math.CT — Category Theory
Acyclic Networks: Causal Comparisons, Convex Minors, and Gauge Descent
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.