nLab
dualizable object

Context

Monoidal categories

Duality

Contents

Idea

A (left/right) dual to an object in a monoidal category 𝒞\mathcal{C} is a left/right adjoint to the object regarded as a morphism in the delooping 2-category B𝒞\mathbf{B}\mathcal{C}. If a dual exists, the object is called dualizable.

Being dualizable may often be thought of as a category-theoretic notion of finiteness for objects in a monoidal category. For instance, a vector space is dualizable in Vect with its standard tensor product just when it is a finite-dimensional vector space; and a spectrum is dualizable in the stable homotopy category with its smash product just when it is a finite spectrum.

A more precise intuition is that an object is dualizable if its “size” is no larger than the “additivity” of the monoidal category. Since Vect and the stable homotopy category are finitely additive, but not infinitely so, dualizability there is a notion of finiteness. This is the case for many monoidal categories in which one considers dualizability. However, in a monoidal category which is not additive at all, such as Set (or any cartesian monoidal category), only the terminal object is dualizable—whereas in an “infinitely additive” monoidal category such as Rel or SupLat, many “infinite” objects are dualizable. (In RelRel, all objects are dualizable.)

Warning

There are other notions of “dual object”, distinct from this one. See for example dual object in a closed category, and also the discussion at category with duals.

In a monoidal category

Definition

Definition

An object AA in a monoidal category CC is dualizable if it has an adjoint when regarded as a morphism in the one-object delooping bicategory BC\mathbf{B}C corresponding to CC. Its adjoint in BC\mathbf{B}C is called its dual in CC and often written as A *A^*.

If CC is braided then left and right adjoints in BC\mathbf{B}C are equivalent; otherwise one speaks of AA being left dualizable or right dualizable.

Remark

Unfortunately, conventions on left and right vary and sometimes contradict their use for adjoints. A common convention is that a right dual of AA is an object A *A^* equipped with a unit (or coevaluation)

i:IAA *i: I \to A \otimes A^*

and counit (or evaluation)

e:A *AIe : A^* \otimes A \to I

satisfying the ‘triangle identities’ familiar from the concept of adjunction. With this convention, if \otimes in CC is interpreted as composition in BC\mathbf{B} C in diagrammatic order, then right duals in CC are the same as right adjoints in BC\mathbf{B}C — whereas if \otimes in CC is interpreted as composition in BC\mathbf{B} C in classical ‘Leibnizian’ order, then right duals in CC are the same as left adjoints in BC\mathbf{B} C.

Of course, in a symmetric monoidal category, there is no difference between left and right duals.

Definition

A dualizable object AA, def. 1, for which the structure unit/counit maps between AA *A \otimes A^\ast and the unit object are isomorphisms is called an invertible object.

Definition

If every object of CC has a left and right dual, then CC is called a rigid monoidal category or an autonomous monoidal category. If moreover it is symmetric, it is called a compact closed category.

See category with duals for more discussion.

Definition

Given a morphism f:XYf \colon X \to Y between two dualizable objects in a symmetric monoidal category, the corresponding dual morphism

f *:Y *X * f^\ast \colon Y^\ast \to X^\ast

is the one obtained by ff by composing the duality unit, the counit and the braiding

Examples

Example

Let VV be a finite-dimensional vector space over a field kk, and let V *=Hom(V,k)V^* = Hom(V,k) be its usual dual vector space. We can define ε:V *Vk\varepsilon\colon V^* \otimes V \to k to be the obvious pairing. If we also choose a finite basis {v i}\{v_i\} of VV, and let {v i *}\{v_i^*\} be the dual basis of V *V^*, then we can define η:kVV *\eta\colon k \to V\otimes V^* by sending 11 to iv iv i *\sum_i v_i \otimes v_i^*. It is easy to check the triangle identities, so V *V^* is a dual of VV in Vect kVect_k.

Example

Let MM be a finite-dimensional manifold, choose an embedding M nM\hookrightarrow \mathbb{R}^n for some nn, and let Th(NX)Th(N X) be the Thom spectrum of the normal bundle of this embedding. Then the Thom collapse map defines an η\eta which exhibits Th(NX)Th(N X) as a dual of Σ + M\Sigma_+^\infty M in the stable homotopy category. This is a version of Spanier-Whitehead duality.

Example

A C*-algebra is a Poincaré duality algebra if it is a dualizable object in the symmetric monoidal category KK with dual its opposite algebra.

See at KK-theory – Poincare duality.

Example

For EE an E-∞ ring, then in the (∞,1)-category of (∞,1)-modules EModE Mod the dualizable objects coincide with the compact objects and the perfect objects.

See at (∞,1)-category of (∞,1)-modules – Compact generation for more.

Properties

Trace

Dualizable objects support a good abstract notion of trace. (…)

Relation to cobordism hypothesis

Dualizable objects in an symmetric monoidal (∞,1)-category are already fully dualizable objects. The cobordism hypothesis implies that there is a canonical O(1)/2O(1) \simeq \mathbb{Z}/2\mathbb{Z}-action on the ∞-groupoid of dualizable objects, and this is just the dualizing operation. See at cobordism hypothesis – Framed version – Implications: Canonical O(n)-action.

In a closed category

In a closed category (𝒞,[,],1)(\mathcal{C}, [-,-], 1) the dual to an object X𝒞X \in \mathcal{C} is defined to be the internal hom into the unit object

𝔻X[X,1]. \mathbb{D}X \coloneqq [X,1] \,.

In a closed monoidal category

In a closed monoidal category 𝔻X\mathbb{D}X is also called the weak dual of XX (e.g. Becker-Gottlieb, p. 5), to contrast with the monoidal dual as above, which would then be called the strong dual . If the induced morphism X𝔻𝔻XX \to \mathbb{D}\mathbb{D}X is an equivalence this weak dual is called a reflexive weak dual.

If 𝒞\mathcal{C} is a compact closed category, def. 3, then the weak dual 𝔻X\mathbb{D}X is also the strong dual object X *X^\ast to XX in the above monoidal sense. Here dualization exhibits 𝒞\mathcal{C} as a star-autonomous category (𝔻()=() *\mathbb{D}(-) = (-)^\ast is the star-operation)

In a symmetric monoidal (,n)(\infty,n)-category

Definition

An object in a symmetric monoidal (∞,n)-category CC is called dualizable if it is so as an object in the ordinary symmetric monoidal homotopy category Ho(C)Ho(C).

This appears as (Lurie, def. 2.3.5).

Remark

This means that an object in CC is dualizable if there exists unit and counit 1-morphism that satisfy the triangle identity up to homotopy. The definition does not demand that this homotopy is coherent (that it satisfies itself higher order relations up to higher order k-morphisms).

If the structure morphisms of the adjunction of a dualizable object have themselves all adjoints, then the object is called a fully dualizable object.

Remark

As before, we may equivalently state this after delooping the monoidal structure and passing to the (,n+1)(\infty,n+1)-category BC\mathbf{B}C. Then CC has duals for objects precisely if BC\mathbf{B}C has all adjoints.

finite objects:

geometrymonoidal category theorycategory theory
perfect module(fully-)dualizable objectcompact object

References

Duals in a closed/monoidal category are a very classical notion. A history of the basic definitions and applications in stable homotopy theory/higher algebra is in

A large number of further examples can be found in

The notion of duals in a symmetric monoidal (,n)(\infty,n)-category is due to section 2.3 of

Revised on September 2, 2014 17:14:18 by Urs Schreiber (82.136.246.44)