dualizable object


Monoidal categories




A (left/right) dual to an object in a monoidal category π’ž\mathcal{C} is a left/right adjoint to the object regarded as a morphism in the delooping 2-category Bπ’ž\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 precisly if it is a finite-dimensional vector space; and a spectrum is dualizable in the stable homotopy category with its smash product precisely if it is a finite spectrum.

A more precise intuition is that an object is dualizable if its β€œsize” is no larger than the β€œadditivity” 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 dualizableβ€”whereas in an β€œinfinitely additive” monoidal category such as Rel or SupLat, many β€œinfinite” objects are dualizable. (In RelRel, all objects are dualizable.)


Beware that there are other notions of β€œdual 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



An object AA in a monoidal category (π’ž,βŠ—,1)(\mathcal{C}, \otimes, 1) is dualizable if it has an adjoint when regarded as a morphism in the one-object delooping bicategory Bπ’ž\mathbf{B}\mathcal{C} corresponding to π’ž\mathcal{C}. Its adjoint in Bπ’ž\mathbf{B}\mathcal{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.

Explicitly this means the following:

A right duality between objects A,A *∈(π’ž,βŠ—,1)A, A^\ast \in (\mathcal{C}, \otimes, 1)

consists of

  1. a morphism of the form

    ev A:A *βŠ—A⟢1 ev_A\;\colon\;A^\ast \otimes A \longrightarrow 1

    called the counit of the duality, or the evaluation map;

  2. a morphism of the form

    i A:1⟢AβŠ—A * i_A \;\colon\; 1 \longrightarrow A \otimes A^\ast

    called the unit or coevaluation map

such that

  • (triangle identity) the following diagrams commute

    A *βŠ—(AβŠ—A *) ⟡id A *βŠ—i A A *βŠ—1 ≃ Ξ± A *,A,A * βˆ’1↓ ↓ ≃ β„“ A βˆ’1∘r A (A *βŠ—A)βŠ—A * ⟢ev AβŠ—id A * 1βŠ—A * \array{ A^\ast \otimes (A \otimes A^\ast) &\overset{id_{A^\ast} \otimes i_A}{\longleftarrow}& A^\ast \otimes 1 \\ {}^{\mathllap{\alpha^{-1}_{A^\ast,A, A^\ast}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\ell^{-1}_A \circ r_A}}_{\mathrlap{\simeq}} \\ (A^\ast \otimes A) \otimes A^\ast &\underset{ev_A \otimes id_{A^\ast}}{\longrightarrow}& 1 \otimes A^\ast }


    (AβŠ—A *)βŠ—A ⟡i AβŠ—id A 1βŠ—A ≃ Ξ± A,A *,A↓ ↓ ≃ r A βˆ’1βˆ˜β„“ A AβŠ—(A *βŠ—A) ⟢id AβŠ—ev A AβŠ—1 \array{ (A \otimes A^\ast) \otimes A &\overset{i_A \otimes id_A}{\longleftarrow}& 1 \otimes A \\ {}^{\mathllap{\alpha_{A,A^\ast, A}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r_A^{-1}\circ \ell_A}}_{\mathrlap{\simeq}} \\ A \otimes (A^\ast \otimes A) &\underset{id_A \otimes ev_A}{\longrightarrow}& A \otimes 1 }

    where Ξ±\alpha denotes the associator of the monoidal category π’ž\mathcal{C}, and β„“\ell and rr denote the left and right unitors, respectively.


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:Iβ†’AβŠ—A *i: I \to A \otimes A^*

and counit (or evaluation)

e:A *βŠ—Aβ†’Ie : A^* \otimes A \to I

satisfying the β€˜triangle identities’ familiar from the concept of adjunction. With this convention, if βŠ—\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 β€” whereas if βŠ—\otimes in CC is interpreted as composition in BC\mathbf{B} C in classical β€˜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.


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


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.


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

f *:Y *β†’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…



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 Ξ΅:V *βŠ—Vβ†’k\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 Ξ·:kβ†’VβŠ—V *\eta\colon k \to V\otimes V^* by sending 11 to βˆ‘ iv iβŠ—v 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.


Let MM be a finite-dimensional manifold, choose an embedding Mβ†ͺℝ 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 Ξ·\eta which exhibits Th(NX)Th(N X) as a dual of Ξ£ + ∞M\Sigma_+^\infty M in the stable homotopy category. This is a version of Spanier-Whitehead duality.


A C*-algebra is a PoincarΓ© 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.


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.



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)≃℀/2β„€O(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 (π’ž,[βˆ’,βˆ’],1)(\mathcal{C}, [-,-], 1) the dual to an object Xβˆˆπ’žX \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 X→𝔻𝔻XX \to \mathbb{D}\mathbb{D}X is an equivalence this weak dual is called a reflexive weak dual.

If π’ž\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 π’ž\mathcal{C} as a star-autonomous category (𝔻(βˆ’)=(βˆ’) *\mathbb{D}(-) = (-)^\ast is the star-operation)

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


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).


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.


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.

In a linearly distributive category

In a linearly distributive category, duality is naturally defined by mixing the two tensors (βŠ—,⊀)(\otimes,\top) and (β…‹,βŠ₯)(\parr,\bot): the unit is i:βŠ€β†’Aβ…‹A *i : \top \to A \parr A^* and the counit is ev:A *βŠ—Aβ†’βŠ₯ev:A^* \otimes A \to \bot. The triangle identities make sense by inserting the linear distributivities; they assert that the following composites are identities:

Aβ‰…βŠ€βŠ—Aβ†’i(Aβ…‹A *)βŠ—Aβ†’Ξ΄Aβ…‹(A *βŠ—A)β†’evAβ…‹βŠ₯β‰…A A \cong \top \otimes A \xrightarrow{i} (A \parr A^*) \otimes A \xrightarrow{\delta} A \parr (A^* \otimes A) \xrightarrow{ev} A \parr \bot \cong A
A *β‰…A *βŠ—βŠ€β†’iA *βŠ—(Aβ…‹A *)β†’Ξ΄(A *βŠ—A)β…‹A *β†’evβŠ₯β…‹A *β‰…A *. A^* \cong A^* \otimes \top \xrightarrow{i} A^* \otimes (A \parr A^*) \xrightarrow{\delta} (A^* \otimes A) \parr A^* \xrightarrow{ev} \bot \parr A^* \cong A^*.

A symmetric linearly distributive category is (symmetric) star-autonomous if and only if all objects have duals in this sense. The same is true in the non-symmetric case if we require both left and right duals.

This notion of duality generalizes to that of linear adjoints in a linear bicategory, and also to dual objects in a polycategory.

finite objects:

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


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 November 7, 2017 05:15:48 by Mike Shulman (