gaia.core.ends_coends module

Ends and Coends for GAIA Framework - Integral Calculus for Generative AI

Implements Section 7 from paper.md: “The Coend and End of GAIA: Integral Calculus for Generative AI”

THEORETICAL FOUNDATIONS: - Section 7.1: Ends and Coends as categorical integrals - Section 7.2: Sheaves and Topoi in GAIA - Section 7.3: Topological Embedding of Simplicial Sets - Section 7.4: The Geometric Transformer Model - Section 7.5: The End of GAIA: Monads and Categorical Probability

This completes the theoretical foundations of GAIA with integral calculus over categories, enabling topological vs probabilistic generative systems and geometric transformers.

class gaia.core.ends_coends.CategoricalIntegral(name)

Bases: ABC

Abstract base for categorical integrals (ends and coends)

From (MAHADEVAN,2024) Section 7.1: Ends and coends as limits and colimits over twisted arrow categories

__init__(name)

abstractmethod compute_integral(functor, category)

Compute the categorical integral

abstractmethod verify_universal_property()

Verify the universal property of the integral

class gaia.core.ends_coends.End(functor, name='End')

Bases: CategoricalIntegral

End of a functor F: C^op × C → D

∫_c F(c,c) - limit over twisted arrow category

From paper.md: “The End of GAIA represents the terminal object in the category of natural transformations”

__init__(functor, name='End')

compute_integral(functor=None, category=None)

Compute end ∫_c F(c,c)

This is the limit of the diagram F(c,c) over all objects c

verify_universal_property()

Verify universal property of end

For any wedge α: X → F, there exists unique h: X → ∫F such that the diagram commutes

class gaia.core.ends_coends.Coend(functor, name='Coend')

Bases: CategoricalIntegral

Coend of a functor F: C^op × C → D

∫^c F(c,c) - colimit over twisted arrow category

From paper.md: “Coends represent the initial object for generative processes in GAIA”

__init__(functor, name='Coend')

compute_integral(functor=None, category=None)

Compute coend ∫^c F(c,c)

This is the colimit of the diagram F(c,c) over all objects c

verify_universal_property()

Verify universal property of coend

For any cowedge α: F → X, there exists unique h: ∫^F → X such that the diagram commutes

class gaia.core.ends_coends.Sheaf(base_space, name='Sheaf')

Bases: object

Sheaf on a topological space for GAIA

From (MAHADEVAN,2024) Section 7.2: “Sheaves and Topoi in GAIA”

Enables local-to-global reasoning in generative AI

__init__(base_space, name='Sheaf')

add_section(open_set, section)

Add section over open set

add_restriction_map(larger_set, smaller_set, restriction_map)

Add restriction map from larger to smaller open set

verify_sheaf_axioms()

Verify sheaf axioms: 1. Identity: restriction to same set is identity 2. Composition: restrictions compose properly 3. Locality: sections agree on overlaps 4. Gluing: local sections glue to global sections

global_sections()

Get global sections (sections over entire base space)

class gaia.core.ends_coends.Topos(base_space, name='GAIATopos')

Bases: object

Topos for GAIA - category of sheaves

From (MAHADEVAN,2024) Section 7.2: Elementary topos structure for generative AI reasoning

__init__(base_space, name='GAIATopos')

add_sheaf(name, sheaf)

Add sheaf to topos

add_morphism(source_name, target_name, morphism)

Add morphism between sheaves

create_subobject_classifier()

Create subobject classifier Ω

This is the sheaf of truth values, crucial for topos structure

verify_topos_axioms()

Verify elementary topos axioms: 1. Finite limits and colimits 2. Exponentials (internal hom) 3. Subobject classifier

class gaia.core.ends_coends.TopologicalEmbedding(simplicial_set, name='TopologicalEmbedding')

Bases: object

Topological embedding of simplicial sets

From (MAHADEVAN,2024) Section 7.3: “Topological Embedding of Simplicial Sets”

Connects discrete simplicial structure to continuous topology

__init__(simplicial_set, name='TopologicalEmbedding')

compute_geometric_realization()

Compute geometric realization |X| of simplicial set X

This creates a topological space from the simplicial set

verify_embedding_properties()

Verify embedding preserves simplicial structure

class gaia.core.ends_coends.CategoricalProbability(name='CategoricalProbability')

Bases: object

Categorical probability using monads

From (MAHADEVAN,2024) Section 7.5: “The End of GAIA: Monads and Categorical Probability”

Provides probabilistic reasoning in categorical framework

__init__(name='CategoricalProbability')

create_probability_monad()

Create probability monad P: Set → Set

P(X) = probability distributions over X

verify_monad_laws()

Verify monad laws: 1. Left unit: μ ∘ Pη = id 2. Right unit: μ ∘ ηP = id 3. Associativity: μ ∘ Pμ = μ ∘ μP

gaia.core.ends_coends.create_complete_gaia_system(vocab_size=10000, hidden_dim=512)

Create complete GAIA system with all theoretical components