Sintesi categorica

A competitive game can be described using the language of category theory using algebraic constructions from the resource space of the game.

Definition. A resource space is a ringed Hilbert space (X,O_X), where every ring O_X(U) of the structure sheaf is homomorphic to a ring of sequences with positive integer terms, equipped with a countering form % : X²-> R which satisfies the following properties:

1) x%­y = 1/y%­x for all x,y ≠ e;

2) e%­x = 0 for all x ≠ e;

3) e%e = 1.

where e is the zero vector in X.

Let Res denote the category of resource spaces with ringed space homomorphism which preserve the countering structure.

Definition. Let R be a resource space. A metagame theory on R with countering potential V, or a V-metagame theory, is the scalar QED theory with potential V with the tensor algebra T(R) as state space. The complex scalar field of the theory is called the usage field, and the associated quanta are called usage (or antiusage) particles.

Definition. Let r ∈ O_R be a section of the structure sheaf and let C={V_i}_{i ∈ I} be a cover of R such that V_i ∩ V_j = Ø for i ≠ j. Define a sequence of forgetful functors to the category Hilb of Hilbert spaces

P_C,k : Res -> Hilb

given by

P_C,k(R) = U_{i ∈ I} (V_i)^(r(V_i))_k

where (r(V_i))_k is the k-th number of the sequence (r_n)_{n ∈ N} ∈ S_i obtained from the ring homomorphism

O_R(V_i) -> S_i

A rule choice for a resource space R is a choice of a such a section and cover. Then the state space of the game with a rule choice (r,C) is given by

K(R) := F(⊕_k T^k(P_C,k(R)))

where F is a sheaf of Hilbert spaces, each of which is called a structure space, and T^k is the k-graded part of the tensor algebra functor T, i.e. T(V) = ⊕_k T^k(V) for every vector space V.

A game of state space K(R) is said to be of global structure W if F is the constant sheaf associating a structure space W to every open set of K(R).

A game of global structure space W and of state space K(R) is said to be of pure structure if K(R) is isomorphic to W. Examples of pure structure games include card games and chess.

Definition. Let H be a Hilbert space, and define c = c(H) to be 0 for discrete topology on H, and 1 otherwise. Define the strategy space functor S_c by S_c(H) = L²(H)/H if c=1, and S_c(H) = T(H)/H if c=0.

S_c(K(R)) is the state space of the strategies of the game. A strategy is thus a path in the game state space K(R), which starts from a fixed game state called the initial state. Moves are elements of the endomorphism group End(K(R)) on K(R).

Definition. The strategy theory on S_c(K(R)) is a bosonic string theory restricted to level zero excitations defined on K(R). The action is given by the Polyakov action and the scattering amplitudes are given by the pinned path integral.

Giving the dynamics of a V-metagame theory on T(R) to the fixed initial state of S_c(K(R)) gives the full quantum game-theoretic framework studied in this field-theoretic approach to the subject.