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 $\langle C,\otimes, I,\multimap\rangle$ with an object $\bot$ such that the canonical morphism

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

which is the transpose of the evaluation map

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

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

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

A strictly weak $*$-autonomous category is a weak $*$-autonomous category such that for at least one object $A$, $d_{A}$ is not an isomorphism. Equivalently, a strictly weak $*$-autonomous category is a weak $*$-autonomous category which is not a $*$-autonomous category.

Example

• For every field $\mathbb{K}$, the category $Vec_{\mathbb{K}}$ is a strict weak $*$-autonomous category.
• For every commutative ring $R$, the category $Mod_{R}$ is a strict weak $*$-autonomous category.

Related concepts

*-autonomous category

closed monoidal category