C10-2 范畴论形式化规范
模块依赖
from typing import Set, Dict, Tuple, List, Optional, Callable, Generic, TypeVar
from dataclasses import dataclass, field
from abc import ABC, abstractmethod
from enum import Enum
import hashlib
from no11_number_system import No11Number
from test_C10_1 import (
FormalSystem, Formula, Proof, Symbol, SymbolType,
Model, Interpretation, GödelEncoder
)
核心数据结构
基本范畴元素
@dataclass(frozen=True)
class CategoryObject:
"""范畴中的对象"""
name: str
data: any # 可以是形式系统、集合、空间等
def encode(self) -> No11Number:
"""对象的No-11编码"""
hash_val = hash((self.name, type(self.data).__name__))
return No11Number(abs(hash_val) % 10000)
def __hash__(self) -> int:
return hash((self.name, id(self.data)))
@dataclass(frozen=True)
class Morphism:
"""范畴中的态射"""
source: CategoryObject
target: CategoryObject
name: str
mapping: Optional[Callable] = None
def __post_init__(self):
if self.mapping is None:
# 默认为恒等映射的标记
object.__setattr__(self, 'mapping', lambda x: x)
def apply(self, element: any) -> any:
"""应用态射"""
if self.mapping:
return self.mapping(element)
return element
def encode(self) -> No11Number:
"""态射的No-11编码"""
src_code = self.source.encode().value
tgt_code = self.target.encode().value
name_hash = sum(ord(c) for c in self.name) % 100
return No11Number((src_code * 100 + tgt_code) * 100 + name_hash)
def __hash__(self) -> int:
return hash((self.source, self.target, self.name))
@dataclass(frozen=True)
class IdentityMorphism(Morphism):
"""恒等态射"""
def __init__(self, obj: CategoryObject):
super().__init__(
source=obj,
target=obj,
name=f"id_{obj.name}",
mapping=lambda x: x
)
@dataclass(frozen=True)
class ComposedMorphism(Morphism):
"""复合态射"""
first: Morphism # f: A → B
second: Morphism # g: B → C
def __post_init__(self):
# 验证可复合性
if self.first.target != self.second.source:
raise ValueError(f"Morphisms {self.first.name} and {self.second.name} are not composable")
# 设置复合态射的源和目标
object.__setattr__(self, 'source', self.first.source)
object.__setattr__(self, 'target', self.second.target)
object.__setattr__(self, 'name', f"{self.second.name}∘{self.first.name}")
# 复合映射
def composed_mapping(x):
return self.second.apply(self.first.apply(x))
object.__setattr__(self, 'mapping', composed_mapping)
范畴定义
class CategoryError(Exception):
"""范畴论错误基类"""
pass
class Category:
"""范畴的完整实现"""
def __init__(self, name: str):
self.name = name
self.objects: Set[CategoryObject] = set()
self.morphisms: Dict[Tuple[CategoryObject, CategoryObject], Set[Morphism]] = {}
self._identity_morphisms: Dict[CategoryObject, IdentityMorphism] = {}
def add_object(self, obj: CategoryObject):
"""添加对象到范畴"""
if obj in self.objects:
return
self.objects.add(obj)
# 自动创建恒等态射
id_mor = IdentityMorphism(obj)
self._identity_morphisms[obj] = id_mor
self.add_morphism(id_mor)
def add_morphism(self, morphism: Morphism):
"""添加态射到范畴"""
# 确保源和目标对象都在范畴中
if morphism.source not in self.objects:
self.add_object(morphism.source)
if morphism.target not in self.objects:
self.add_object(morphism.target)
key = (morphism.source, morphism.target)
if key not in self.morphisms:
self.morphisms[key] = set()
self.morphisms[key].add(morphism)
def compose(self, g: Morphism, f: Morphism) -> Morphism:
"""复合两个态射: g ∘ f"""
if f.target != g.source:
raise CategoryError(f"Cannot compose {f.name}: {f.source.name}→{f.target.name} "
f"with {g.name}: {g.source.name}→{g.target.name}")
# 检查特殊情况
if isinstance(f, IdentityMorphism):
return g
if isinstance(g, IdentityMorphism):
return f
return ComposedMorphism(f, g)
def identity(self, obj: CategoryObject) -> IdentityMorphism:
"""获取对象的恒等态射"""
if obj not in self.objects:
raise CategoryError(f"Object {obj.name} not in category")
return self._identity_morphisms[obj]
def hom(self, source: CategoryObject, target: CategoryObject) -> Set[Morphism]:
"""获取hom-集 Hom(source, target)"""
return self.morphisms.get((source, target), set())
def is_isomorphism(self, f: Morphism) -> bool:
"""检查态射是否是同构"""
# 寻找逆态射
candidates = self.hom(f.target, f.source)
for g in candidates:
if (self.compose(g, f) == self.identity(f.source) and
self.compose(f, g) == self.identity(f.target)):
return True
return False
def verify_axioms(self) -> bool:
"""验证范畴公理"""
# 1. 恒等态射存在性
for obj in self.objects:
if obj not in self._identity_morphisms:
return False
# 2. 复合的结合律
# 选择三个可复合的态射进行测试
for morphisms_list in self.morphisms.values():
for f in morphisms_list:
for g_set in self.morphisms.get((f.target, f.target), []):
for g in g_set:
for h_set in self.morphisms.get((g.target, g.target), []):
for h in h_set:
# 验证 h∘(g∘f) = (h∘g)∘f
left = self.compose(h, self.compose(g, f))
right = self.compose(self.compose(h, g), f)
if left != right:
return False
# 3. 单位律
for mor in sum(self.morphisms.values(), []):
id_src = self.identity(mor.source)
id_tgt = self.identity(mor.target)
if (self.compose(mor, id_src) != mor or
self.compose(id_tgt, mor) != mor):
return False
return True
def encode(self) -> No11Number:
"""范畴的No-11编码"""
obj_sum = sum(obj.encode().value for obj in self.objects)
mor_sum = sum(mor.encode().value for mors in self.morphisms.values() for mor in mors)
return No11Number((obj_sum + mor_sum) % 100000)
函子定义
@dataclass
class Functor:
"""函子 F: C → D"""
name: str
source: Category
target: Category
object_map: Dict[CategoryObject, CategoryObject] = field(default_factory=dict)
morphism_map: Dict[Morphism, Morphism] = field(default_factory=dict)
def map_object(self, obj: CategoryObject) -> CategoryObject:
"""对象的函子映射"""
if obj not in self.object_map:
raise CategoryError(f"Object {obj.name} not in functor domain")
return self.object_map[obj]
def map_morphism(self, mor: Morphism) -> Morphism:
"""态射的函子映射"""
if mor not in self.morphism_map:
# 尝试从已知映射推导
if isinstance(mor, IdentityMorphism):
# F(id_A) = id_F(A)
f_obj = self.map_object(mor.source)
return self.target.identity(f_obj)
elif isinstance(mor, ComposedMorphism):
# F(g∘f) = F(g)∘F(f)
f_first = self.map_morphism(mor.first)
f_second = self.map_morphism(mor.second)
return self.target.compose(f_second, f_first)
else:
raise CategoryError(f"Morphism {mor.name} not in functor domain")
return self.morphism_map[mor]
def verify_functoriality(self) -> bool:
"""验证函子性质"""
# 1. 保持恒等态射
for obj in self.source.objects:
if obj in self.object_map:
id_obj = self.source.identity(obj)
f_id = self.map_morphism(id_obj)
expected_id = self.target.identity(self.map_object(obj))
if f_id != expected_id:
return False
# 2. 保持态射复合
for mor_set in self.source.morphisms.values():
for f in mor_set:
if f not in self.morphism_map:
continue
for g_set in self.source.hom(f.target, f.target):
for g in g_set:
if g not in self.morphism_map:
continue
# 验证 F(g∘f) = F(g)∘F(f)
composed = self.source.compose(g, f)
f_composed = self.map_morphism(composed)
f_f = self.map_morphism(f)
f_g = self.map_morphism(g)
expected = self.target.compose(f_g, f_f)
if f_composed != expected:
return False
return True
def is_faithful(self) -> bool:
"""检查函子是否忠实"""
# 单射性检查：不同的态射映射到不同的态射
mapped_morphisms = set()
for mor in self.morphism_map:
f_mor = self.morphism_map[mor]
if f_mor in mapped_morphisms:
return False
mapped_morphisms.add(f_mor)
return True
def is_full(self) -> bool:
"""检查函子是否满"""
# 对每对映射后的对象，检查是否所有态射都来自原范畴
for obj1 in self.object_map:
for obj2 in self.object_map:
f_obj1 = self.map_object(obj1)
f_obj2 = self.map_object(obj2)
# 目标范畴中的所有态射
target_hom = self.target.hom(f_obj1, f_obj2)
# 源范畴中对应态射的像
source_hom = self.source.hom(obj1, obj2)
mapped_hom = {self.map_morphism(f) for f in source_hom if f in self.morphism_map}
if mapped_hom != target_hom:
return False
return True
自然变换定义
@dataclass
class NaturalTransformation:
"""自然变换 η: F ⇒ G"""
name: str
source: Functor # F: C → D
target: Functor # G: C → D
components: Dict[CategoryObject, Morphism] = field(default_factory=dict)
def __post_init__(self):
# 验证源函子和目标函子有相同的域和陪域
if self.source.source != self.target.source:
raise CategoryError("Source functors must have same domain")
if self.source.target != self.target.target:
raise CategoryError("Target functors must have same codomain")
def component_at(self, obj: CategoryObject) -> Morphism:
"""获取在对象处的分量 η_A: F(A) → G(A)"""
if obj not in self.components:
raise CategoryError(f"No component at object {obj.name}")
return self.components[obj]
def verify_naturality(self) -> bool:
"""验证自然性条件"""
C = self.source.source # 源范畴
D = self.source.target # 目标范畴
# 对源范畴中的每个态射 f: A → B
for mor_set in C.morphisms.values():
for f in mor_set:
A, B = f.source, f.target
# 检查分量是否都存在
if A not in self.components or B not in self.components:
continue
# 获取相关的对象和态射
FA = self.source.map_object(A)
FB = self.source.map_object(B)
GA = self.target.map_object(A)
GB = self.target.map_object(B)
Ff = self.source.map_morphism(f) # F(f): F(A) → F(B)
Gf = self.target.map_morphism(f) # G(f): G(A) → G(B)
eta_A = self.component_at(A) # η_A: F(A) → G(A)
eta_B = self.component_at(B) # η_B: F(B) → G(B)
# 验证交换性: η_B ∘ F(f) = G(f) ∘ η_A
left = D.compose(eta_B, Ff)
right = D.compose(Gf, eta_A)
if left != right:
return False
return True
def is_isomorphism(self) -> bool:
"""检查是否是自然同构"""
# 每个分量都必须是同构
for obj, component in self.components.items():
if not self.source.target.is_isomorphism(component):
return False
return True
极限和余极限
@dataclass
class Cone:
"""锥：极限的候选"""
apex: CategoryObject
diagram: Dict[CategoryObject, CategoryObject] # 图表
projections: Dict[CategoryObject, Morphism] # 投影态射
def verify_commutativity(self, category: Category) -> bool:
"""验证锥的交换性"""
# 对图表中的每个态射，验证相应的三角交换
return True
@dataclass
class Limit:
"""极限"""
cone: Cone
universal_property: Callable[[Cone], Morphism] # 泛性质
def is_product(self) -> bool:
"""检查是否是积"""
return len(self.cone.diagram) == 2
def is_equalizer(self) -> bool:
"""检查是否是等化子"""
# 特殊的极限类型
return False
@dataclass
class Colimit:
"""余极限"""
cocone: 'Cocone'
universal_property: Callable[['Cocone'], Morphism]
def is_coproduct(self) -> bool:
"""检查是否是余积"""
return len(self.cocone.diagram) == 2
def is_coequalizer(self) -> bool:
"""检查是否是余等化子"""
return False
伴随函子
@dataclass
class Adjunction:
"""伴随 F ⊣ G"""
left: Functor # F: C → D
right: Functor # G: D → C
unit: NaturalTransformation # η: Id_C ⇒ G∘F
counit: NaturalTransformation # ε: F∘G ⇒ Id_D
def verify_triangle_identities(self) -> bool:
"""验证三角恒等式"""
# 1. (ε * F) ∘ (F * η) = id_F
# 2. (G * ε) ∘ (η * G) = id_G
return True
def hom_isomorphism(self, c: CategoryObject, d: CategoryObject) -> Tuple[Callable, Callable]:
"""同构 Hom_D(F(c), d) ≅ Hom_C(c, G(d))"""
def forward(f: Morphism) -> Morphism:
# f: F(c) → d 映射到 G(f) ∘ η_c: c → G(d)
pass
def backward(g: Morphism) -> Morphism:
# g: c → G(d) 映射到 ε_d ∘ F(g): F(c) → d
pass
return forward, backward
2-范畴结构
class TwoCategory:
"""2-范畴"""
def __init__(self, name: str):
self.name = name
self.objects: Set[Category] = set() # 0-胞
self.morphisms: Dict[Tuple[Category, Category], Set[Functor]] = {} # 1-胞
self.two_morphisms: Dict[Tuple[Functor, Functor], Set[NaturalTransformation]] = {} # 2-胞
def vertical_composition(self, beta: NaturalTransformation,
alpha: NaturalTransformation) -> NaturalTransformation:
"""垂直复合 β • α"""
if alpha.target != beta.source:
raise CategoryError("Natural transformations not vertically composable")
# (β • α)_A = β_A ∘ α_A
components = {}
for obj in alpha.components:
components[obj] = alpha.source.target.compose(
beta.component_at(obj),
alpha.component_at(obj)
)
return NaturalTransformation(
name=f"{beta.name}•{alpha.name}",
source=alpha.source,
target=beta.target,
components=components
)
def horizontal_composition(self, beta: NaturalTransformation,
alpha: NaturalTransformation) -> NaturalTransformation:
"""水平复合 β * α"""
# 需要函子复合
pass
Topos结构
class ElementaryTopos(Category):
"""初等topos"""
def __init__(self, name: str):
super().__init__(name)
self.terminal_object: Optional[CategoryObject] = None
self.subobject_classifier: Optional[CategoryObject] = None
self.truth_morphism: Optional[Morphism] = None
def has_finite_limits(self) -> bool:
"""检查是否有有限极限"""
return True
def has_exponentials(self) -> bool:
"""检查是否有指数对象"""
return True
def internal_logic(self) -> 'InternalLogic':
"""获取内部逻辑"""
return InternalLogic(self)
def is_boolean(self) -> bool:
"""检查是否是布尔topos"""
# 子对象格是布尔代数
return False
范畴等价
@dataclass
class Equivalence:
"""范畴等价"""
functor: Functor # F: C → D
quasi_inverse: Functor # G: D → C
unit_iso: NaturalTransformation # η: Id_C ⇒ G∘F (自然同构)
counit_iso: NaturalTransformation # ε: F∘G ⇒ Id_D (自然同构)
def verify_equivalence(self) -> bool:
"""验证等价条件"""
# 1. η 和 ε 都是自然同构
if not self.unit_iso.is_isomorphism() or not self.counit_iso.is_isomorphism():
return False
# 2. 函子是本质满的
# 3. 函子是忠实的
return True
算法实现
Collapse函子
class CollapseFunctor(Functor):
"""Collapse函子的特殊实现"""
def __init__(self, category: Category):
super().__init__(
name="Collapse",
source=category,
target=category
)
self._compute_collapse_mapping()
def _compute_collapse_mapping(self):
"""计算collapse映射"""
# 移除冗余结构
for obj in self.source.objects:
collapsed_obj = self._collapse_object(obj)
self.object_map[obj] = collapsed_obj
for mor_set in self.source.morphisms.values():
for mor in mor_set:
collapsed_mor = self._collapse_morphism(mor)
self.morphism_map[mor] = collapsed_mor
def _collapse_object(self, obj: CategoryObject) -> CategoryObject:
"""对象的collapse"""
# 如果对象包含形式系统，移除冗余公理
if isinstance(obj.data, FormalSystem):
collapsed_system = self._collapse_formal_system(obj.data)
return CategoryObject(f"collapsed_{obj.name}", collapsed_system)
return obj
def _collapse_morphism(self, mor: Morphism) -> Morphism:
"""态射的collapse"""
# 简化复合态射链
if isinstance(mor, ComposedMorphism):
# 尝试简化
return self._simplify_composition(mor)
return mor
def _collapse_formal_system(self, system: FormalSystem) -> FormalSystem:
"""形式系统的collapse"""
# 实现来自C10-1
pass
def _simplify_composition(self, mor: ComposedMorphism) -> Morphism:
"""简化复合态射"""
# 递归简化
return mor
Yoneda嵌入
class YonedaEmbedding:
"""Yoneda嵌入 Y: C → [C^op, Set]"""
def __init__(self, category: Category):
self.category = category
self.presheaf_category = self._construct_presheaf_category()
def _construct_presheaf_category(self) -> Category:
"""构造预层范畴"""
presheaves = Category(f"[{self.category.name}^op, Set_no11]")
# 每个对象A产生预层Hom(-, A)
for obj in self.category.objects:
presheaf = self._hom_presheaf(obj)
presheaves.add_object(presheaf)
return presheaves
def _hom_presheaf(self, obj: CategoryObject) -> CategoryObject:
"""构造Hom(-, A)预层"""
def presheaf_data(x: CategoryObject) -> Set[Morphism]:
return self.category.hom(x, obj)
return CategoryObject(f"Hom(-,{obj.name})", presheaf_data)
def embed(self, obj: CategoryObject) -> CategoryObject:
"""嵌入对象"""
return self._hom_presheaf(obj)
def yoneda_lemma(self, presheaf: CategoryObject, obj: CategoryObject) -> bool:
"""验证Yoneda引理：Nat(Hom(-,A), F) ≅ F(A)"""
# 自然变换与元素的双射
return True
接口规范
范畴论系统接口
class CategoryTheoryInterface:
"""范畴论系统的标准接口"""
def create_category(self, name: str) -> Category:
"""创建范畴"""
pass
def create_functor(self, name: str, source: Category, target: Category) -> Functor:
"""创建函子"""
pass
def create_natural_transformation(self, name: str,
source: Functor,
target: Functor) -> NaturalTransformation:
"""创建自然变换"""
pass
def compute_limit(self, diagram: Dict[CategoryObject, Morphism]) -> Limit:
"""计算极限"""
pass
def find_adjunction(self, left: Functor, right: Functor) -> Optional[Adjunction]:
"""寻找伴随"""
pass
def check_equivalence(self, C: Category, D: Category) -> Optional[Equivalence]:
"""检查范畴等价"""
pass
验证规范
公理验证
def verify_category_axioms(category: Category) -> bool:
"""验证范畴公理"""
# 1. 态射复合的结合律
# 2. 恒等态射的存在性
# 3. 单位律
return category.verify_axioms()
函子性验证
def verify_functor_properties(functor: Functor) -> bool:
"""验证函子性质"""
# 1. 保持恒等态射
# 2. 保持态射复合
return functor.verify_functoriality()
自然性验证
def verify_naturality(nat: NaturalTransformation) -> bool:
"""验证自然性"""
return nat.verify_naturality()
错误处理规范
所有范畴操作必须进行严格的错误检查：
- 对象错误: 对象必须在范畴中
- 态射错误: 态射的源和目标必须正确
- 复合错误: 只有可复合的态射才能复合
- 函子错误: 映射必须保持结构
- 自然性错误: 必须满足交换图
性能要求
- 对象查找: O(1) 使用集合
- 态射查找: O(1) 使用字典
- 复合运算: O(1) 对简单态射
- 公理验证: O(n³) 其中n是态射数
- 等价判定: 指数级，需要优化
测试规范
每个范畴组件必须通过以下测试：
- 基础测试: 对象和态射的创建
- 复合测试: 态射复合的正确性
- 公理测试: 范畴公理的满足
- 函子测试: 函子性质的保持
- 极限测试: 极限的构造和泛性质
- 高级测试: 伴随、等价等高级概念