FinGames
Contents
1. Definition§
Definition 1.1 (Finite games)§
FinGames is the full subcategory of Games whose objects have finitely many positions.
2. Properties§
Proposition 2.1 (Finite limits and colimits)§☆
FinGames has finite nonempty limits and all finite colimits.
Proof§
Nonempty finite products embed in a bounded-height part of the cofree game on the product of the carriers, which is finite; equalizers are subgames (Appendix B.2–B.3). Colimits are created on underlying sets by Lemma B.2.
Proposition 2.2 (No terminal object)§☆
FinGames has no terminal object. In particular, it is not a coherent category.
Proof§
Game morphisms preserve height, the length of a longest play. For any finite game, a chain of greater height admits no morphism into it.
3. Topology§
Let $J$ be the Grothendieck topology generated by finite jointly surjective families. The empty family covers the empty game.
Proposition 3.1 (Subcanonical topology)§☆
For every game $X$, the presheaf $A\mapsto\mathbf{Games}(A,X)$ on FinGames is a sheaf for $J$. In particular, $J$ is subcanonical.
Proof§
The finite-game presentation identifies these presheaves with the images of games under the canonical embedding into Game topos.
4. Related concepts§
5. References§
- Ryuya Hora, Games as recursive coalgebras: A categorical view on the Nim-sum, Appendix B.