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

=== We present several constructions that realize O in specific categories. ===

==== Given C = Set, one can define O as the free monad on the endofunctor G(X) = A × X? That’s not quite. For Obatala we want self-reference and nesting. An abstract free construction: ====
* Let T be the free monad generated by signatures describing agent structure (state, memory, simulation).
* Impose equations that identify nested simulation structure (collapse property).
* Then O := T / ≈ (quotient by those equations).

Explicitly building this requires a formal grammar; SDELang provides this in the engineering implementation.

==== Let C = Set. For an agent type A we can define: ====

<syntaxhighlight>O A := Tree(A)

</syntaxhighlight>

finite (or possibly infinite) trees whose leaves are A and whose internal nodes are labeled merge or simulate. η_A(a) := leaf(a). μ flattens nested trees by grafting. join is binary tree merge. sim interprets the tree as a joint trajectory by recursively composing simulations of subtrees with coordination rules.

This Tree monad (free monad over signature with merge etc.) is an explicit realization of Obatala: nested agents and organizations are trees.

==== If S is the state monad S X = (σ → (X × σ)), and P the probability monad D (distributions), we often want to combine O with S and P: ====
* O ∘ S is a candidate, but to make S ∘ O vs O ∘ S behave, we need a distributive law λ : S O ⇒ O S (see §9).