False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-04

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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.
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FALSE: a braiding suffices to define a trace

Statement

A braiding on a monoidal category, without rigidity, suffices to define a categorical trace.

Facts & Assumptions

Given: The symmetric monoidal category ${V e c t}_{k}$ of all vector spaces and its infinite-dimensional object $k [x]$ .

[L1]

The trace formulas require evaluation and coevaluation maps (What is needed before a trace can be written).

[L2]

The symmetric monoidal category of all vector spaces has an infinite-dimensional object with no categorical dual (Not every monoidal category is rigid).

Refutation

technique · direct

1.1

The usual symmetry makes ${V e c t}_{k}$ braided, but [L2] shows that its object $k [x]$ has no evaluation and coevaluation satisfying the zig-zag identities.

givenL2

2.1

Consequently the trace composites described in [L1] cannot even be formed for $k [x]$ . The braiding supplies swaps but does not supply the missing duality maps.

L1step 1.1

3.1

Therefore braiding without rigidity does not suffice to define categorical trace. The statement is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Sections 8.9-8.10 (standard reference, not scraped)