nLab dual object in a closed category

Conventional meaning

Let $V$ be a closed category (for example a closed monoidal category), with the internal hom denoted as $A \multimap B$ and the unit object as $I$,

Definition

The (“weak”) dual $\mathbb{D}A$ of an object $A \in V$ is the internal hom out of $A$ into the unit object

$$\mathbb{D}A \coloneqq A \multimap I$$

This applies for example to categories of modules over a commutative ring, to the category of Banach spaces, etc., with their usual closed monoidal structures.

Warning

A dual in the sense above might not be a dual object in other established senses of the word. For example,

• The dual $X^\ast$ of a Banach space $X$ is generally not the dual in any sense that guarantees $X^\ast \otimes -$ is left or right adjoint to $X \otimes -$, unless $X$ is finite-dimensional. (Here the monoidal product is the projective tensor product, which makes the closed category of Banach spaces into a closed monoidal category.) In particular, there is generally no appropriate unit of the form $I \to X^\ast \otimes X$ (where $I$ is the ground field).

For a compact closed category, a dual object (in one or another monoidal category sense; see category with duals) is the same as a dual in the closed category sense. But in general, monoidal duals are a stronger notion than duals in the closed category sense.

Of course, where the monoidal product is not braided or symmetric monoidal, due care must be exercised in distinguishing between between different types of monoidal dual (left or right dual).

• If $V$ is a star-autonomous category, “dual object” has a different meaning: it usually is taken to mean $A^\ast \cong A \multimap \bot$ where $\bot$ is the dualizing object. In a $\ast$-autonomous category where $\bot$ and $I = \top$ are not isomorphic (e.g., a Boolean algebra), this gives something genuinely different from “dual” in the sense of this page. But again, these two meanings, $A \multimap \top$ and $A \multimap \bot$, coincide in the case where the $\ast$-autonomous category is compact closed.

Applications

Closed duals play a central role in six operations yoga, notably in Verdier duality and in the Wirthmüller isomorphism.

Related concepts

• dualizing object in a closed category

• dual object

• duality