LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

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.

Normalized comparison isomorphisms are transitive

Statement

Let $t, u, w$ be signed words representing the same braid. Then $γ_{u, w} \circ γ_{t, u} = γ_{t, w} in {Hom}_{K^{b}} (F (t), F (w)) = Q \cdot [γ_{t, w}] .$ In particular $γ_{t, t} = i d$ and $γ_{u, t} γ_{t, u} = i d$ , so the maps $γ$ form a transitive system of homotopy equivalences between the word complexes of a fixed braid; consequently the multiplication comparisons of the next theorem are well defined on chosen representatives.

Facts & Assumptions

Given: Signed words $t, u, w$ with the same product, the word complexes $F (t), F (u), F (w)$ of The Rouquier complex of a braid word, and the normalized maps $γ_{t, u}, γ_{u, w}, γ_{t, w}$ of Derived comparisons give unique normalized homotopy maps.

[F1]

Uniqueness. For words $a, b$ with the same product, ${Hom}_{K^{b}} (F (a), F (b))$ is one-dimensional in internal degree $0$ , the localization map to ${Hom}_{D^{b}}$ is an isomorphism, and $γ_{a, b}$ is the unique homotopy class whose derived image is the comparison $c_{a, b}$ of the graph models. (Derived comparisons give unique normalized homotopy maps)

[F2]

Transitivity of the comparisons. The derived comparisons satisfy $c_{u, w} c_{t, u} = c_{t, w}$ and $c_{t, t} = i d$ ; they are the multiplication isomorphisms of the words through the standard graph bimodules. (Canonical comparisons between standard graph tensor products)

Proof

technique · direct

1.1F1

The composite $γ_{u, w} \circ γ_{t, u}$ is an element of ${Hom}_{K^{b}} (F (t), F (w))$ , which by [F1] is one-dimensional in internal degree $0$ ; its derived image is $c_{u, w} c_{t, u}$ because localization is a functor on the homotopy categories in which the $γ$ become isomorphisms.

2.1F1F2step 1.1

By [F2] $c_{u, w} c_{t, u} = c_{t, w}$ , which is the derived image of $γ_{t, w}$ by [F1]; two elements of the one-dimensional space with the same nonzero derived image coincide, so $γ_{u, w} γ_{t, u} = γ_{t, w}$ .

3.1F1F2step 2.1∎

Taking $u = t = w$ and using $c_{t, t} = i d$ gives $γ_{t, t} = i d$ by the same uniqueness argument; then $γ_{t, u}$ and $γ_{u, t}$ are mutually inverse homotopy equivalences because both composites equal the corresponding identity maps, and the identity is the unique degree-zero endomorphism class whose derived image is the normalized identity comparison.

Depends on

Used by

Dependency tree · two levels

24 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

Raphaël Rouquier, Categorification of the braid groups, arXiv:math/0409593v1 (30 September 2004), §3 "The 2-braid group" (standard reference, not scraped)