Generalized games framework

Question

How far does the recursive-coalgebra framework extend from sets and the finite-powerset functor to a locally presentable category $\mathcal C$ and an accessible endofunctor $T:\mathcal C\to\mathcal C$? This is Question 5.9 of Games as recursive coalgebras.

Separate the claims

The framework contains several ingredients: recursive evaluation into algebras; a terminal recursive coalgebra; game-combining operations; and universal factorizations of values. A construction of one ingredient is not by itself a construction of all the others for arbitrary $(\mathcal C,T)$.

One concrete extension is already in the paper: replacing $\mathcal P_{\mathrm{fin}}$ with $\mathcal P_{\mathrm{fin}}\times\mathcal P_{\mathrm{fin}}$ recovers short partisan forms (Proposition 5.4). Note 3 states the remaining boundary for values and strategies.

The factorization argument recorded in note 2 is another partial direction; it does not, on its own, establish the entire framework for every locally presentable base and accessible endofunctor.

Next step

Formulate one generalization theorem at a time. State the hypotheses needed for recursive evaluation and the terminal object, then separately give the additional monoidal or value data used by a Bouton-type factorization. Test those hypotheses against the impartial and partisan examples before making the universal claim.