nLab dualizable object



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory




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 [Pareigis 1976 p. 113] for objects in a monoidal category. For instance:

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. For instance, since Vect and the stable homotopy category are finitely additive, but not infinitely so, dualizability there is indeed a notion of plain finiteness, as 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 β€” 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 (4), 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^*.

(This notion, though not the terminology, is due to Lindner 1978.)

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—,πŸ™)A, A^\ast \in (\mathcal{C}, \otimes, \mathbb{1})

consists of

  1. a morphism of the form

    (1)ev A:A *βŠ—AβŸΆπŸ™ ev_A \;\colon\; A^\ast \otimes A \longrightarrow \mathbb{1}

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

  2. a morphism of the form

    (2)i A:πŸ™βŸΆAβŠ—A * i_A \;\colon\; \mathbb{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—πŸ™ ≃ Ξ± A *,A,A * βˆ’1↓ ↓ ≃ β„“ A * βˆ’1∘r A * (A *βŠ—A)βŠ—A * ⟢ev AβŠ—id A * πŸ™βŠ—A * \array{ A^\ast \otimes (A \otimes A^\ast) &\overset{id_{A^\ast} \otimes i_A}{\longleftarrow}& A^\ast \otimes \mathbb{1} \\ {}^{\mathllap{\alpha^{-1}_{A^\ast,A, A^\ast}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\ell^{-1}_{A^\ast} \circ r_{A^\ast}}}_{\mathrlap{\simeq}} \\ (A^\ast \otimes A) \otimes A^\ast &\underset{ev_A \otimes id_{A^\ast}}{\longrightarrow}& \mathbb{1} \otimes A^\ast }


    (AβŠ—A *)βŠ—A ⟡i AβŠ—id A πŸ™βŠ—A ≃ Ξ± A,A *,A↓ ↓ ≃ r A βˆ’1βˆ˜β„“ A AβŠ—(A *βŠ—A) ⟢id AβŠ—ev A AβŠ—πŸ™, \array{ (A \otimes A^\ast) \otimes A &\overset{i_A \otimes id_A}{\longleftarrow}& \mathbb{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 \mathbb{1} \mathrlap{\,,} }

    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:πŸ™β†’AβŠ—A * i \,\colon\, \mathbb{1} \to A \otimes A^*

and counit (or evaluation)

e:A *βŠ—Aβ†’πŸ™ e \,\colon\, A^* \otimes A \to \mathbb{1}

satisfying the β€˜triangle identities’ familiar from adjunctions.

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.


There are various equivalent definitions of dualizability, some of which are apparently weaker than the explicit definition in terms of both unit and counit, or which assume only one of them together with a universal property for it.

Regarding the equivalent characterization via adjointable tensor products, beware Rem. below.


A dualizable object AA, def. , 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…



(finite-dimensional vector spaces)
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.


(dualizable modules)
More generally, in the symmetric monoidal category of modules over a commutative ring, dualizable objects are precisely finitely generated projective modules. [Dold & Puppe 1984, Ex. 1.4]. See at dualizable module for more.


(dualizable chain complexes) Yet more generally, in the category of chain complexes of modules over a commutative ring, with respect to the tensor product of chain complexes, an object is dualizable iff it is a bounded chain complex of dualizable modules, hence (by Ex. ) a bounded chain complex of finitely generated projective modules [Dold & Puppe 1984, Prop. 1.6].


(Spanier-Whitehead duality)
Let MM be a finite-dimensional smooth 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.


(adjoint endofunctors)
Given any category π’ž\mathscr{C}, the endo-functor category End(π’ž)≔Fun(π’ž,π’ž)End(\mathcal{C}) \,\coloneqq\, \text{Fun}(\mathscr{C},\mathscr{C}) canonically becomes a monoidal category, with tensor product given by composition of functors and horizontal composition of natural transformations (whiskering).

Under this identification, an object in End(π’ž)End(\mathcal{C}) is dualizable iff it is an adjoint functor, with evaluation and co-evaluation given by the counit of the adjunction and the unit of the adjunction, respectively.


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.


Relation to adjunctable tensor products


In a symmetric monoidal category, the following are equivalent:

  • an object AA is dual to A *A^\ast in the sense of Def. , with evaluation map ev A:A *βŠ—Aβ†’πŸ™ev_A \,\colon\, A^\ast \otimes A \to \mathbb{1} (1),

  • the map

    (3)Hom(X,YβŠ—A *)β†’(-)βŠ—AHom(XβŠ—A,YβŠ—A *βŠ—A)β†’id YβŠ—ev A∘(-)Hom(XβŠ—A,Y) Hom\big( X,\, Y \otimes A^\ast \big) \xrightarrow{\; (\text{-}) \otimes A \;} Hom\big( X \otimes A ,\, Y \otimes A^\ast \otimes A \big) \xrightarrow{\; id_Y \otimes ev_A \circ (\text{-}) \;} Hom\big( X \otimes A ,\, Y \big)

    is an isomorphism (a bijection of hom sets) for all objects XX, YY.

This is Dold & Puppe 1984, Thm. 1.3 (b) and (c).


Prop. says in particular that for dual objects AA, A *A^\ast the map (3) exhibits the tensor product functor (-)βŠ—A *(\text{-}) \otimes A^\ast as a right adjoint to (-)βŠ—A(\text{-}) \otimes A (hence: an internal hom-functor):

(-)βŠ—A⊣(-)βŠ—A *. (\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \,.

If the ambient category is indeed closed monoidal with internal hom denoted [βˆ’,βˆ’][-,-] and unit denoted by πŸ™\mathbb{1}, this means that (-)βŠ—A *≃[A,-](\text{-}) \otimes A^\ast \,\simeq\, [A,\text{-}] (by essential uniqueness of adjoints) and hence in particular that:

A *β‰ƒπŸ™βŠ—A *≃[A,πŸ™]. A^\ast \;\simeq\; \mathbb{1} \otimes A^\ast \,\simeq\, [A,\mathbb{1}] \,.

But since (by symmetry) the object A≃(A *) *A \,\simeq\, (A^{\ast})^\ast is also the dual object to A *A^\ast, there is

  1. a further right adjoint to the internal hom (an β€œamazing right adjoint”)

  2. which coincides with the left adjoint to make an ambidextrous adjunction:

(-)βŠ—A⊣(-)βŠ—A *⊣(-)βŠ—A. (\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \;\; \dashv \;\; (\text{-}) \otimes A \,.


(Tensor-adjunctability does not imply dualizability)
Prop. does not claim that for AA to be dualizable it is sufficient that (-)βŠ—A(\text{-}) \otimes A has a right adjoint.

A counterexample is indicated by Noah Snyder in math.SE:a/692318, referring to Exp. 2.20 in arXiv:1406.4204.


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 (weak) dual to an object Xβˆˆπ’žX \in \mathcal{C} is defined to be the internal hom into the unit object

(4)𝔻X≔[X,1]. \mathbb{D}X \coloneqq [X,1] \,.

In a closed monoidal category

In a closed monoidal category 𝔻X\mathbb{D}X (4) is also called the weak dual of XX [Becker & Gottlieb, p. 5] (in contrast with the monoidal dual of Def. , which would then be called the strong dual [Dold & Puppe 1984 Def. 1.2]).


(reflexive dualizable object)
If the adjunct

XβŸΆπ”»π”»X X \longrightarrow \mathbb{D}\mathbb{D}X


XβŠ—π”»X→σ𝔻XβŠ—Xβ†’evπŸ™ X \otimes \mathbb{D}X \xrightarrow{\;\; \sigma \;\;} \mathbb{D}X \otimes X \xrightarrow{\;\; ev \;\;} \mathbb{1}

is an isomorphism (identifying XX with its double dual) then XX is called reflexive as a weak dualizable object. [Deligne & Milne 1982 p 111, Dold & Puppe 1984 Def. 1.2]


If π’ž\mathcal{C} is a compact closed category, def. , 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).


The property of XX being dualizable can be expressed as a property of the weak dual, namely that the induced map 𝔻XβŠ—Xβ†’[X,X]\mathbb{D}X \otimes X \to [X,X] is an isomorphsim.

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


The strong-duality of finite-dimensional vector spaces is the lead-in example used in

to motivate category theory in the first place.

Further original articles with dedicated discussion of the notion of dualizable objects in monoidal categories:

Early history with an eye towards formulating Becker-Gottlieb transfer:

Further developments:


See also:

Further examples:

On self-dual objects and the corresponding inner products and dagger-structure:

  • Peter Selinger, Autonomous categories in which A≃A *A \simeq A^\ast, talk at QPL 2012 (pdf)

Monoidal categories with freely adjoint duals:

The notion of fully dualizable objects in a symmetric monoidal ( ∞ , n ) (\infty,n) -category:

On the connection between dualisability, finiteness, and enrichment, see:

Last revised on February 26, 2024 at 06:21:11. See the history of this page for a list of all contributions to it.