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    "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.",
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    "owner": {
      "name": "hora",
      "handle": "ryuya"
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    "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.",
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        "id": "goal_71391ac0-c899-41e4-b6ed-e9d825d41001",
        "title": "To understand why Nim-sum works",
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      {
        "id": "goal_4d2683f0-360b-4edd-b553-ca70400f6943",
        "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory",
        "body": "",
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  "notes": [
    {
      "id": "n_sXsnDLGfziJq",
      "number": 42,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/42",
      "title": "Exact completion of games and the Leibniz obstruction",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "## Current result and proof status\n\nThe category of games is not itself a topos; see [the original question](https://sumire.aiformath.jp/p/2610.00003/n/16). The proposed replacement is its exact completion, presented as sheaves on finite games. This note records that construction and a concrete obstruction to its proposed differential structure: the Leibniz law below gives **6 maps on one side and 2 on the other**.\n\nThe exact-completion and Day-tensor arguments are in the linked manuscripts. The finite map counts below are explicit; their interpretation as a sheaf counterexample uses the embedding and closed-tensor results. The open questions below remain separate from those results.\n\n- [Regularity, the site, and the noneffective relation (Japanese)](https://sumire.aiformath.jp/p/2610.00003/main/assets/topos-checks-1-4-proofs-2026-09-26.pdf).\n- [Exact completion as a sheaf topos (Japanese)](https://sumire.aiformath.jp/p/2610.00003/main/assets/explain-map-games-coherent-sheaves-2026-09-26.pdf).\n- [Properties and points of the topos (Japanese)](https://sumire.aiformath.jp/p/2610.00003/main/assets/explain-topos-properties-2026-09-26.pdf).\n- [Conway addition and ex/reg completion (Japanese)](https://sumire.aiformath.jp/p/2610.00003/main/assets/explain-exreg-conway-day-2026-09-26.pdf).\n- [The Day-tensor descent proof (Japanese)](https://sumire.aiformath.jp/p/2610.00003/main/assets/explain-day-convolution-sheaf-2026-09-27.pdf); this supplies the corrected proof required by the preceding manuscript.\n- [The species comparison and Leibniz counterexample (Japanese)](https://sumire.aiformath.jp/p/2610.00003/main/assets/explain-species-nominal-analogy-2026-09-27.pdf).\n\n## Definitions and the exact-completion construction\n\nLet \\(\\mathrm{Games}=\\operatorname{RecCoalg}(P_f)\\), with maps satisfying\n\\(\\operatorname{succ}(f(x))=f[\\operatorname{succ}(x)]\\).\nLet \\(\\mathrm{Games}_{\\mathrm{fin}}\\) be the full subcategory on finite carriers, and let \\(J_{\\mathrm{coh}}\\) be generated by finite jointly surjective families. Put\n\\[\n\\mathcal E=\\operatorname{Sh}(\\mathrm{Games}_{\\mathrm{fin}},J_{\\mathrm{coh}}),\n\\qquad N(X)=\\mathrm{Games}(-,X)|_{\\mathrm{Games}_{\\mathrm{fin}}}.\n\\]\nWrite \\(\\mathrm{Star}_m\\) for a root with \\(m\\) terminal children.\n\nThe construction asserts that Games is regular, with epimorphisms equal to surjections and regular epimorphisms, stable under pullback. It is infinitary extensive but not exact: \\(\\Delta\\cup\\Gamma_\\sigma\\subseteq\\mathrm{Star}_2\\times\\mathrm{Star}_2\\), with \\(\\sigma\\) swapping the two terminal children, is a six-element noneffective equivalence relation. Its ex/reg completion is\n\\[\n\\operatorname{Map}(\\mathrm{Games})\\simeq\\mathcal E.\n\\]\nThe proof uses that every subobject of \\(N(X)\\) comes from a subgame and that \\(N\\) preserves coproducts, surjections, and filtered colimits. The subobject classifier is \\(\\Omega_{\\mathcal E}=N(\\Omega_{\\mathrm{Games}})\\), described using truth-closed hereditarily finite sets.\n\nAn exact-completion object \\((X,R)\\) retains a game together with witnesses of its equivalence relation. A quotient is already represented in Games when its relation is a kernel pair. In particular,\n\\(Q=N(\\mathrm{Star}_2)/(\\Delta\\cup\\Gamma_\\sigma)\\)\nis outside the image of \\(N\\); it has \\(N(\\mathrm{Star}_1)\\) as a quotient, not as a claimed retract.\n\n## Properties asserted by the construction\n\n- The terminal object is \\(N(V)\\), where \\(V\\) is the game of all hereditarily finite sets. It is not quasi-compact, so \\(\\mathcal E\\) is not coherent. The subterminal objects \\(N(V_{<n})\\) give an open cover by coherent topoi\n  \\(\\mathcal E_n\\simeq\\operatorname{Sh}((\\mathrm{Games}_{\\mathrm{fin}})_{<n},J_{\\mathrm{coh}})\\).\n  Thus \\(\\mathcal E\\) has enough points. Its proposed geometric theory combines the coherent theories at each finite height with the infinitary assertion that every position has some finite height.\n- \\(\\mathcal E_1\\simeq\\mathrm{Set}\\) is a dense open point and \\(\\operatorname{Sh}_{\\neg\\neg}(\\mathcal E)\\simeq\\mathrm{Set}\\). The point is given by the terminal-position functor. The one-position no-move game is projective; so is every finite discrete no-move game. No uniqueness or classification of connected projectives is claimed here.\n- \\(\\mathcal E\\) is not Boolean: the terminal-position subgame of \\(\\mathrm{Star}_1\\) has no complement. It is not localic: \\(N(\\mathrm{Star}_2)\\) is not covered by subterminal objects. Its proposed localic reflection is the presheaf topos on \\((V_\\omega,\\in^+)\\).\n- A proposed family of points, indexed by nonempty perfect Stone spaces \\(S\\), is\n  \\(F_S(X)=\\{(x,e):\\operatorname{succ}(x)\\text{ consists of terminal positions},\\ e:S\\twoheadrightarrow\\operatorname{succ}(x)\\text{ continuous}\\}\\).\n  This is not a classification of all points.\n\n## Conway addition and the tensor\n\nConway addition \\(+\\) is not left exact: the comparison involving\n\\((\\mathrm{Star}_2\\times_{\\mathrm{Star}_1}\\mathrm{Star}_2)+\\mathrm{Star}_1\\)\nand\n\\((\\mathrm{Star}_2+\\mathrm{Star}_1)\\times_{\\mathrm{Star}_1+\\mathrm{Star}_1}(\\mathrm{Star}_2+\\mathrm{Star}_1)\\)\nhas respectively \\(7\\) and \\(7\\cdot127\\) roots. Therefore its descent to the completion cannot simply use the 2-functoriality of ex/reg completion on left-exact functors.\n\nThe proposed alternative is the sheafified Day tensor \\(\\oplus\\) on \\(\\mathcal E\\). It is symmetric monoidal closed, with \\(N\\) strong monoidal and\n\\(N(A)\\oplus N(B)\\cong N(A+B)\\).\nFor finite \\(A,B\\), their presheaf Day tensor is already the representable sheaf \\(N(A+B)\\). The closed structure uses a separate descent argument proving that \\(F(-+B)\\) is a sheaf; cover preservation alone does not suffice. No claim is made here that arbitrary presheaf Day convolution is already a sheaf. See [Reflected Day convolution](https://sumire.aiformath.jp/p/2610.00003/n/37) for the distinction between convolution and its reflection.\n\nSpecies with Cauchy product and nominal sets with separated product suggest comparisons through Day convolution on finite objects; those analogies do not by themselves identify the game tensor or its differential structure.\n\n## The proposed Leibniz law fails for this operator\n\nThe symbol \\(\\sqcup\\) denotes coproduct. Use the sheafified Day tensor \\(\\oplus\\) described above and set\n\\[\n\\begin{gathered}\nD=[N(\\mathrm{Star}_1),-]_{\\oplus},\\\\\n(DF)(A)=F(A+\\mathrm{Star}_1).\n\\end{gathered}\n\\]\nHere the tensor unit is \\(I=N(\\{*\\})\\), where \\(\\{*\\}\\) is the one-position game with no moves (the unit for Conway addition); \\(\\mathrm{Star}_1\\) is not the tensor unit.\n\nThe evaluation formula follows from the internal-hom adjunction and Yoneda: for a finite game \\(A\\),\n\\[\n(DF)(A)\\cong\\operatorname{Hom}_{\\mathcal E}(N(A),[N(\\mathrm{Star}_1),F]_{\\oplus})\n\\cong\\operatorname{Hom}_{\\mathcal E}(N(A+\\mathrm{Star}_1),F)\n\\cong F(A+\\mathrm{Star}_1).\n\\]\nThis uses the stated closed Day tensor and its tensor formula on representables.\n\nTake \\(F=G=N(\\mathrm{Star}_1)\\). A coalgebra morphism preserves height. Thus\n\\(\\mathrm{Games}(A+\\mathrm{Star}_1,\\mathrm{Star}_1)\\) has one element when every position of \\(A\\) is terminal, and is empty otherwise. This is exactly \\(N(\\{*\\})(A)\\), so \\(DF=DG=I\\). Using the stated strong monoidality and coproduct preservation of \\(N\\), the proposed law\n\\[\nD(F\\oplus G)\\cong (DF\\oplus G)\\sqcup(F\\oplus DG)\n\\]\nwould therefore identify, at \\(A=\\mathrm{Star}_2\\),\n\\[\n\\mathrm{Games}(\\mathrm{Star}_2+\\mathrm{Star}_1,\n                    \\mathrm{Star}_1+\\mathrm{Star}_1)\n\\quad\\text{and}\\quad\n\\mathrm{Games}(\\mathrm{Star}_2,\n                    \\mathrm{Star}_1\\sqcup\\mathrm{Star}_1).\n\\]\n\nThe first set has \\(6\\) elements. Its source has one height-two root, three height-one positions, and two terminal positions; the target has one height-two root, two height-one positions, and one terminal position. Height preservation fixes the root and terminal images. Each of the three height-one positions can map to either target position, and the coalgebra equation\n\\(\\operatorname{succ}(f(x))=f[\\operatorname{succ}(x)]\\) at the root requires both images to occur. There are \\(2^3-2=6\\) such maps.\n\nThe second set has \\(2\\) elements. The root of \\(\\mathrm{Star}_2\\) chooses one of the two coproduct components; both terminal children must then map to that component's terminal position. Hence the two evaluated sheaves cannot be isomorphic.\n\nThis refutes the displayed Leibniz isomorphism for this particular \\(D\\) and \\(\\oplus\\). It does not refute the successor formula defining Conway addition, or every possible differential structure on games. The remaining question is a replacement rule accounting for the four additional maps in this example. The finite map counts can be checked directly from the coalgebra equation; the identification with these sheaf values still uses the Day-tensor and embedding properties stated above.\n\n## Remaining questions\n\n1. Find a replacement for the failed Leibniz law, accounting for the four additional maps in this example. Compare [Differential structures of games](https://sumire.aiformath.jp/p/2610.00003/n/6) and the differential 2-rig axioms.\n2. Classify the points of \\(\\mathcal E\\). Do Stone-game constructions exhaust them? Is \\(\\mathcal E\\) a presheaf topos? The expectation recorded here is negative, not a proof.\n3. Determine when the tensor internal hom\n   \\[\n   [NX,NY]_{\\oplus}(A)=\\int_B\\operatorname{Set}(\\mathrm{Games}(B,X),\\mathrm{Games}(A+B,Y))\n   \\]\n   lies in the image of \\(N\\); compare [Internal hom for Conway addition](https://sumire.aiformath.jp/p/2610.00003/n/4).\n4. Is there an ex/reg analogue of Menni's ex/lex criterion using weak dependent products and a generic proof? The proposed sufficient conditions for a Grothendieck-topos completion are regularity, infinitary extensivity, and a small generating family. The earlier comparison with Worrell, Corollary 5.5.7, still requires its precise hypotheses to be matched. The remembered realizability comparison \\(\\mathrm{Eff}\\simeq\\mathrm{Asm}_{\\mathrm{ex/reg}}\\) is not used as evidence here.\n\nThe distinction between ex/reg and ex/lex matters: ex/reg completion is idempotent on exact categories, whereas ex/lex completion is not generally idempotent. The existing proof manuscripts and their hypothesis checks are needed before the general criterion above can be reused as a theorem.",
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