# nLab (infinity,n)-category with duals

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Idea

An (∞,n)-category with adjoints and a dual object for every object.

## Definition

###### Definition

Let $C$ be an (∞,n)-category. We say that

• $C$ has adjoints for morphisms if in its homotopy 2-category every morphism has a left adjoint and a right adjoint;

• for $1 that $C$ has adjoints for k-morphisms if for every pair $X,Y\in C$ of objects, the hom-(∞,n-1)-category $C\left(X,Y\right)$ has adjoints for $\left(k-1\right)$-morphisms.

• $C$ has adjoints if it has adjoints for k-morphisms with $0.

If $C$ is in addition a symmetric monoidal (∞,n)-category we say that

Finally we say that

• $C$ has duals if it has duals for objects and has adjoints.

This is (Lurie, def. 2.3.13, def. 2.3.16).

## References

Revised on October 31, 2012 22:48:41 by Urs Schreiber (82.169.65.155)