nLab dualizing object in a closed category

This entry is about dualizing objects in closed (monoidal) categories in the sense of homological algebra and stable homotopy theory (e.g. dualizing modules). For a more general concept see at dualizing object.

Idea

A dualizing object $D$ in a closed category $\mathcal{C}$ is an object such that the internal hom $[-,D] \colon \mathcal{C} \to \mathcal{C}^{op}$ into it serves as an involutive duality operation on $\mathcal{C}$, or at least on a suitable full subcategory $\mathcal{C}_D \hookrightarrow \mathcal{C}$, i.e. it induces an equivalence of categories $[-,D] \colon \mathcal{C}_D \to \mathcal{C}_D^{op}$.

Definition

Let $\mathcal{C}$ be a closed category, $D\in \mathcal{C}$ an object, and $\mathcal{C}_D \hookrightarrow \mathcal{C}$ a full subcategory. We say that $D$ is a dualizing object on $\mathcal{C}_D$ if the induced functor

$$[-,D] : \mathcal{C}_D \to \mathcal{C}_D^{op}$$

is an equivalence of categories. Note that this includes the assumption that $[-,D]$ maps $\mathcal{C}_D$ into itself, which is not automatic.

Note that we do not in general assume $D\in \mathcal{C}_D$. In stable homotopy theory the subcategory in question is typically that of homotopy types with finite homotopy groups (e.g. for Anderson duality). Specifically in homological algebra one speaks also of dualizing modules. (See for instance (Heard-Stojanoska 14, def. 3.1 and Lurie, section 4.2).) Notably in a Grothendieck-Verdier context $(f^\ast \dashv f_\ast)$, $(f_! \dashv f^!)$ of six operations the functor $f^!$ typically preserves dualizing objects in this sense, which is a crucial ingredient of Verdier duality.

If $\mathcal{C} = \mathcal{C}_D$, we say that $D$ is a global dualizing object. A biclosed monoidal category $\mathcal{C}$ with a global dualizing object is one definition of a star-autonomous category.

In closed monoidal categories

If $\mathcal{C}$ is a closed symmetric monoidal category, then $[-,D]$ is adjoint to itself on the right, i.e. we have a natural isomorphism $\mathcal{C}(A,[B,D]) \cong \mathcal{C}(B,[A,D])$. Thus, if $[-,D]$ maps $\mathcal{C}_D$ into itself, then its restriction to $\mathcal{C}_D$ is an equivalence of categories if and only if the unit and counit of this adjunction are isomorphisms. But these unit and counit are both the “double-dualization” map

$$A \to [[A,D],D]$$

(the adjunct of the evaluation map $[A,D] \otimes A \to D$, which in turn is the adjunct of the identity map $[A,D] \to [A,D]$) for $A\in \mathcal{C}_D$; so $D$ is a dualizing object on $\mathcal{C}_D$ if and only if $[-,D]$ preserves $\mathcal{C}_D$ and these maps are isomorphisms for all $A\in \mathcal{C}_D$.

If $\mathcal{C}$ is a non-symmetric biclosed monoidal category, with two internal-homs $(A\otimes -) \dashv [A,-]$ and $(-\otimes A) \dashv \langle A,-\rangle$, then $[-,D]$ is adjoint on the right not to itself but to $\langle-,D\rangle$. A similar argument then shows that $D$ is dualizing if and only if $[-,D]$ and $\langle-,D\rangle$ both map $\mathcal{C}_D$ to itself, and both double-dualization maps

$$A\to \langle [A,D],D\rangle\qquad and \qquad A \to [\langle A,D\rangle,D]$$

are isomorphisms.

Examples

In a cartesian closed category

A cartesian closed category that with a global dualizing object is necessarily just a preorder. This statement is often known as Joyal‘s lemma, recalled for instance in Abramsky 09. It can be slightly strengthened as follows:

Proposition

A cartesian closed category $C$ that is self-dual (carries an equivalence $N: C^{op} \to C$) is necessarily a preorder (whose posetal reflection is then a Heyting algebra). If the self-duality comes from a dualizing object, then the Heyting algebra is a Boolean algebra.

Proof

For the first statement, let $1$ be terminal; then $N(1)$ is an initial object $0$, and similarly $N$ takes finite products to finite coproducts (coproducts are necessary for a Heyting algebra). So it remains to show $C$ is a preorder. Let $x, y$ be any two objects. The number of morphisms $x \to y$ is the number of morphisms $1 \to [x, y]$, which is the number of morphisms $N([x, y]) \to N(1) = 0$. But this is at most one since the initial object is strict (if there is any $z \to 0$, then $z$ is a retract of $0 \times z \cong 0$, hence $z \cong 0$; thus there is at most one morphism $z \to 0$).

The second statement is immediate: if $d$ is the dualizing object, then

$$d \cong [1, d] = N(1) \cong 0$$

so that $N(x) = [x, 0]$ is the negation and the hypothesis becomes the condition that double negation on the Heyting algebra is the identity, i.e., the Heyting algebra is a Boolean algebra.

In the category of spectra – Anderson duality

In the stable (infinity,1)-category of spectra, the sphere spectrum (which induces Spanier-Whitehead duality on spectra which are dualizable objects with respect to the smash product of spectra) is not a dualizing object. However, the Anderson spectrum $I_{\mathbb{Z}}$ is a dualizing object on a suitable subcategory of finite spectra (Lurie, Example 4.3.9). The duality operation $[-,I_{\mathbb{Z}}]$ that it induces is Anderson duality.

For instance, the Anderson dual of KU is (complex conjugation-equivariantly) the 4-fold suspension spectrum $\Sigma^4 KU$ (Heard-Stojanoska 14, theorem 8.2); and tmf$[1/2]$ is Anderson dual to its 21-fold suspension (Stojanoska 12)

In the category of suplattices

In the category Sup of suplattices, the opposite $\Omega^{op}$ of the poset $\Omega$ of truth values is a global dualizing object (and hence $Sup$ is star-autonomous). The internal-hom $[A,B]$ is the suplattice of sup-preserving maps $A\to B$, hence $[A,\Omega^{op}]$ is the suplattice of contravariant maps $A \to \Omega^{op}$ that take suprema in $A$ to infima in $\Omega$. But by the adjoint functor theorem for posets, any such map is representable by an element of $A$; thus $[A,\Omega^{op}] \cong A^{op}$. The double-dualization is therefore isomorphic to the identity $A\cong (A^{op})^{op}$.

Chu spaces

The Chu construction is a (co)universal way of “making any object into a dualizing object”.

Related concepts

• dual object in a closed category
• star-autonomous category
• Chu construction
• ambimorphic object

References

General discussion is in

• Mitya Boryachenko, Vladimir Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier (arXiv:1108.6020)

• Jacob Lurie, section 4.2 of Representability Theorems

Reviews of the general concept and then discussion of Anderson duality is in

• Vesna Stojanoska, Duality for Topological Modular Forms, Doc. Math. 17 (2012) 271-311 (arXiv:1105.3968)

• Drew Heard, Vesna Stojanoska, K-theory, reality, and duality (arXiv:1401.2581)

Discussion in the context of the linear logic/quantum logic of quantum physics is in

• Samson Abramsky, No-Cloning in categorical quantum mechanics, (2008) in I. Mackie and S. Gay (eds), Semantic Techniques for Quantum Computation , Cambridge University Press (arXiv:0910.2401)