In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object ${\displaystyle \bot }$. The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality.

Let C be a symmetric monoidal closed category ${\displaystyle \langle C,\otimes ,I,\Rightarrow \rangle }$. For any pair of objects, in particular A and ${\displaystyle \bot }$, there exists a morphism

defined as the image by the bijection defining the monoidal closure

of the evaluation map:

where ${\displaystyle \gamma }$ is the symmetry of the tensor product. An object ${\displaystyle \bot }$ of the category C is called dualizing when the associated morphism ${\displaystyle \partial _{A,\bot }}$ is an isomorphism for every object A of the category C.

Equivalently, a *-autonomous category is a symmetric monoidal category C together with a functor ${\displaystyle (-)^{*}:C^{\mathrm {op} }\to C}$ such that for every object A there is a natural isomorphism ${\displaystyle A\cong {A^{**}}}$, and for every three objects A, B and C there is a natural bijection

The dualizing object of C is then defined by ${\displaystyle \bot =I^{*}}$. The equivalence of the two definitions is shown by identifying ${\displaystyle A^{*}=A\Rightarrow \bot }$.

Compact closed categories are *-autonomous, with the monoidal unit as the dualizing object. Conversely, if the unit of a *-autonomous category is a dualizing object then there is a canonical family of maps