||
|
||
assert!(!verification.is_topos());
|
||
}
|
||
}
|