nLab dualizable object

Idea

A (left/right) dual to an object in a monoidal category $\mathcal{C}$ is a left/right adjoint to the object regarded as a morphism in the delooping 2-category $\mathbf{B}\mathcal{C}$. 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 vector space is dualizable in Vect with its standard tensor product of vector spaces precisely if it is a finite-dimensional vector space (cf. Exp. ),

• 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 “size” is no larger than the “additivity” 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 — whereas in an “infinitely additive” monoidal category such as Rel or SupLat, many “infinite” objects are dualizable. (In $Rel$, all objects are dualizable.)

Remark

Beware that there are other notions of “dual 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

Definition

Definition

An object $A$ in a monoidal category $(\mathcal{C}, \otimes, 1)$ is dualizable if it has an adjoint when regarded as a morphism in the one-object delooping bicategory $\mathbf{B}\mathcal{C}$ corresponding to $\mathcal{C}$. Its adjoint in $\mathbf{B}\mathcal{C}$ is called its dual in $C$ and often written as $A^*$.

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

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

Explicitly this means the following:

A right duality between objects $A, A^\ast \in (\mathcal{C}, \otimes, \mathbb{1})$

consists of

1. a morphism of the form

   (1)$$ev_A \;\colon\; A^\ast \otimes A \longrightarrow \mathbb{1}$$

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

2. a morphism of the form

   (2)$$i_A \;\colon\; \mathbb{1} \longrightarrow A \otimes A^\ast$$

   called the unit or coevaluation map

such that

• (triangle identity) the following diagrams commute

  $$\array{ A^\ast \otimes (A \otimes A^\ast) &\overset{id_{A^\ast} \otimes i_A}{\longleftarrow}& A^\ast \otimes \mathbb{1} \\ {}^{\mathllap{\alpha^{-1}_{A^\ast,A, A^\ast}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\ell^{-1}_{A^\ast} \circ r_{A^\ast}}}_{\mathrlap{\simeq}} \\ (A^\ast \otimes A) \otimes A^\ast &\underset{ev_A \otimes id_{A^\ast}}{\longrightarrow}& \mathbb{1} \otimes A^\ast }$$

  and

  $$\array{ (A \otimes A^\ast) \otimes A &\overset{i_A \otimes id_A}{\longleftarrow}& \mathbb{1} \otimes A \\ {}^{\mathllap{\alpha_{A,A^\ast, A}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r_A^{-1}\circ \ell_A}}_{\mathrlap{\simeq}} \\ A \otimes (A^\ast \otimes A) &\underset{id_A \otimes ev_A}{\longrightarrow}& A \otimes \mathbb{1} \mathrlap{\,,} }$$

  where $\alpha$ denotes the associator of the monoidal category $\mathcal{C}$, and $\ell$ and $r$ denote the left and right unitors, respectively.

Remark

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

$$i \,\colon\, \mathbb{1} \to A \otimes A^*$$

and counit (or evaluation)

$$e \,\colon\, A^* \otimes A \to \mathbb{1}$$

satisfying the ‘triangle identities’ familiar from adjunctions.

With this convention, if $\otimes$ in $C$ is interpreted as composition in $\mathbf{B} C$ in diagrammatic order, then right duals in $C$ are the same as right adjoints in $\mathbf{B}C$ — whereas if $\otimes$ in $C$ is interpreted as composition in $\mathbf{B} C$ in classical ‘Leibnizian’ order, then right duals in $C$ are the same as left adjoints in $\mathbf{B} C$.

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

Remark

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.

Definition

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

Definition

If every object of $C$ has a left and right dual, then $C$ 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 $f \colon X \to Y$ between two dualizable objects in a symmetric monoidal category, the corresponding dual morphism

$$f^\ast \colon Y^\ast \to X^\ast$$

is the one obtained by $f$ by composing the duality unit, the counit and the braiding…

Examples

Definition

(finite-dimensional vector spaces)
Let $V$ be a finite-dimensional vector space over a field $k$, and let $V^* = Hom(V,k)$ be its usual dual vector space. We can define $\varepsilon\colon V^* \otimes V \to k$ to be the obvious pairing. If we also choose a finite basis $\{v_i\}$ of $V$, and let $\{v_i^*\}$ be the dual basis of $V^*$, then we can define $\eta\colon k \to V\otimes V^*$ by sending $1$ to $\sum_i v_i \otimes v_i^*$. It is easy to check the triangle identities, so $V^*$ is a dual of $V$ in $Vect_k$.

Example

(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.

Example

(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].

Example

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

Definition

(adjoint endofunctors)
Given any category $\mathscr{C}$, the endo-functor category $End(\mathcal{C}) \,\coloneqq\, \text{Fun}(\mathscr{C},\mathscr{C})$ 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 $End(\mathcal{C})$ 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.

Example

(Poincaré duality algebras)
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 $E$ an E-∞ ring, then in the (∞,1)-category of (∞,1)-modules $E Mod$ the dualizable objects coincide with the compact objects and the perfect objects.

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

Properties

Relation to adjunctable tensor products

Proposition

In a symmetric monoidal category, the following are equivalent:

• an object $A$ is dual to $A^\ast$ in the sense of Def. , with evaluation map $ev_A \,\colon\, A^\ast \otimes A \to \mathbb{1}$ (1),

• the map

  (3)$$Hom\big( X,\, Y \otimes A^\ast \big) \xrightarrow{\; (\text{-}) \otimes A \;} Hom\big( X \otimes A ,\, Y \otimes A^\ast \otimes A \big) \xrightarrow{\; id_Y \otimes ev_A \circ (\text{-}) \;} Hom\big( X \otimes A ,\, Y \big)$$

  is an isomorphism (a bijection of hom sets) for all objects $X$, $Y$.

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

Remark

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

$$(\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \,.$$

If the ambient category is indeed closed monoidal with internal hom denoted $[-,-]$ and unit denoted by $\mathbb{1}$, this means that $(\text{-}) \otimes A^\ast \,\simeq\, [A,\text{-}]$ (by essential uniqueness of adjoints) and hence in particular that:

$$A^\ast \;\simeq\; \mathbb{1} \otimes A^\ast \,\simeq\, [A,\mathbb{1}] \,.$$

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

1. a further right adjoint to the internal hom (an “amazing right adjoint”)

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

$$(\text{-}) \otimes A \;\; \dashv \;\; (\text{-}) \otimes A^\ast \;\; \dashv \;\; (\text{-}) \otimes A \,.$$

Remark

(Tensor-adjunctability does not imply dualizability)
Prop. does not claim that for $A$ to be dualizable it is sufficient that $(\text{-}) \otimes A$ 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. See also this n-Café discussion.

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 $O(1) \simeq \mathbb{Z}/2\mathbb{Z}$-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 $(\mathcal{C}, [-,-], 1)$ the (weak) dual to an object $X \in \mathcal{C}$ is defined to be the internal hom into the unit object

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

In a closed monoidal category

In a closed monoidal category $\mathbb{D}X$ (4) is also called the weak dual of $X$ [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]).

Definition

(reflexive dualizable object)
If the adjunct

$$X \longrightarrow \mathbb{D}\mathbb{D}X$$

of

$$X \otimes \mathbb{D}X \xrightarrow{\;\; \sigma \;\;} \mathbb{D}X \otimes X \xrightarrow{\;\; ev \;\;} \mathbb{1}$$

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

Remark

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

Remark

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

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

Definition

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

This appears as (Lurie, def. 2.3.5).

Remark

This means that an object in $C$ 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 $(\infty,n+1)$-category $\mathbf{B}C$. Then $C$ has duals for objects precisely if $\mathbf{B}C$ has all adjoints.

In a linearly distributive category

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

$$A \cong \top \otimes A \xrightarrow{i} (A \parr A^*) \otimes A \xrightarrow{\delta} A \parr (A^* \otimes A) \xrightarrow{ev} A \parr \bot \cong A$$

$$A^* \cong A^* \otimes \top \xrightarrow{i} A^* \otimes (A \parr A^*) \xrightarrow{\delta} (A^* \otimes A) \parr A^* \xrightarrow{ev} \bot \parr A^* \cong A^*.$$

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.

Related concepts

• dualizable module

• dualizing object

• self-dual object

• fully dualizable object

• rigid monoidal category,

• star-autonomous category

finite objects:

geometry	monoidal category theory	category theory
perfect module	(fully-)dualizable object	compact object

References

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

• Samuel Eilenberg, Saunders MacLane, first page of: General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 2 (1945) 231-294 [doi:10.1090/S0002-9947-1945-0013131-6, jstor:1990284]

to motivate category theory in the first place.

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

• Bodo Pareigis, p. 113 in: Non-additive ring and module theory IV: The Brauer group of a symmetric monoidal category, Lecture Notes in Mathematics 549 (1976) [doi:10.1007/BFb0077339]

  (calls dualizable objects “finite objects”)

• Thomas S. Ligon, Galois-Theorie in monoidalen Kategorien, Munich (1978) [pdf, doi:10.5282/edoc.14958]

  transl: Galois theory in monoidal categories (2019) [pdf, doi:10.5282/edoc.24952]

  (follows Pareigis 1976)

• Harald Lindner, Adjunctions in monoidal categories, Manuscripta Mathematica 26 1-2 (1978) 123-139 [doi:10.1007/BF01167969]

  (discusses dual objects in a monoidal category as adjunctions in its delooping 2-category)

• Pierre Deligne, James Milne: §1 pp. 110 of: Tannakian categories, in: Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer (1982) [doi:10.1007/978-3-540-38955-2_4, webpage]

• Albrecht Dold, Dieter Puppe, §1 of: Duality, Trace and Transfer, Proceedings of the Steklov Institute of Mathematics, 154 (1984) 85–103 [mathnet:tm2435, pdf]

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

• James Becker, Daniel Gottlieb, from p. 5. in A History of Duality in Algebraic Topology [pdf, pdf]

Further developments:

• Mark Hovey, John Palmieri, Neil Strickland: Def. 1.1.2 and Thm. 2.1.3 in: Axiomatic stable homotopy theory, Memoirs Amer. Math. Soc. 610 (1997) [ISBN:978-1-4704-0195-5, pdf]

• L. Gaunce Lewis, Peter May, Mark Steinberger (with contributions by J.E. McClure): §III.1 in: Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics 1213 (1986) [pdf, doi:10.1007/BFb0075778]

Review:

• Peter May, §2 in: Picard Groups, Grothendieck Rings, and Burnside Rings of Categories, Advances in Mathematics 163 1 (2001) 1-16 [pdf, doi:10.1006/aima.2001.1996]

• Dominic Culver, Mitchell Faulk, Duality Notes, lecture notes for West Coast Algebraic Topology Summer School (2014) [pdf, pdf mathtube]

  (towards fully dualizable objects)

• §3.1 in: Chris Heunen, Jamie Vicary, Categories for Quantum Theory, Oxford University Press 2019 [ISBN:9780198739616]

  based on:

  Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics (2012) [pdf, pdf]

  (in the context of quantum information theory via dagger-compact categories)

See also:

• Wikipedia, Dual object

Further examples:

• Kate Ponto and Mike Shulman, Traces in symmetric monoidal categories, (arXiv:1107.6032)

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

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

Monoidal categories with freely adjoint duals:

• Kevin Coulembier, Ross Street, Michel van den Bergh, Freely adjoining monoidal duals [arXiv:2004.09697]

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

• Jacob Lurie, Section 2.3 On the Classification of Topological Field Theories

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

• Chris Heunen, Semimodule enrichment, Electronic Notes in Theoretical Computer Science 218 (2008) 193-208 [doi:10.1016/j.entcs.2008.10.012]