Cohesive-Chain is a predicate that holds when four functors form the characteristic adjunction chain of a cohesive context, binding local and global perspectives into a unified framework.
Formal Signature
CohesiveChain : (Obj→Obj, Obj→Obj, Obj→Obj, Obj→Obj) → Truth
Definition
CohesiveChain(Shape, Discrete, Codiscrete, Global) holds when:
- Shape is left adjoint to Discrete
- Discrete is left adjoint to Codiscrete
- Codiscrete is left adjoint to Global
- Each functor satisfies its preservation axiom (Shape preserves colimits, Discrete preserves finite limits, Codiscrete preserves finite colimits, Global preserves limits)
- Stabilizes-Between relations are preserved by Discrete and Codiscrete
This chain of four adjunctions is the signature structure of cohesive type theory. It ensures that an object can be viewed at every level of connectivity --- from its pure shape down to its globally visible points --- and that these views are related by canonical transformations.
Derivational context
Cohesive-Chain arises in Movement IV: Geometric Cohesion as the structural signature of cohesive coexistence. When multiple relational units must coexist, local and global perspectives must translate without loss. The chain of four operations (Shape, Discrete, Codiscrete, Global) provides exactly this translation — any relational configuration can be viewed at every level of connectivity, and these views are canonically related. In mathematical terms, this corresponds to a quadruple adjunction characteristic of Lawvere’s axiomatic cohesion, but relationally it is the architecture that binds immediate and extended perspectives into a single framework.
Relations to Other Terms
- Shape --- the leftmost functor; extracts global form
- Discrete --- maximally disconnects
- Codiscrete --- maximally connects
- Global --- the rightmost functor; extracts global sections
- Stabilizes-Between --- the mediating-identity condition preserved by the chain