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Acyclic Networks: Causal Comparisons, Convex Minors, and Gauge Descent

Primarily human-written textHuman understanding: all partsmath.CT — Category Theorymath.CO — Combinatoricsmath.QA — Quantum Algebra

Contributed by Zhenyu Sun ↗

Version 1 / Nov 05, 2026 / CC BY 4.0

Abstract

We study natural transformations, minors, and symmetric monoidal decorations of finite directed acyclic networks. For any weakly connected small source category, a natural transformation into a free acyclic path category is determined by one component. If more than one transformation exists, common prefixes and suffixes give a canonical factorization through two constant functors and identify the transformation set with a single set of directed paths. Consequences include boundary rigidity, a decision algorithm, and a fundamental-group obstruction to non-uniqueness. For fusion–embedding minors, right fillings exist exactly when the graph pushout is acyclic. A support invariant detects a nonexact minor under the full gauge congruence, while path-convex embeddings admit two-sided fillings. Inverse-image pullbacks give canonical partial-map normal forms, a scanning algorithm, and a coherent presentation of the cartesian quotient. For open networks, wire splicing gives units, and diagram groupoids satisfy descent by homotopy pullback over the interface groupoid, including through-wires. This descent is invariant under strong symmetric monoidal equivalence and compatible with admissible event contractions within each piece. Finite-field examples yield orbit classifications, stabilizer weights, and boundary convolution formulas. Roof-refinement spaces realize spheres whose higher homotopy is lost under groupoid completion.

Provenance statement

The manuscript was primarily human-written. OpenAI Codex was used for language editing, LaTeX formatting, and independent consistency checks. Exact finite computations were run in Python. The contributor has checked and understood all parts of the contribution.

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Version history

  1. v1 · 36 pagesSubmitted by Zhenyu SunInitial depositCurrentNov 05, 2026