Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

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* Agt is a (small) V-enriched category: Hom(A,B) ∈ V (captures affinity or more general graded relation).
* Observations are modeled by a Grothendieck fibration: ``<code> π : Obs → World </code>`` each agent A has perception section P_A : World → Obs with π ○ P_A = Id.

===== - O : Agt → Agt with η : Id ⇒ O, μ : O O ⇒ O. =====
* O supports: - reflect ≡ η - μ flattening of nested agency - join_W : O A × O A → O A (affinity-parameterized merge) - sim : O A → Traj(A) (natural in A)

Monad laws hold for all A:

<syntaxhighlight>μ ∘ O η = id,    μ ∘ η_O = id,    μ ∘ O μ = μ ∘ μ_O

</syntaxhighlight>

===== - Aff : Agt^op × Agt → V (a V-profunctor). =====
* Composition defined by coend:

<syntaxhighlight>Aff^{(2)}(A,C) := (Aff ∘ Aff)(A,C) = ∫^{B∈Agt} Aff(A,B) ⊗ Aff(B,C).

</syntaxhighlight>
* Iteration:

<syntaxhighlight>Aff^{(n)}(A,Z) ≅ ∫^{B1...B_{n-1}} ∏_{i=0}^{n-1} Aff(B_i,B_{i+1})
where B_0=A, B_n=Z.

</syntaxhighlight>

This yields multiplicative / path-based transitive affinity when V is a quantale with ⊗ product and coends computed as joins (sup).