Alphabeta MathHow statement and proof provenance work
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The left unitor of a tensor product is determined by the associator
Statement
In any monoidal category,
for all objects .
Facts & Assumptions
Given: A monoidal category and objects .
The monoidal-category axioms are the pentagon and triangle from Monoidal category.
Proposition 2.2.4 of EGNO proves that, under those axioms and with the same associator orientation as this page, the left unitor satisfies
Proof
The hypotheses of [F1] are exactly those in [L1], so [F1] applies to the present monoidal category.
Therefore the displayed identity holds for every pair .
Depends on
Used by
- The two unitors agree on the tensor unit Corollary
- FALSE: the standard unit-constraint identities must all be imposed as independent axioms False statement
- How Mac Lane's original coherence conditions reduce to this page's two axioms Remark
- The unit-constraint redundancies are cited mathematically through EGNO Remark
Dependency tree · two levels
4 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, Proposition 2.2.4 (standard reference, not scraped)