Relational Machine Carrier
What this is
The RelationalMachineCarrier is a G_s-coalgebra in R: a sheaf X ∈ R equipped with a family of stepping maps, one per step s ∈ Σ and history t ∈ T.
The carrier is what holds current state and advances it. It is the covariant pole of the Passage: restriction maps go backward along histories (from longer to shorter); carrier stepping maps go forward (from shorter to longer).
The carrier name comes from the Chu construction: the points object A in a Chu space (A, r, X) is the input-side object — the carrier of atomic content. In our system, the atomic class A from generative-act fills this role: A ⊆ Obj(R) is always generated, and every carrier is built from atomic content upward.
Mathematical grounding
The full construction is in directed-comonad. Fix a step s ∈ Σ. The directed comonad G_s on R sends F ↦ F(s★−). A RelationalMachineCarrier is a G_s-coalgebra: a pair (X, γ) where X ∈ R and γ : X → G_s X is a natural transformation. Concretely, γ is a family of maps γ_t : X(t) → X(s★t), natural in t — the RelationalMachineSteppingMap of this carrier.
The stepping maps must satisfy two laws derived from the comonad structure:
Counit law — ε_X ∘ γ = id_X. The counit ε_X(t) : X(s★t) → X(t) is the restriction map back to the shorter history. Stepping forward then restricting back recovers the original state. γ is a section of ε_X.
Coassociativity law — γ_{s★t} ∘ γ_t = ν_X(t). Two sequential applications of γ cohere with the comonad comultiplication ν_X. Stepping twice by s is the same as applying the comultiplication first.
In the FARS
In a FlatfileAgentialResourceSystem locale, the RelationalMachineCarrier is the collection of entity files — the markdown files holding the locale’s current state. The stepping maps are instantiated each session: the improvement skill reads the carrier at history t, processes incoming steps, and writes updated files, producing the carrier at history s★t.
A RelationalMachineCarrier MUST have stepping maps γ_t : X(t) → X(s★t) for each step s ∈ Σ and each history t. It MUST satisfy the counit law — stepping forward then restricting back is the identity. It MUST satisfy the coassociativity law — sequential stepping coheres with comultiplication. It MUST contain elements of the atomic class A — the carrier must be built from atomic content.