# final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?

In the paper Coalgebraic Games and Strategies F. Honsell, M. Lenisa, and R. Redamalla use the functor $$F_A$$(X) = $${\mathscr{P}_{<κ}}$$(A×X) to define games coalgebraically. This is a functor from the category of $$Set^* \to Set^*$$ where $$Set^*$$ is the category of sets satisfying the antifoundation axioms.

They state that the final $$F_A$$ coalgebra is ($$G_A$$,id) and that a game is to be understood as an element of $$G_A$$, but I don't know where the final coalgebra construction for $$F_A$$ is explained in detail. Why would id be the coalgebraic function?

They write

The elements of the final coalgebra $$G_A$$ are the minimal graphs up-to bisimilarity.

To add a bit more context as to how this relates to games:

We consider a general notion of 2-player game of perfect information, where the two players are called Left (L) and Right (R). A game x is identified with its initial position; at any position, there are moves for L and R, taking to new positions of the game. By abstracting from superficial features of positions, games can be viewed as elements of the final coalgebra for the functor F$$_A$$(X) = $$\mathscr{P}_{<κ}$$(A×X), where A is a parametric set of atoms which encode information on moves and positions, i.e. move names, and the player who has moved, and $$\mathscr{P}_{<κ}$$ is the set of all subsets of cardinality < κ. The coalgebra structure captures, for any position, the moves of the players and the corresponding next positions.

as for the meaning of $$\mathscr{P}_{<κ}$$ this is explained in a later article "Multigames and strategies, coalgebraically"

$$\mathscr{P}_{<κ}$$is the set of all subsets of cardinality < κ, where κ can be ω, if only finitely branching games are considered, or it can be an inaccessible cardinal, if we are interested in more general games.

• $\kappa=\omega$ in this case, I guess? – Asaf Karagila Jan 4 at 21:26
• I think it is actually finite in this case. I added a bit more info above. – Henry Story Jan 4 at 21:39
• I'm confused. You said "finite power set", namely the set of all finite subsets. Then how could $\kappa$ be finite? – Asaf Karagila Jan 4 at 21:40
• I added an explanation of ${\mathscr{P}_{<κ}}$ taken from another article. It can be smaller the w or larger, depending on what game you want. – Henry Story Jan 4 at 23:40
• They are using/assuming a strong form of anti-foundation which allows them to pretend they are iterating the operation $X\rightarrow \mathcal{P}(A\times X)$. In reality what I think they intend is for the objects of $G_A$ to be the sets (witnessed to exist by this strong form of anti-foundation) whose membership relation codes the iterates of some labeled transitional system. – Not Mike Jan 5 at 11:00