dual object in a closed category


Conventional meaning

Let VV be a closed category (for example a closed monoidal category), with the internal hom denoted as ABA \multimap B and the unit object as II,


The dual 𝔻A\mathbb{D}A of an object AVA \in V is the internal hom out of AA into the unit object

𝔻AAI \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.


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

  • The dual X *X^\ast of a Banach space XX is generally not the dual in any sense that guarantees X *X^\ast \otimes - is left or right adjoint to XX \otimes -, unless XX 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 IX *XI \to X^\ast \otimes X (where II 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 VV is a star-autonomous category, “dual object” has a different meaning: it usually is taken to mean A *AA^\ast \cong A \multimap \bot where \bot is the dualizing object. In a *\ast-autonomous category where \bot and I=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, AA \multimap \top and AA \multimap \bot, coincide in the case where the *\ast-autonomous category is compact closed.


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

Last revised on February 27, 2014 at 06:01:16. See the history of this page for a list of all contributions to it.