final coalgebra of the 𝓟${_{<κ}}$(A×X) endo-functor in $Set^*$?
$begingroup$
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.
PS I posted this question on mathoverflow yesterday, thinking I was on math.stackexchange...
category-theory game-theory coalgebras
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
$endgroup$
add a comment |
$begingroup$
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.
PS I posted this question on mathoverflow yesterday, thinking I was on math.stackexchange...
category-theory game-theory coalgebras
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
$endgroup$
add a comment |
$begingroup$
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.
PS I posted this question on mathoverflow yesterday, thinking I was on math.stackexchange...
category-theory game-theory coalgebras
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
$endgroup$
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.
PS I posted this question on mathoverflow yesterday, thinking I was on math.stackexchange...
category-theory game-theory coalgebras
category-theory game-theory coalgebras
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
edited Jan 5 at 18:19
Henry Story
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
asked Jan 5 at 17:43
Henry StoryHenry Story
25818
25818
add a comment |
add a comment |
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