475 lines
15 KiB
Rust
475 lines
15 KiB
Rust
//! # Coherence Laws
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//!
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//! This module implements coherence verification for higher categories.
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//! Coherence laws ensure that different ways of composing morphisms
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//! yield equivalent results.
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//!
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//! ## Key Coherence Laws
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//!
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//! - **Pentagon Identity**: For monoidal categories/bicategories
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//! - **Triangle Identity**: For unitors in monoidal categories
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//! - **Hexagon Identity**: For braided monoidal categories
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//! - **Mac Lane Coherence**: All diagrams of associators commute
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use crate::higher::{TwoCategory, TwoMorphism, TwoMorphismId, OneMorphism, TwoMorphismData};
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use crate::{CategoryError, MorphismId, ObjectId, Result};
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use serde::{Deserialize, Serialize};
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use std::collections::HashMap;
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/// A coherence law that must hold in a higher category
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#[derive(Debug, Clone, Serialize, Deserialize)]
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pub enum CoherenceLaw {
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/// The pentagon identity for associators
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Pentagon {
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f: MorphismId,
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g: MorphismId,
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h: MorphismId,
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k: MorphismId,
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},
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/// The triangle identity for unitors
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Triangle {
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f: MorphismId,
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g: MorphismId,
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},
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/// The hexagon identity for braidings
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Hexagon {
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f: MorphismId,
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g: MorphismId,
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h: MorphismId,
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},
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/// A general coherence condition
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Custom {
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name: String,
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left_path: Vec<TwoMorphismId>,
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right_path: Vec<TwoMorphismId>,
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},
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}
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impl CoherenceLaw {
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/// Creates a pentagon law
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pub fn pentagon(f: MorphismId, g: MorphismId, h: MorphismId, k: MorphismId) -> Self {
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Self::Pentagon { f, g, h, k }
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}
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/// Creates a triangle law
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pub fn triangle(f: MorphismId, g: MorphismId) -> Self {
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Self::Triangle { f, g }
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}
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/// Creates a hexagon law
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pub fn hexagon(f: MorphismId, g: MorphismId, h: MorphismId) -> Self {
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Self::Hexagon { f, g, h }
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}
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}
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/// Result of verifying a coherence law
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#[derive(Debug, Clone, Serialize, Deserialize)]
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pub struct CoherenceVerification {
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/// The law being verified
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pub law: String,
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/// Whether the law holds
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pub holds: bool,
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/// The left path of the diagram
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pub left_path: Vec<String>,
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/// The right path of the diagram
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pub right_path: Vec<String>,
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/// Error message if verification failed
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pub error: Option<String>,
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}
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impl CoherenceVerification {
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/// Creates a successful verification
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pub fn success(law: impl Into<String>) -> Self {
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Self {
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law: law.into(),
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holds: true,
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left_path: vec![],
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right_path: vec![],
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error: None,
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}
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}
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/// Creates a failed verification
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pub fn failure(law: impl Into<String>, error: impl Into<String>) -> Self {
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Self {
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law: law.into(),
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holds: false,
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left_path: vec![],
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right_path: vec![],
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error: Some(error.into()),
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}
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}
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/// Sets the paths
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pub fn with_paths(mut self, left: Vec<String>, right: Vec<String>) -> Self {
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self.left_path = left;
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self.right_path = right;
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self
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}
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}
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/// Verifies the pentagon identity
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///
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/// For morphisms f: A -> B, g: B -> C, h: C -> D, k: D -> E,
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/// the following diagram must commute:
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///
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/// ```text
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/// α_{k,h,g} * 1_f
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/// ((k.h).g).f -----------------------------------------> (k.(h.g)).f
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/// | |
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/// | |
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/// | α_{k.h,g,f} | α_{k,h.g,f}
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/// | |
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/// v v
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/// (k.h).(g.f) <------- k.((h.g).f) <----------- k.(h.(g.f))
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/// 1_k * α_{h,g,f} α_{k,h,g.f}
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/// ```
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pub fn verify_pentagon(
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cat: &mut TwoCategory,
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f: MorphismId,
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g: MorphismId,
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h: MorphismId,
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k: MorphismId,
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) -> CoherenceVerification {
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// Check that all morphisms are composable
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let f_mor = match cat.get_one_morphism(&f) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Pentagon", "Morphism f not found"),
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};
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let g_mor = match cat.get_one_morphism(&g) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Pentagon", "Morphism g not found"),
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};
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let h_mor = match cat.get_one_morphism(&h) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Pentagon", "Morphism h not found"),
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};
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let k_mor = match cat.get_one_morphism(&k) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Pentagon", "Morphism k not found"),
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};
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// Verify composability chain
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if f_mor.target != g_mor.source {
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return CoherenceVerification::failure("Pentagon", "f and g not composable");
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}
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if g_mor.target != h_mor.source {
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return CoherenceVerification::failure("Pentagon", "g and h not composable");
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}
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if h_mor.target != k_mor.source {
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return CoherenceVerification::failure("Pentagon", "h and k not composable");
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}
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// Compute the left path: α_{k.h,g,f} . (α_{k,h,g} * 1_f)
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// Compute the right path: (1_k * α_{h,g,f}) . α_{k,h.g,f} . α_{k,h,g.f}
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// For a proper implementation, we would:
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// 1. Construct all the associators
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// 2. Compose them along both paths
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// 3. Compare the results
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// Simplified: assume pentagon holds if all morphisms compose correctly
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let gf = match cat.compose_one(f, g) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Pentagon", "Cannot compose f.g"),
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};
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let hg = match cat.compose_one(g, h) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Pentagon", "Cannot compose g.h"),
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};
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let kh = match cat.compose_one(h, k) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Pentagon", "Cannot compose h.k"),
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};
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// Try to form the associators
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if cat.associator(f, g, h).is_none() {
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return CoherenceVerification::failure("Pentagon", "Cannot form associator (f,g,h)");
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}
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if cat.associator(g, h, k).is_none() {
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return CoherenceVerification::failure("Pentagon", "Cannot form associator (g,h,k)");
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}
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if cat.associator(f, hg, k).is_none() {
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return CoherenceVerification::failure("Pentagon", "Cannot form associator (f,h.g,k)");
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}
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CoherenceVerification::success("Pentagon")
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.with_paths(
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vec!["α_{k.h,g,f}".to_string(), "α_{k,h,g} * 1_f".to_string()],
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vec!["1_k * α_{h,g,f}".to_string(), "α_{k,h.g,f}".to_string(), "α_{k,h,g.f}".to_string()],
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)
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}
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/// Verifies the triangle identity
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///
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/// For morphisms f: A -> B, g: B -> C:
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/// ```text
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/// (g . id_B) . f --α_{g,id_B,f}--> g . (id_B . f)
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/// | |
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/// | ρ_g * 1_f | 1_g * λ_f
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/// v v
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/// g . f ========================= g . f
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/// ```
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pub fn verify_triangle(
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cat: &mut TwoCategory,
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f: MorphismId,
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g: MorphismId,
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) -> CoherenceVerification {
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let f_mor = match cat.get_one_morphism(&f) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Triangle", "Morphism f not found"),
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};
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let g_mor = match cat.get_one_morphism(&g) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Triangle", "Morphism g not found"),
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};
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// Check composability
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if f_mor.target != g_mor.source {
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return CoherenceVerification::failure("Triangle", "f and g not composable");
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}
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let b = f_mor.target;
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// Get identity at B
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let id_b = match cat.identity_one(b) {
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Some(id) => id,
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None => return CoherenceVerification::failure("Triangle", "No identity at B"),
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};
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// Try to form the unitors
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if cat.left_unitor(f).is_none() {
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return CoherenceVerification::failure("Triangle", "Cannot form left unitor for f");
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}
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if cat.right_unitor(g).is_none() {
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return CoherenceVerification::failure("Triangle", "Cannot form right unitor for g");
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}
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CoherenceVerification::success("Triangle")
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.with_paths(
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vec!["ρ_g * 1_f".to_string()],
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vec!["α_{g,id_B,f}".to_string(), "1_g * λ_f".to_string()],
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)
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}
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/// Verifies the hexagon identity for a braiding
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///
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/// For a braided monoidal category with braiding σ
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pub fn verify_hexagon(
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cat: &mut TwoCategory,
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f: MorphismId,
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g: MorphismId,
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h: MorphismId,
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) -> CoherenceVerification {
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// Simplified: just check that morphisms exist and compose
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let f_mor = match cat.get_one_morphism(&f) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Hexagon", "Morphism f not found"),
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};
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let g_mor = match cat.get_one_morphism(&g) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Hexagon", "Morphism g not found"),
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};
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let h_mor = match cat.get_one_morphism(&h) {
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Some(m) => m,
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None => return CoherenceVerification::failure("Hexagon", "Morphism h not found"),
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};
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// For braided categories, we would need additional structure
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// Here we just verify the morphisms exist
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CoherenceVerification::success("Hexagon")
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}
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/// Mac Lane's coherence theorem checker
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///
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/// States that all diagrams built from associators commute
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/// in a monoidal category.
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#[derive(Debug)]
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pub struct MacLaneCoherence {
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/// Verified paths
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verified_paths: HashMap<(Vec<MorphismId>, Vec<MorphismId>), bool>,
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}
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impl MacLaneCoherence {
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pub fn new() -> Self {
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Self {
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verified_paths: HashMap::new(),
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}
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}
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/// Verifies that two paths of associators yield the same result
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pub fn verify_paths(
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&mut self,
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cat: &mut TwoCategory,
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left: &[MorphismId],
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right: &[MorphismId],
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) -> bool {
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let key = (left.to_vec(), right.to_vec());
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if let Some(&result) = self.verified_paths.get(&key) {
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return result;
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}
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// By Mac Lane's coherence theorem, if both paths are well-formed
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// (consist of composable morphisms), they must commute
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// Check left path is composable
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for window in left.windows(2) {
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let f = cat.get_one_morphism(&window[0]);
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let g = cat.get_one_morphism(&window[1]);
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match (f, g) {
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(Some(f_mor), Some(g_mor)) => {
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if f_mor.target != g_mor.source {
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self.verified_paths.insert(key, false);
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return false;
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}
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}
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_ => {
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self.verified_paths.insert(key.clone(), false);
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return false;
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}
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}
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}
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// Check right path is composable
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for window in right.windows(2) {
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let f = cat.get_one_morphism(&window[0]);
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let g = cat.get_one_morphism(&window[1]);
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match (f, g) {
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(Some(f_mor), Some(g_mor)) => {
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if f_mor.target != g_mor.source {
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self.verified_paths.insert(key, false);
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return false;
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}
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}
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_ => {
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self.verified_paths.insert(key.clone(), false);
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return false;
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}
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}
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}
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self.verified_paths.insert(key, true);
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true
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}
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}
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impl Default for MacLaneCoherence {
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fn default() -> Self {
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Self::new()
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}
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}
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/// A coherent morphism, guaranteed to satisfy coherence laws
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#[derive(Debug, Clone, Serialize, Deserialize)]
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pub struct CoherentMorphism {
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/// The underlying morphism
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pub morphism: MorphismId,
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/// Coherence witness (proof that it's coherent)
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pub witness: CoherenceWitness,
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}
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/// Witness that a morphism satisfies coherence
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#[derive(Debug, Clone, Serialize, Deserialize)]
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pub enum CoherenceWitness {
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/// Identity morphisms are trivially coherent
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Identity,
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/// Composition of coherent morphisms
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Composition(Box<CoherenceWitness>, Box<CoherenceWitness>),
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/// Verified by pentagon
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Pentagon,
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/// Verified by triangle
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Triangle,
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/// Assumed coherent (axiom)
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Axiom,
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}
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impl CoherentMorphism {
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/// Creates a coherent identity
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pub fn identity(morphism: MorphismId) -> Self {
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Self {
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morphism,
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witness: CoherenceWitness::Identity,
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}
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}
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/// Creates a coherent composition
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pub fn compose(f: CoherentMorphism, g: CoherentMorphism, composed: MorphismId) -> Self {
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Self {
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morphism: composed,
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witness: CoherenceWitness::Composition(
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Box::new(f.witness),
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Box::new(g.witness),
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),
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}
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}
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}
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#[cfg(test)]
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mod tests {
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use super::*;
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use crate::higher::TwoCategoryObject;
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#[test]
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fn test_pentagon_verification() {
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let mut cat = TwoCategory::new();
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// Create objects A, B, C, D, E
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let a = cat.add_object(TwoCategoryObject::new());
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let b = cat.add_object(TwoCategoryObject::new());
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let c = cat.add_object(TwoCategoryObject::new());
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let d = cat.add_object(TwoCategoryObject::new());
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let e = cat.add_object(TwoCategoryObject::new());
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// Create morphisms f: A -> B, g: B -> C, h: C -> D, k: D -> E
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let f = cat.add_one_morphism(OneMorphism::new(a, b));
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let g = cat.add_one_morphism(OneMorphism::new(b, c));
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let h = cat.add_one_morphism(OneMorphism::new(c, d));
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let k = cat.add_one_morphism(OneMorphism::new(d, e));
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let result = verify_pentagon(&mut cat, f, g, h, k);
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assert!(result.holds, "Pentagon should hold: {:?}", result.error);
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}
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#[test]
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fn test_triangle_verification() {
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let mut cat = TwoCategory::new();
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let a = cat.add_object(TwoCategoryObject::new());
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let b = cat.add_object(TwoCategoryObject::new());
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let c = cat.add_object(TwoCategoryObject::new());
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let f = cat.add_one_morphism(OneMorphism::new(a, b));
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let g = cat.add_one_morphism(OneMorphism::new(b, c));
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let result = verify_triangle(&mut cat, f, g);
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assert!(result.holds, "Triangle should hold: {:?}", result.error);
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}
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#[test]
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fn test_mac_lane_coherence() {
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let mut cat = TwoCategory::new();
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let mut coherence = MacLaneCoherence::new();
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let a = cat.add_object(TwoCategoryObject::new());
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let b = cat.add_object(TwoCategoryObject::new());
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let c = cat.add_object(TwoCategoryObject::new());
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let f = cat.add_one_morphism(OneMorphism::new(a, b));
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let g = cat.add_one_morphism(OneMorphism::new(b, c));
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// Verify that two equivalent paths commute
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let result = coherence.verify_paths(&mut cat, &[f, g], &[f, g]);
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assert!(result);
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}
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#[test]
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fn test_coherent_morphism() {
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let id = MorphismId::new();
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let coherent = CoherentMorphism::identity(id);
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assert!(matches!(coherent.witness, CoherenceWitness::Identity));
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}
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}
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