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Dualization in a multitensor category is exact
Statement
For a multitensor category, the chosen left-dual functor is exact.
Facts & Assumptions
Given: A multitensor category .
is an abelian rigid monoidal category (Tensor and multitensor categories).
Chosen left duals define a contravariant anti-monoidal functor (Left duality is a contravariant antimonoidal functor).
An equivalence between abelian categories is exact (An equivalence between abelian categories is exact).
Proof
Rigidity makes the contravariant functor of [L1] a duality: its quasi-inverse is the chosen right-dual functor. Thus it is an equivalence .
Both source and target are abelian, so [L2] makes this equivalence exact. Equivalently, a short exact sequence is carried, with arrows reversed, to a short exact sequence.
Depends on
Used by
Dependency tree · two levels
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Sources
- Etingof, Gelaki, Nikshych, Ostrik, Tensor Categories, Proposition 4.2.9 (standard reference, not scraped)