Expedition Two • Front CMarch 12, 2026
Foundation Securing
Adjoint Functors and Left Adjoints
Summary
This document constructs the left adjoints to the forgetful functors in the Erlangen chain.
The existence of F₁ ⊣ U₁ and F₂ ⊣ U₂ is proven via explicit construction, establishing that the
categorical structure of the Universal Language hierarchy has the expected universal properties.
Status Summary
Proven(3)
- •Left adjoint F₁ ⊣ U₁ by explicit construction
- •Left adjoint F₂ ⊣ U₂ by explicit construction
- •Toy model verification
Conjectured(1)
- •F₃ and F₄ exist via AFT (hypotheses not verified)
Key Ideas
Left Adjoint F₁ ⊣ U₁
ProvenThe left adjoint to the Euclidean → Similarity forgetful functor is constructed explicitly.
Left Adjoint F₂ ⊣ U₂
ProvenThe left adjoint to the Similarity → Affine forgetful functor is constructed explicitly.
Toy Model Construction
ProvenA concrete toy model verifies the adjunction properties in a simplified setting.
Not Yet Addressed
- Full verification of AFT hypotheses for F₃, F₄
- Computational aspects of the adjunctions
Prerequisites
- category-of-languages
Related Open Problems
Source Document
frontier/expedition-two/foundation-securing.md