# nLab dual morphism

### Context

#### Monoidal categories

monoidal categories

duality

# Contents

## 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 composing the duality unit of $X$ (the coevaluation map), the duality counit of $Y$ (the evaluation map)…

###### 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:

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

Revised on July 19, 2013 15:24:08 by Urs Schreiber (89.204.138.251)