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  "exportedAt": "2026-11-04T08:27:11.052Z",
  "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",
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  "project": {
    "id": "2609.00007",
    "url": "https://sumire.aiformath.jp/p/2609.00007",
    "title": "Rieg theory",
    "tagline": "Semirings with exponentiation, connecting arithmetic, logic, and categorical counting.",
    "visibility": "public",
    "owner": {
      "name": "hora",
      "handle": "ryuya"
    },
    "origin": "",
    "source": "kobin:u_EMlmNonOMJFd:slogan:riegs",
    "goal": "Develop a coherent theory of semirings with exponentiation and use it to connect finite and profinite arithmetic, logic, enumerative combinatorics, and category theory.",
    "narrative": "A rieg is a commutative semiring equipped with exponentiation satisfying the usual exponent laws, including \\(x^0=1\\) and hence \\(0^0=1\\). This project studies the structure itself and the examples it connects.\n\nThe research programme is described in [note 7](https://sumire.aiformath.jp/p/2609.00007/n/7). The current questions concern [the subrieg generated by nonnegative rationals](https://sumire.aiformath.jp/p/2609.00007/n/1), [game interpretations of the free rieg](https://sumire.aiformath.jp/p/2609.00007/n/3), [functions represented by unary rieg terms](https://sumire.aiformath.jp/p/2609.00007/n/4), and [finite-valued presheaf categories](https://sumire.aiformath.jp/p/2609.00007/n/5).\n\nOne concrete result narrows the topological problem: [a compact Hausdorff topological rieg with cancellative addition is trivial](https://sumire.aiformath.jp/p/2609.00007/n/9). The question [whether a nontrivial connected compact Hausdorff rieg exists](https://sumire.aiformath.jp/p/2609.00007/n/2) therefore reduces to the noncancellative case, where it has a positive answer.\n\nFor finite-valued presheaves, essentially finite slices give a sufficient condition for a topos. [A partial answer](https://sumire.aiformath.jp/p/2609.00007/n/5) proposes an exact criterion stated in terms of finite presheaves. [The right Kan extension approach](https://sumire.aiformath.jp/p/2609.00007/n/8) is absorbed into that note, with the coreflection \\(X\\mapsto X_{\\mathrm{fin}}\\) onto locally finite presheaves as its comonad.\n\nThe next step is concrete: give a criterion for finite presheaf topoi in terms of the indexing category alone.",
    "history": "",
    "dates": {
      "inspiration": null,
      "sourceRecorded": null,
      "evidence": null,
      "registeredInSumire": null,
      "updatedInSumire": "2026-10-30T14:34:57.980Z",
      "legacyRecordedAt": "2026-03-30T00:00:00.000Z",
      "legacyUpdatedAt": "2026-10-30T14:34:57.980Z"
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    {
      "id": "n_pVrpVp9EBz36",
      "number": 5,
      "url": "https://sumire.aiformath.jp/p/2609.00007/n/5",
      "title": "Finite presheaf topoi",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "kobin:u_EMlmNonOMJFd:idea:tk8psy",
      "body": "## Question\n\nFor a small category \\(\\mathcal C\\), when is the category\n\\[\n\\mathbf{fPSh}(\\mathcal C)=[\\mathcal C^{\\mathrm{op}},\\mathbf{FinSet}]\n\\]\nof pointwise finite-valued presheaves an elementary topos, or locally cartesian closed?\n\nThe motivation is enumerative combinatorics and Burnside riegs. This question was discussed with Jérémie Marquès.\n\n## Known sufficient condition\n\nFinite limits are computed pointwise. If every slice \\(\\mathcal C/c\\) is essentially finite, then the usual presheaf subobject classifier and exponentials are finite-valued, so \\(\\mathbf{fPSh}(\\mathcal C)\\) is an elementary topos. This includes finite categories and groups regarded as one-object categories.\n\nHere are the two relevant ambient formulas. The value \\(\\Omega(c)\\) is the set of sieves on \\(c\\), so the ambient classifier is finite-valued exactly when there are finitely many sieves on each object. For finite-valued \\(F,G\\),\n\\[\n(G^F)(c)=\\operatorname{Nat}(y(c)\\times F,G),\n\\]\nso the ambient exponential is finite-valued exactly when these sets are finite for every \\(c\\).\n\nThese criteria describe when the ambient constructions stay in the finite-valued subcategory. They are not, without an additional argument, a necessary-and-sufficient classification of when that subcategory has its own classifier and exponentials.\n\n## Remaining question and next step\n\nFind a general criterion beyond essentially finite slices. For a proposed criterion, distinguish closure under the ambient presheaf constructions from existence of the required universal objects inside the finite-valued category.\n\n[Note 8](https://sumire.aiformath.jp/p/2609.00007/n/8) records the right Kan extension approach. Its first missing ingredient is the precise adjunction or comonad that would implement the reduction.\n\n## Source\n\n[Original problem 6.0.1](https://github.com/hora-algebra/ryuya-hora-homepage/blob/1b8b74c2b3447bdc744a0cac256c5d5806588adc/scripts/site.js#L941-L951).",
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        "updatedInSumire": "2026-10-30T14:29:33.421Z",
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        "legacyUpdatedAt": "2026-10-30T14:29:33.421Z"
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      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "conjecture",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "open-questions",
      "problemState": "answer-proposed",
      "resolution": "Partial answer. For a presheaf \\(X\\) let \\(X_{\\mathrm{fin}}\\subseteq X\\) consist of the elements that generate a finite-valued subpresheaf; \\(X\\mapsto X_{\\mathrm{fin}}\\) is the left exact coreflection onto locally finite presheaves (for a free monoid it is \\(h_*\\) of Hora, *Topoi of automata I*, arXiv:2411.06358, Lemma 2.20). Every map from a finite presheaf into \\(X\\) lands in \\(X_{\\mathrm{fin}}\\), and a finite presheaf representing \\(\\operatorname{Hom}(-,X)\\) on \\(\\mathbf{fPSh}(\\mathcal C)\\) is necessarily \\(X_{\\mathrm{fin}}\\). Hence: (1) With \\(\\Omega\\) and \\(G^F\\) computed in all presheaves, \\(\\mathbf{fPSh}(\\mathcal C)\\) has a subobject classifier iff \\(\\Omega_{\\mathrm{fin}}\\) is finite-valued, and has an exponential of \\(G\\) by \\(F\\) iff \\((G^F)_{\\mathrm{fin}}\\) is finite-valued; these are then the universal objects. (2) It is a topos iff for every object \\(c\\) and finite \\(X\\) only finitely many \\(R\\subseteq y(c)\\times X\\) have, for every \\(d\\), finitely many pullbacks \\((f\\times X)^*R\\) along \\(f:d\\to c\\). (3) If all hom-sets are finite, it is a topos iff every finite presheaf on each slice \\(\\mathcal C/c\\) has finitely many subpresheaves. Essentially finite slices are not necessary: for an infinite group \\(G\\) and \\(M=G\\sqcup\\{0\\}\\), \\(\\mathbf{fPSh}(M)\\) is a topos (\\(M\\) has three right ideals and \\(\\operatorname{Sub}(M\\times X)\\) has at most \\(4^{|X|}\\) elements), although \\(\\operatorname{End}(0)=M\\) in \\(M/{\\ast}\\). A subobject classifier is not enough: on nonempty finite sets \\(\\Omega\\) is finite-valued, but the presheaf of equivalence relations has the subpresheaves “at most \\(k\\) classes” for all \\(k\\), so it has no power object. For the monoid \\((\\mathbb N,+)\\) there is no subobject classifier, since every ideal \\(n+\\mathbb N\\) lies in \\(\\Omega_{\\mathrm{fin}}\\) (nor is it cartesian closed: [Notes on Rieg Theory](https://sumire.aiformath.jp/p/2609.00007?view=main), Example 4.1.8). Open: a criterion in terms of \\(\\mathcal C\\) alone, local cartesian closedness, and whether for finite hom-sets the condition in (3) forces essentially finite slices. Not yet verified by a human.",
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      "ai": {
        "use": "assisted",
        "tools": "Codex, Claude",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
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