nLab dual morphism

Idea

The notion of dual morphism is the generalization to arbitrary monoidal categories of the notion of dual linear map in the category Vect of vector spaces.

Definition

Definition

Given a morphism $f \colon X \to Y$ between two dualizable objects in a monoidal category $(\mathcal{C}, \otimes)$, the corresponding dual morphism

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

is the one obtained by $f$ by using the duality unit of $X$ (the coevaluation map) and the duality counit of $Y$ (the evaluation map) as follows:

$$Y^* \to Y^*\otimes I \to Y^*\otimes X\otimes X^* \to Y^*\otimes Y\otimes X^* \to I\otimes X^* \to X^*$$

Remark

This notion is a special case of the the notion of mate in a 2-category.

Namely if $K \coloneqq \mathbf{B}_\otimes \mathcal{C}$ is the delooping 2-category of the monoidal category $(\mathcal{C}, \otimes)$, then objects of $\mathcal{C}$ correspond to morphisms of $K$, dual objects correspond to adjunctions and morphisms in $\mathcal{C}$ correspond to 2-morphisms in $K$. Under this identification a morphism $f \colon X \to Y$ in $\mathcal{C}$ may be depicted as a 2-morphism of the form

$$\array{ \ast &\stackrel{\mathbb{1}}{\to}& \ast \\ {}^{\mathllap{Y}}\downarrow &\swArrow_{\mathrlap{f}}& \downarrow^{\mathrlap{X}} \\ \ast &\underset{\mathbb{1}}{\to}& \ast }$$

and duality on morphisms is then given by the mate bijection

$$\array{ \ast & \overset{\mathbb{1}}{\to} & \ast \\ {}^{\mathllap{Y}} \downarrow & \swArrow_{\mathrlap{f}} & \downarrow^{\mathrlap{X}} \\ \ast & \underset{\mathbb{1}}{\to} & \ast } \;\;\;\;\; \mapsto \;\;\;\;\; \array{ \ast & \overset{\mathbb{1}}{\to} & \ast \\ {}^{\mathllap{X^\ast}} \downarrow & \swArrow_{\mathrlap{f^\ast}} & \downarrow^{\mathrlap{Y^\ast}} \\ \ast & \underset{\mathbb{1}}{\to} & \ast } \;\;\;\; \coloneqq \;\;\;\; \array{ \ast & \overset{Y^\ast}{\to} & \ast & \overset{\mathbb{1}}{\to} & \ast & \overset{\mathbb{1}}{\to} & \ast \\ {}^\mathllap{\mathbb{1}}\downarrow & \swArrow_{\epsilon_Y} & {}^{\mathllap{Y}} \downarrow & \swArrow_{f} & \downarrow^{\mathrlap{X}} & \swArrow_{\eta_X} & \downarrow \mathrlap{1} \\ \ast & \underset{\mathbb{1}}{\to} & \ast & \underset{\mathbb{1}}{\to} & \ast & \underset{X^\ast}{\to} & \ast } \,.$$

Examples

Example

In $\mathcal{C} =$ Vect with its standard tensor product monoidal structure, a dual object is a dual vector space and a dual morphism is a dual linear map.

Example

If $A$, $B$ are C*-algebras which are Poincaré duality algebras, hence dualizable objects in the KK-theory-category, then for $f \colon A \to B$ a morphism it is K-oriented, the corresponding Umkehr map is (postcomposition) with the dual morphism of its opposite algebra version:

$$f! \coloneqq (f^op)^\ast \,.$$

See at KK-theory – Push-forward in KK-theory.

Example

More generally, twisted Umkehr maps in generalized cohomology theory are given by dual morphisms in (∞,1)-category of (∞,1)-modules. See at twisted Umkehr map for more.

Related concepts

• trace