{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:28:54.141Z", "scope": "Reader-visible project fields and native live notes only; referenced notes in other projects, deleted notes and private AI context are excluded.", "submission": "not_submitted", "mathematicalCorrespondence": "not_automatically_checked", "project": { "id": "2610.00003", "url": "https://sumire.aiformath.jp/p/2610.00003", "title": "Games as recursive coalgebras", "tagline": "Recursive coalgebras, game values, and the categorical structure of combinatorial games.", "visibility": "public", "owner": { "name": "hora", "handle": "ryuya" }, "origin": "", "source": "", "goal": "Explain how recursive game positions, game values, and operations such as Conway addition fit together, and identify what this explains about the Nim-sum.", "narrative": "The starting point is [Games as recursive coalgebras: A categorical view on the Nim-sum](https://arxiv.org/abs/2510.22886v3), available in [Main](https://sumire.aiformath.jp/p/2610.00003?view=main). The paper develops impartial games as recursive finite-powerset coalgebras, game values as recursive evaluations, and Bouton monoids as universal factorizations of game values. Conway addition is symmetric monoidal closed.\n\nThe next question is how much more explicit this framework can become. [The internal hom](https://sumire.aiformath.jp/p/2610.00003/n/4) has a proposed concrete cofree-subgame construction and proof in [note 29](https://sumire.aiformath.jp/p/2610.00003/n/29); its relationship to playing strategies remains separate. [Differential structures](https://sumire.aiformath.jp/p/2610.00003/n/6) asks for a common explanation of the Leibniz-type move formula and the Rota–Baxter-type law, with an explanation of Nim-sum as the target.\n\nTwo extensions have different starting points. [Partisan games](https://sumire.aiformath.jp/p/2610.00003/n/3) already recover short game forms, but values and strategies need additional structure. [A double category of games](https://sumire.aiformath.jp/p/2610.00003/n/1) still requires a precise strategy class and compatible squares; counterexamples for arbitrary relations do not settle that question.\n\nA settled boundary is that [Game itself is not a topos](https://sumire.aiformath.jp/p/2610.00003/n/16), by Remark B.27 of the paper. Questions about a topos completion are therefore distinct from that answered question.\n\nTo contribute, choose one of these linked questions and work against its stated remaining claim. For the internal hom, compute a small explicit example or clarify its game-theoretic meaning. For the double category, specify legal strategies and test compatibility with coalgebra maps before claiming a general construction or obstruction.", "history": "", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:29:39.051Z", "updatedInSumire": "2026-10-30T14:35:21.579Z", "legacyRecordedAt": "2026-10-07T16:29:39.050Z", "legacyUpdatedAt": "2026-10-30T14:35:21.579Z" }, "goals": [ { "id": "goal_71391ac0-c899-41e4-b6ed-e9d825d41001", "title": "To understand why Nim-sum works", "body": "", "noteIds": [] }, { "id": "goal_4d2683f0-360b-4edd-b553-ca70400f6943", "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory", "body": "", "noteIds": [] } ], "goalDependencies": [] }, "notes": [ { "id": "n_vOlFGtk83nXW", "number": 29, "url": "https://sumire.aiformath.jp/p/2610.00003/n/29", "title": "A proposed internal hom for Conway addition", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "This note gives a **proposed construction and proof** for the explicit internal hom asked for in [note 4](https://sumire.aiformath.jp/p/2610.00003/n/4). Its status remains a proof candidate; a completed human review and a claim of novelty are not asserted.\n\nThe scope is impartial recursive finite-powerset coalgebras with exact coalgebra morphisms. Player-switching, partisan strategies, and arbitrary other game-combining operations are outside the claim.\n\n## 1. Games, Conway addition, and cofree games\n\nAn object \\(X\\) of \\(\\mathbf{Game}\\) consists of a set \\(UX\\) and an option map\n\\[\n\\theta_X:UX\\longrightarrow\\mathcal P_{\\mathrm{fin}}(UX)\n\\]\nwith no infinite path. A morphism \\(f:X\\to Y\\) satisfies\n\\[\n\\theta_Y(f(x))=f[\\theta_X(x)].\n\\]\nConway addition has carrier \\(UG\\times UK\\) and options\n\\[\n\\theta_{G+K}(g,k)\n=(\\theta_G(g)\\times\\{k\\})\\cup(\\{g\\}\\times\\theta_K(k)).\n\\]\nOn maps it is the underlying product map, giving a bifunctor.\n\nFor a set \\(S\\), the cofree game \\(R(S)\\) has positions \\(t=(T_t,a_t)\\), recursively formed from a finite set of child positions \\(T_t\\) and a label \\(a_t\\in S\\). Its option map is \\(t\\mapsto T_t\\), and its counit sends \\(t\\) to \\(a_t\\). These are Definition B.13 and Proposition B.14 of [Games as recursive coalgebras, v3](https://arxiv.org/abs/2510.22886v3).\n\nA label map \\(\\ell:UK\\to S\\) has a unique cofree transpose. Each position \\(k\\) has finite rank: otherwise the finitely branching tree of paths from \\(k\\) would have an infinite branch by König's lemma. Rank recursion therefore defines\n\\[\n\\ell^\\flat(k)\n=\\bigl(\\{\\ell^\\flat(k'):k'\\in\\theta_K(k)\\},\\ell(k)\\bigr).\n\\]\nIt is a game morphism with counit \\(\\ell\\). Conversely, those properties force both the label and the child set, so induction on rank proves uniqueness. The entire carrier \\(UK\\) need not be finite.\n\n## 2. The compatible subgame\n\nFix \\(G,H\\), and put \\(S=\\operatorname{Set}(UG,UH)\\). A label \\(a_w\\) is now a function \\(UG\\to UH\\).\n\nLet \\(A(G,H)\\) consist of positions \\(u\\) of \\(R(S)\\) such that every descendant \\(w\\), including \\(u\\), satisfies\n\\[\n\\theta_H(a_w(g))\n=\\{a_w(g'):g'\\in\\theta_G(g)\\}\n \\cup\\{a_v(g):v\\in T_w\\}\n\\qquad(g\\in UG). \\tag{E}\n\\]\nEvery child has the same property, so this is a subgame.\n\nDefine\n\\[\n\\operatorname{ev}:G+A(G,H)\\to H,\\qquad\n\\operatorname{ev}(g,u)=a_u(g).\n\\]\nThe image of the Conway options is the right-hand side of (E), so evaluation is an exact coalgebra morphism.\n\nThis is the largest subgame of \\(R(S)\\) on which label evaluation is a morphism. Indeed, for any other such subgame \\(B\\), the morphism equation gives (E) at each position. Since \\(B\\) contains its descendants, \\(B\\subseteq A(G,H)\\).\n\n## 3. Images and currying\n\nThe image of a game morphism \\(h:K\\to R(S)\\) is a subgame, since\n\\[\n\\theta_R(h(k))=h[\\theta_K(k)]\\subseteq h[UK].\n\\]\nRestricting the codomain gives a surjective game morphism onto that image.\n\nTake \\(f:G+K\\to H\\), and let \\(h_f:K\\to R(S)\\) be the cofree transpose of\n\\[\n\\ell_f(k)(g)=f(g,k).\n\\]\nIts labels and children satisfy\n\\[\na_{h_f(k)}(g)=f(g,k),\\qquad\nT_{h_f(k)}=\\{h_f(k'):k'\\in\\theta_K(k)\\}.\n\\]\nThe morphism equation for \\(f\\) is\n\\[\n\\theta_H(f(g,k))\n=\\{f(g',k):g'\\in\\theta_G(g)\\}\n \\cup\\{f(g,k'):k'\\in\\theta_K(k)\\}.\n\\]\nThis is (E) at \\(h_f(k)\\). Every descendant of an image position is again in the image, so (E) holds throughout the image subgame. Therefore \\(h_f\\) factors uniquely as\n\\[\n\\lambda f:K\\to A(G,H).\n\\]\n\n## 4. Uncurrying and the inverse equations\n\nFor \\(h:K\\to A(G,H)\\), define\n\\[\n\\operatorname{uncurry}(h)\n=\\operatorname{ev}\\circ(\\mathrm{id}_G+h).\n\\]\nThis is a game morphism, with value \\(a_{h(k)}(g)\\) at \\((g,k)\\).\n\nStarting with \\(f\\), the label of \\(\\lambda f(k)\\) is \\(g\\mapsto f(g,k)\\), so\n\\[\n\\operatorname{uncurry}(\\lambda f)=f.\n\\]\nStarting with \\(h\\), its composite into \\(R(S)\\) has exactly the labels used for the cofree transpose of \\(\\operatorname{uncurry}(h)\\). Uniqueness of that transpose gives\n\\[\n\\lambda(\\operatorname{uncurry}(h))=h.\n\\]\nThe proposed construction thus gives\n\\[\n\\mathbf{Game}(G+K,H)\\cong\\mathbf{Game}(K,A(G,H)).\n\\]\n\n## 5. Naturality\n\nTake \\(p:G'\\to G\\), \\(q:K'\\to K\\), and \\(r:H\\to H'\\). The label map\n\\[\n\\Phi_{p,r}(a)=r\\circ a\\circ p\n\\]\ninduces \\(R(\\Phi_{p,r})\\) by recursively changing labels and taking images of child sets. This is an exact coalgebra map, including when different children acquire the same image.\n\nFor a compatible position \\(w\\), exactness of \\(p,r\\) gives\n\\[\n\\begin{aligned}\n\\theta_{H'}(r(a_w(p(g'))))\n&=r[\\theta_H(a_w(p(g')))]\\\\\n&=\\{r(a_w(p(g''))):g''\\in\\theta_{G'}(g')\\}\\\\\n&\\quad\\cup\\{r(a_v(p(g'))):v\\in T_w\\}.\n\\end{aligned}\n\\]\nThis is (E) for the transformed position. Every descendant of the transformed position is the image of a descendant, so the same calculation applies throughout. The map therefore restricts to\n\\[\nA(p,r):A(G,H)\\to A(G',H').\n\\]\nIdentities and composition follow from the label maps and the functoriality of \\(R\\).\n\nBoth sides of\n\\[\n\\lambda\\bigl(r\\circ f\\circ(p+q)\\bigr)\n=A(p,r)\\circ\\lambda(f)\\circ q\n\\]\nhave label \\(g'\\mapsto r(f(p(g'),q(k')))\\) at \\(k'\\). Uniqueness of the cofree transpose proves the equation. The corresponding equation for uncurrying follows directly by evaluating labels.\n\n## 6. Size and empty games\n\nThe labels form a set \\(S\\); the finite recursively labelled positions form a set \\(R(S)\\); and \\(A(G,H)\\) is a subset.\n\nIf \\(UG=\\varnothing\\), then \\(S\\) is a singleton and (E) is vacuous, so \\(A(G,H)=R(1)\\), the terminal game. If \\(UG\\ne\\varnothing\\) and \\(UH=\\varnothing\\), then \\(S=\\varnothing\\) and \\(A(G,H)=\\varnothing\\). These cases agree with the proposed adjunction.\n\nThe full category is essential: \\(R(1)\\) has infinite carrier. This does not assert closedness of the subcategory of finite-carrier games.\n\n## 7. What remains\n\nThe candidate describes positions, moves, evaluation, currying, and naturality. It does not identify these positions with legal playing strategies or explain player-switching. Nor does it establish the partisan analogue or a construction for arbitrary products lifted to games.\n\nThe next tasks are human review of the proof in the stated category, explicit small examples, a game-theoretic interpretation, and comparison with prior mapping-object constructions. Failure to locate an identical theorem would not establish novelty.", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:36:44.473Z", "updatedInSumire": "2026-10-18T12:42:01.622Z", "legacyRecordedAt": "2026-10-07T16:36:44.473Z", "legacyUpdatedAt": "2026-10-18T12:42:01.622Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "note", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "unassigned", "problemState": "unknown", "resolution": "", "verificationEvidence": { "ai": "", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "Codex", "additions": "" }, "originalQuestion": "", "references": [ { "url": "https://arxiv.org/abs/2510.22886v3", "verification": "not_checked", "service": "arxiv", "identifier": "arXiv:2510.22886v3", "version": "v3", "pinned": true, "correspondence": { "status": "reader_memo_unverified", "assumptions": "", "conclusions": "", "evidence": "", "differences": "", "unchecked": "", "declaration": "" } } ] } ], "noteDependencies": [], "guides": { "arxiv": "https://info.arxiv.org/help/submit/index.html", "formal_conjectures": "https://github.com/google-deepmind/formal-conjectures/blob/main/CONTRIBUTING.md", "mathlib": "https://leanprover-community.github.io/contribute/index.html" }, "search": { "arxiv": "https://arxiv.org/search/", "formal_conjectures": "https://google-deepmind.github.io/formal-conjectures/browse/", "mathlib": "https://leanprover-community.github.io/mathlib4_docs/" } }