{
  "id": "20261026-0002",
  "version": "v1",
  "title": "Monoidal Token Spaces: Initial Sources and Tensor Adjoints",
  "authors": [
    {
      "name": "Wuming Pan",
      "affiliation": "Sichuan University"
    }
  ],
  "date": "2026-10-26",
  "type": "Research paper",
  "pages": 24,
  "subjects": [
    "Category theory",
    "Token Space",
    "Monoidal categories"
  ],
  "summary": "A criterion for preservation of initial sources, transport for free-monoid presentations, and a classification of tensor factors admitting right adjoints.",
  "abstract": "A monoidal token space is the predicate category of a monoid-valued record functor, equipped with its tensor over coproduct. We study general record functors, free-monoid presentations, and the classical Token Space of words over a carrier and a fixed core. Tensoring by a fixed factor preserves every nonempty initial source exactly when left factors are unique and the product-image inclusion is cartesian. A free-group example separates these conditions. For free-monoid presentations whose generator functor preserves two specified coproduct pullback squares, tensoring preserves every one-arrow initial lift, so only uniqueness remains. Polynomial generator functors satisfy this hypothesis; freeness alone does not suffice. In classical Token Space, the criterion becomes unambiguity of a language product, and compatible factors have copretopological forgetful functors. Independently, in every monoidal token space, tensoring has a right adjoint exactly for factors with empty carrier. These factors form a powerset quantale whose residuals act on the whole category. Ordered signatures delimit the constructions, and finite automata give effective tests for regular heaps.",
  "release_note": "The v1 publication imprint was updated on 26 October 2026 to use the date-and-sequence identifier requested by the author. The former random identifier remains a permanent alias to this record. Scholarly text, body placement and pagination are unchanged. The mathematical manuscript remains revision V2.1.",
  "provenance": "AI-assisted drafting, revision and computational checks. Complete mathematical proofs are presented in the manuscript.",
  "verification": "The source archive includes finite verification programs and recorded results. These bounded checks support reproducibility and do not replace the general proofs. No external peer-review status is claimed.",
  "related": "20261026-0001",
  "namespace": "core",
  "number": 2,
  "file_revision": "date-id-20261026",
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  "identifier_profile": "1.1",
  "registration_date": "2026-10-26",
  "sequence": 2,
  "key": "core/20261026-0002",
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  "version_pid": "1id1:core:20261026-0002@v1",
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  "version_url": "https://1id1.net/id/core/20261026-0002/v1/",
  "language": "en",
  "status": "preprint",
  "rights": "Copyright retained by the author. No additional reuse licence specified.",
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