AI 中文总结
该文定义策略博弈范畴与对应预层，分析不同映射类型下预层良定义的条件，明确纳什均衡等预层良定义的具体要求。
AI 中文摘要
我们定义了一个策略博弈的范畴，其中态射是玩家集合之间的映射与具有特定性质的策略组合集合之间的映射构成的对，且证明该范畴是良定义的。针对策略组合集合之间的映射，我们考虑三种情形：相对于每个玩家的偏好关系，它们可以是保序、反序或序嵌入映射。我们定义了一个取值于集合范畴的博弈范畴上的预层，该预层将每个策略博弈映射到一个策略组合集合，并给出了该预层良定义的等价条件。针对集合范畴中的态射，我们考虑两种情形：它们可以是关系或映射。我们定义了一个预层，该预层将每个策略博弈映射到纳什均衡（对应帕累托有效策略组合）的集合，并证明当且仅当策略组合集合之间的映射是反序或序嵌入映射（对应策略组合集合之间的映射是序嵌入映射）且集合范畴中的态射是关系时，该预层是良定义的。
英文摘要
We define a category of strategic games in which a morphism is a pair of a map between sets of players and a map between sets of strategy profiles with specific properties and show that this category is well-defined. Three cases are considered for the maps between sets of strategy profiles: they may be order-preserving, order-reflecting or order-embedding with respect to each player's preference relation. We define a presheaf on the category of games valued in a category of sets that sends each strategic game to a set of strategy profiles and present conditions for this presheaf to be well-defined. Two cases are considered for the morphisms in the category of sets: they may be relations or maps. We define a presheaf that sends each strategic game to the set of Nash equilibria (resp. Pareto efficient strategy profiles) and show that this presheaf is well-defined if and only if the maps between sets of strategy profiles are order-reflecting or order-embedding (resp. order-embedding), and the morphisms in the category of sets are relations.
