nLab weak (star)-autonomous category

Idea

Some symmetric monoidal categories have a notion of duality without being a *-autonomous category. However, they can be a weak **-autonomous category.

Definition

Definition

A weak **-autonomous category is a symmetric closed monoidal category C,,I,\langle C,\otimes, I,\multimap\rangle with an object \bot such that the canonical morphism

d A:A(A), d_A: A \to (A \multimap \bot) \multimap \bot ,

which is the transpose of the evaluation map

ev A,:(A)A, ev_{A,\bot}: (A \multimap \bot) \otimes A \to \bot ,

is a monomorphism for all AA. (Here, \multimap denotes the internal hom.)

A **-autonomous category is a weak **-autonomous category such that for every object AA, d A:A(A)d_{A}: A \to (A \multimap \bot) \multimap \bot is an isomorphism.

A strict weak **-autonomous category is a weak **-autonomous category such that for at least one object AA, d Ad_{A} is not an isomorphism. Equivalently, a strict weak **-autonomous category is a **-autonomous category which is not a **-autonomous category.

Example

  • For every field 𝕂\mathbb{K}, the category Vec 𝕂Vec_{\mathbb{K}} is a strict weak **-autonomous category.
  • For every commutative ring RR, the category Mod RMod_{R} is a strict weak **-autonomous category.

*-autonomous category

closed monoidal category

Last revised on August 1, 2022 at 14:56:16. See the history of this page for a list of all contributions to it.