{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:27:09.631Z",
  "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": "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"
    },
    "goals": [],
    "goalDependencies": []
  },
  "notes": [
    {
      "id": "n_y5PlwhcXQ9m1",
      "number": 8,
      "url": "https://sumire.aiformath.jp/p/2609.00007/n/8",
      "title": "A right Kan extension approach to finite presheaves",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "kobin:u_EMlmNonOMJFd:idea:pcsyhq",
      "body": "## Aim\n\nFind a criterion for \\([\\mathcal C^{\\mathrm{op}},\\mathbf{FinSet}]\\) to be an elementary topos or locally cartesian closed, as asked in [Finite presheaf topoi](https://sumire.aiformath.jp/p/2609.00007/n/5).\n\n## Proposed approach\n\nUse a left-exact comonad to reduce the question to existence of suitable right Kan extensions. The motivating idea is to understand which indexing categories have limits for every finite-set-valued diagram, then impose corresponding conditions on comma categories.\n\nThis is an approach to formulate, not an established reduction. A particular comonad and adjunction have not yet been specified. In particular, no right adjoint to the inclusion of finite-valued presheaves into all presheaves is assumed here.\n\n## Next step\n\nSpecify the categories, functors, and adjunction involved. Write the proposed right Kan extension pointwise as a limit, then prove both its finiteness and the implication for exponentials or the subobject classifier. Compare the resulting condition with the existing sufficient condition that every slice \\(\\mathcal C/c\\) is essentially finite.\n\nThe underlying finite-presheaf question was discussed with Jérémie Marquès.",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
        "evidence": null,
        "registeredInSumire": null,
        "updatedInSumire": "2026-10-30T14:31:08.017Z",
        "legacyRecordedAt": "2026-08-16T00:00:00.000Z",
        "legacyUpdatedAt": "2026-10-30T14:31:08.017Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "conjecture",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "unassigned",
      "problemState": "withdrawn",
      "resolution": "Absorbed into [[2609.00007/5]]. The left exact comonad can be taken to be \\(X\\mapsto X_{\\mathrm{fin}}\\), the coreflection onto locally finite presheaves (elements generating finite-valued subpresheaves). Every map from a finite presheaf factors through it, so the subobject classifier and the exponentials of \\(\\mathbf{fPSh}(\\mathcal C)\\), when they exist, are \\(\\Omega_{\\mathrm{fin}}\\) and \\((G^F)_{\\mathrm{fin}}\\); the resulting criterion is in the resolution of that note. Not yet verified by a human.",
      "verificationEvidence": {
        "ai": "",
        "lean": "",
        "human": ""
      },
      "aiClaimModel": "",
      "formalization": "unknown",
      "humanReview": "unreviewed",
      "verification": "unverified",
      "ai": {
        "use": "assisted",
        "tools": "Codex, Claude",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
    }
  ],
  "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/"
  }
}