nLab
rigid monoidal category

Idea

A rigid (monoidal) category, also called an autonomous (monoidal) category is a kind of category with duals. Specifically, all of its objects are dualisable on both the left and the right.

Definition

A monoidal category is rigid if every object has duals on both sides. If only one type of dual exists, we speak of left rigid (or left autonomous) or right rigid categories.

Conventions differ regarding which type of duals are which. One convention is as follows: a right dual of an object $V$ in a monoidal category $M$ is an object $V^{*}$ equipped with unit $\eta :1\to V^{*}\otimes V$ and counit maps $ϵ:V\otimes V^{*}\to 1$ satisfying the triangle identities (the snake diagrams), while a left dual is is the dual notion. This convention fits in with the standardized conventions regarding adjoint functors: anendofunctor $F:C\to C$ has a right adjoint $F^{*}:C\to C$ if and only if $F^{*}$ is a right dual of $F$ in the monoidal category $\mathrm{End}(C)$.

Remarks

Note that this definition only asserts the existence of the dual objects. It does not assert that specific duals have been chosen. However, the choice of duals is unique up to unique isomorphism, justifying reference to ‘the’ dual of an object; in fact, this extends to a contravariant anafunctor ${}^{*}:M\to M$. (Using the axiom of choice to pick duals for every object at once, we can make this into a strict functor.)

Nor does this definition assert that the right dual of an object is isomorphic to its left dual: this need not be the case in general, though it is true in a braided monoidal category, and thus automatically also in a symmetric monoidal category. (A rigid monoidal category which is also symmetric is sometimes called compact closed or simply “compact”.)

In practice, algebraic geometers are the most frequent users of the term ‘rigid’, and they focus on the symmetric monoidal case, so they ignore the difference between right and left duals.

Related concepts

• (∞,n)-category with duals