1id1Abstract
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.
Cite this version
Wuming Pan. Monoidal Token Spaces: Initial Sources and Tensor Adjoints. 1id1, v1, 26 October 2026. 1id1:core:20261026-0002@v1. https://1id1.net/id/core/20261026-0002/v1/
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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.
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