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

==== Lemma 1. Define ====

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

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Then Aff^{(2)} : Agt^op ⊗ Agt → V is a V-profunctor (i.e. natural in both variables).

Proof.
Composition via coend yields a V-functor in each variable since the integrand Aff(A,B) ⊗ Aff(B,C) is V-natural in A (contravariantly) and C (covariantly), and coend is a colimit in V that preserves these naturalities. Concretely:
* For any f : A' → A in Agt, naturality in the first argument arises from the map </code>`<code> Aff(A,B) → Aff(A',B)    (given by precomposition) </code>`<code> and since (−) ⊗ Aff(B,C) is a V-functor and coends are colimits, we get an induced map </code>`<code> ∫^B Aff(A,B) ⊗ Aff(B,C)  →  ∫^B Aff(A',B) ⊗ Aff(B,C). <syntaxhighlight>- Similarly for postcomposition (maps on C).

Hence Aff^{(2)} is a V-profunctor. □

==== We will use the standard Fubini / Tonelli style identity for coends (valid in complete cocomplete V such as a quantale): ====

Lemma 2 (Fubini for coends). For V complete cocomplete and for triples of variables,

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∫^{b} ∫^{c} X(b,c)  ≅  ∫^{b,c} X(b,c)  ≅  ∫^{c} ∫^{b} X(b,c)

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and more generally iterated coends can be rearranged/flattened.

Sketch proof / justification. Coends are colimits indexed by small categories; iterated colimits over product categories commute when the target category V is cocomplete and tensor ⊗ preserves colimits in each variable (true for quantales). Thus the coend over product category B × C equals iterated coends in either order. This is standard; see enriched category texts (Kelly) for full statement. □

==== Theorem 3 (Associativity). For Aff as above, composition is associative up to canonical V-isomorphism; i.e., for any agents A, D: ====

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(Aff ∘ (Aff ∘ Aff))(A,D) ≅ ((Aff ∘ Aff) ∘ Aff)(A,D)

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and both are isomorphic to the triple coend

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∫^{B,C} Aff(A,B) ⊗ Aff(B,C) ⊗ Aff(C,D).

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Proof.

Compute the left side:

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Aff ∘ (Aff ∘ Aff)  evaluated at (A,D)
= ∫^{X} Aff(A,X) ⊗ (Aff ∘ Aff)(X,D)
= ∫^{X} Aff(A,X) ⊗ ( ∫^{Y} Aff(X,Y) ⊗ Aff(Y,D) ).

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By Fubini (Lemma 2) and monoidality of ⊗ distributing over coends (quantale property), we can exchange/flatten coends:

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≈ ∫^{X,Y} Aff(A,X) ⊗ Aff(X,Y) ⊗ Aff(Y,D).

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Similarly the right side ((Aff ∘ Aff) ∘ Aff)(A,D) reduces to the same coend:

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((Aff ∘ Aff) ∘ Aff)(A,D)
= ∫^{Y} (Aff ∘ Aff)(A,Y) ⊗ Aff(Y,D)
= ∫^{Y} ( ∫^{X} Aff(A,X) ⊗ Aff(X,Y) ) ⊗ Aff(Y,D)
≈ ∫^{X,Y} Aff(A,X) ⊗ Aff(X,Y) ⊗ Aff(Y,D).

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Therefore the two parenthesizations are canonically isomorphic via the natural associativity isomorphisms of coends and of ⊗, and the associator is coherent because it arises from universal properties. This proves associativity up to canonical isomorphism. □

==== We need to specify units. In enriched profunctor bicategory, units are given by representable profunctors (Yoneda). For Agt (viewed as V-category with homs Agt(A,B)), the identity profunctor Id : Agt^op ⊗ Agt → V is ====

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Id(A,B) := Agt(A,B)   ∈ V

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(If Agt is ordinary and we want a Kronecker style identity for affinity composition, we can alternatively use a δ profunctor δ(A,B) = 1 iff A=B, else bottom; both are valid choices depending on modeling.)

Theorem 4 (Left and right unit). For any V-profunctor Φ : Agt ⇸ Agt,

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Id ∘ Φ ≅ Φ ≅ Φ ∘ Id

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canonically.

Proof (sketch). Use definition:

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(Id ∘ Φ)(A,C) = ∫^{B} Id(A,B) ⊗ Φ(B,C) = ∫^{B} Agt(A,B) ⊗ Φ(B,C).

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By enriched Yoneda lemma (or coend calculus), the coend collapses and yields Φ(A,C) (intuitively, integrating Agt(A,B) against B reproduces evaluation at A). Concretely, the universal property of Yoneda provides a canonical iso:

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∫^{B} Agt(A,B) ⊗ Φ(B,C) ≅ Φ(A,C).

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Similarly on the right. This is the standard unit law for enriched profunctors. □

(If you use the Kronecker δ identity profunctor, the argument is simpler: coend with δ picks out the appropriate summand.)

==== Given Aff as a base affinity profunctor, define Aff^{(n)} := Aff ∘ Aff ∘ ... ∘ Aff (n times). Then by repeated application of Theorem 3 and the Fubini identity: ====

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Aff^{(n)}(A,Z) ≅ ∫^{B_1,...,B_{n-1}} Aff(A,B_1) ⊗ Aff(B_1,B_2) ⊗ ... ⊗ Aff(B_{n-1},Z).

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This gives the natural notion of n-step affinity aggregated across intermediate agents.

==== If V = [0,1] with ⊗ = × and coend interpreted as sup of products (or an appropriate integral), then composition yields the intuitive inequality: ====

Proposition 5. For all A,C,

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Aff^{(2)}(A,C) ≤ sup_{B} ( Aff(A,B) ⊗ Aff(B,C) ).

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and in many semantics where coend collapses to ∨ of ⊗, we get equality:

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Aff^{(2)}(A,C) = sup_{B} ( Aff(A,B) ⊗ Aff(B,C) ).

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Interpretation/proof sketch. In quantale coends often compute as joins (∨) of ⊗ terms, hence the composite is the supremum over intermediate paths of their product weights — i.e., path score equals best intermediate path. This matches the intuitive "max-over-intermediates" transitive affinity.

==== The construction Φ,Ψ ↦ Ψ ∘ Φ is functorial at the level of V-profunctors and V-natural transformations: given θ : Φ ⇒ Φ' and χ : Ψ ⇒ Ψ', there is an induced χ ◦ θ : Ψ ∘ Φ ⇒ Ψ' ∘ Φ' obtained by composing the integrand maps and using coend universality. This yields the bicategory structure of V-profunctors. ====

==== - The quantale V choice determines the algebra of composition: - If V uses ⊗ = × and coend = sup, composition takes Aff^{(2)}(A,C) = sup_B Aff(A,B)·Aff(B,C) — “best intermediary path” model. - If V uses ⊗ = min and coend = inf, composition becomes min over joins, etc. Choose according to whether affinity aggregates via bottleneck or by best path or by sum. ====
* If you prefer additive path accumulation (sum of contributions), use V = [0,∞] with ⊗ = + and ∨ = sup or integrals; composition then sums over intermediate contributions (coend becomes sum/integral).
* Unit profunctor choice affects whether composition preserves original affinity exactly (representable yields Yoneda identity) or uses Kronecker delta identity (discrete identity). Use representable for category-theoretic purity.

==== Let V = ([0,1], ≤) with ⊗ = ×, coends computed as sup (∨) over intermediate agents. Then: ====

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Aff^{(2)}(A,C) = sup_{B ∈ Agt} ( Aff(A,B) × Aff(B,C) ).

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Associativity then says:

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sup_{B,C} Aff(A,B)·Aff(B,C)·Aff(C,D)   is independent of parenthesization,

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i.e., Aff^{(3)}(A,D) = sup_{paths A→...→D} product(weights).

This matches the standard best-path multiplicative model (probability of two-step successful transmission is max over intermediates of product of step probabilities).

==== - Affinity Aff : Agt^op × Agt → V is exactly a V-profunctor. ====
* Composition of affinities is given by the enriched coend </syntaxhighlight> (Aff ∘ Aff)(A,C) = ∫^{B} Aff(A,B) ⊗ Aff(B,C), </code>`` which encodes n-step transitive affinity as iterated coends.
* Under the standard quantale assumptions on V (completeness, monoidality distributing over joins), coends satisfy Fubini rules and composition is associative (up to canonical iso) and unital (units = representables / Yoneda), giving the bicategory of V-profunctors.
* Concrete formulas (sup×, sum, min etc.) arise by choosing the quantale (and coend semantics) appropriate for your modeling needs in Kogi/Nandi: e.g., sup× gives best-path multiplicative affinity; sum gives additive contributions.

If you’d like I can:
* produce a formalization sketch of these proofs in Coq/Lean (using enriched category libraries) for V being a quantale like ([0,1],×,∨); or
* instantiate the abstract formulas into a runnable algebra (Python/OCaml) that computes Aff^{(n)} via paths / coend approximations over a finite agent graph; or
* extend the proofs to show contractivity / fixed-point results for affinity dynamics (e.g., when W_{t+1} = F(W_t) has a unique fixed point under given quantale norms).

Which of those would you like next?