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The unit is projective exactly when the tensor category is semisimple
Statement
A tensor category is semisimple if and only if its unit object is projective.
Facts & Assumptions
Given: A tensor category .
Tensoring a projective object with a dualizable object preserves projectivity (Tensoring with a dualizable object preserves projectives).
Semisimple means every object is a finite direct sum of simple objects (Semisimple objects and semisimple abelian categories).
Proof
If is projective, every is projective by [F1]. In this finite-length abelian setting, all objects projective implies that all short exact sequences split, hence every object is a direct sum of its simple factors and is semisimple in the sense of [F2].
Conversely, in a semisimple abelian category every epimorphism splits after decomposing its codomain into simples; thus every object, in particular , is projective.
Depends on
Used by
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Dependency tree · two levels
7 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
- Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Corollary 4.2.13 (standard reference, not scraped)