ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Typed object, 1-morphism, and 2-morphism defect composition

Example

In a chosen 2+1-dimensional defect theory modeled by a fusion 2-category, label objects $A, B, C$ as surface defects, 1-morphisms $f : A \to B$ and $g : B \to C$ as line junctions, and 2-morphisms $α : f \Rightarrow f^{'}$ as point junctions. This is a type dictionary conditional on that chosen theory, not a theorem identifying every algebraic object with a physical defect.

Facts & Assumptions

Given: A weak 2-category with the source, target, and composition data in Weak linear 2-category.

[F1]

Fusion 2-category: Tensor product is part of the monoidal structure; composition is the separately typed 1-morphism composition of the underlying 2-category.

[F2]

Weak linear 2-category: A 1-morphism has a specified source and target, and horizontal composition is defined only for matching endpoints.

Verification

1.1F2givenalgebra

If $f : A \to B$ and $g : B \to C$ , their categorical composite has type $g \circ f : A \to C$ . A point junction $α : f \Rightarrow f^{'}$ lies in the single hom category $B (A, B)$ , and vertical composition is defined only when the target 1-morphism of one 2-cell is the source 1-morphism of the next.

2.1F1F2step 1.1algebra∎

For $α : f \Rightarrow f^{'}$ in $B (A, B)$ and $β : g \Rightarrow g^{'}$ in $B (B, C)$ , functorial horizontal composition gives $β * α : g \circ f \Rightarrow g^{'} \circ f^{'}$ in $B (A, C)$ . The object tensor $A \otimes C$ has a different type from the composite $g \circ f$ ; no physical realization or identification of these operations follows from this typing calculation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Décoppet and Yu, Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories, §1 (standard reference, not scraped)
Douglas and Reutter, Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds, §2.1 (standard reference, not scraped)