{ "schemaVersion": "1.0", "repository": "1id1", "url": "https://1id1.net", "updated": "2026-10-26", "records": [ { "id": "20261026-0001", "version": "v1", "title": "Categorical Studies of Token Structures: A Historical Perspective", "authors": [ { "name": "Wuming Pan", "affiliation": "Sichuan University" } ], "date": "2026-10-26", "type": "Survey", "pages": 10, "subjects": [ "Category theory", "Token Space", "History of mathematics" ], "summary": "A historical map of the relational, topological, algebraic and linguistic ideas behind Token Space, separating inherited structure from questions for further research.", "abstract": "A token structure is a set of carrier elements together with a chosen collection of finite records over those elements and a fixed symbol alphabet, where structure-preserving maps are those functions that carry recorded tokens to recorded tokens. This survey traces the categorical treatment of such structures from four historical sources: categories of relational structures and their homomorphisms; topological and concrete categories, where morphisms are underlying functions subject to a fibrewise admissibility condition; algebraic and coalgebraic accounts of terms, term graphs, sharing and sequences; and the categorical linguistics descending from Lambek's syntactic calculus. We then describe how the Token Space framework of Pan assembles these ingredients into a single category whose objects are carriers with heaps of finite words over carrier and core symbols, and whose fibres are free quantales rather than products of powersets. For each component we state what the earlier theory already provides, what the Token Space presentation adds, and which structural questions remain open. The aim is a map of priorities and dependencies, not an evaluation of applications to machine learning.", "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.", "provenance": "AI-assisted preparation of the publication files. The manuscript includes its own references and attribution of earlier results.", "verification": "Survey manuscript. Registration does not constitute peer review or verification of every historical priority claim.", "related": "20261026-0002", "namespace": "core", "number": 1, "file_revision": "date-id-20261026", "aliases": [ "1id1:core:01m5w1wtvzfsj8gz868kpybwba" ], "identifier_profile": "1.1", "registration_date": "2026-10-26", "sequence": 1, "key": "core/20261026-0001", "schemaVersion": "1.0", "pid": "1id1:core:20261026-0001", "version_pid": "1id1:core:20261026-0001@v1", "url": "https://1id1.net/id/core/20261026-0001/", "version_url": "https://1id1.net/id/core/20261026-0001/v1/", "language": "en", "status": "preprint", "rights": "Copyright retained by the author. No additional reuse licence specified.", "relations": [ { "predicate": "relatedTo", "target": "1id1:core:20261026-0002" } ], "externalIdentifiers": [], "files": [ { "name": "paper.pdf", "url": "https://1id1.net/records/core/20261026-0001/v1/paper.pdf?sha256=cc3074fb70886ae4", "media_type": "application/pdf", "bytes": 412212, "sha256": "cc3074fb70886ae4f557bc1b395c0350367fe6bd6ff47dff15eac043c6d070a8" }, { "name": "source.zip", "url": "https://1id1.net/records/core/20261026-0001/v1/source.zip?sha256=064204b8dcd27baf", "media_type": "application/zip", "bytes": 13225, "sha256": "064204b8dcd27baff52b3eebabddd798f708dc84a241af443e3d385ae1cea372" } ] }, { "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", "aliases": [ "1id1:core:01m5w1wtw0eftsxmq9ks5yg61m" ], "identifier_profile": "1.1", "registration_date": "2026-10-26", "sequence": 2, "key": "core/20261026-0002", "schemaVersion": "1.0", "pid": "1id1:core:20261026-0002", "version_pid": "1id1:core:20261026-0002@v1", "url": "https://1id1.net/id/core/20261026-0002/", "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.", "relations": [ { "predicate": "relatedTo", "target": "1id1:core:20261026-0001" } ], "externalIdentifiers": [], "files": [ { "name": "paper.pdf", "url": "https://1id1.net/records/core/20261026-0002/v1/paper.pdf?sha256=103f2ea6cf6e12e3", "media_type": "application/pdf", "bytes": 392080, "sha256": "103f2ea6cf6e12e356c4c1279dc388a8efe00ac015fa96f935c1d23c624e1d4a" }, { "name": "source.zip", "url": "https://1id1.net/records/core/20261026-0002/v1/source.zip?sha256=97d45e146422d535", "media_type": "application/zip", "bytes": 42563, "sha256": "97d45e146422d535cb02766070ca2146aa132701cf63e5367d1cbbcc08585c95" } ] } ] }