Game topos
Contents
The Game topos is the exact/regular completion of Games, presented as sheaves on finite games.
1. Definition§
Definition 1.1 (Game topos)§
The Game topos is the exact/regular completion $\mathcal{G}=\mathbf{Games}_{\mathrm{ex/reg}}$ of Games.
2. Construction§
Proposition 2.1 (Exact-completion criterion)§☆
The exact/regular completion of a regular, infinitary extensive category with a set of regular generators is a Grothendieck topos.
Proof§
The completion is exact, and its canonical embedding is fully faithful and regular (Lack, Proposition 3.2). By Vitale, p.237, Lemma (3), it is infinitary extensive and the embedding preserves coproducts. The generators remain regular generators since every object is a regular quotient of an object in the image. Apply Giraud's theorem (Worrell, Theorem 5.5.1).
3. Presentation by finite games§
Give FinGames its finite-cover topology $J$.
Theorem 3.1 (Finite-game presentation)§☆
The Game topos is a Grothendieck topos, with an equivalence
Under this equivalence, the canonical embedding of Games is the finite-game nerve.
Proof§
Every game is covered by its finite subgames. Together with regularity and infinitary extensivity, this supplies the hypotheses of the exact-completion criterion.
Apply the Comparison Lemma (Caramello, §3.2) to the images of finite games. The induced topology is $J$: the canonical embedding preserves coproducts and reflects regular epimorphisms, so covers are jointly surjective families; a finite target gives a finite subcover. Full faithfulness identifies the resulting restricted Yoneda functor with the finite-game nerve.
4. Related concepts§
5. References§
- Ryuya Hora, Games as recursive coalgebras: A categorical view on the Nim-sum, §3 and Appendix B.
- Stephen Lack, A note on the exact completion of a regular category, and its infinitary generalizations, Theory and Applications of Categories 5 (1999), 70–80.
- Enrico M. Vitale, Quasi-varieties of presheaves, Theory and Applications of Categories 7 (2000), 236–238.
- James Worrell, On Coalgebras and Final Semantics, DPhil thesis (2000), §5.5.
- Olivia Caramello, Topos-theoretic background, §3.2.