# nLab (infinity,n)-category with duals

Contents

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Idea

An (∞,n)-category with adjoints (see there for more) and a (fully) 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 $1 \lt k \lt n$ that $C$ has adjoints for k-morphisms if for every pair $X,Y \in C$ of objects, the hom-(∞,n-1)-category $C(X,Y)$ has adjoints for $(k-1)$-morphisms.

• $C$ is an (∞,n)-category with adjoints if it has adjoints for k-morphisms with $0 \lt k \lt n$.

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). See at fully dualizable object

## Properties

### Internal language

The internal language of $(\infty,n)$-categories with duals seems plausible to be axiomatized inside opetopic type theory.

## References

For more see at (infinity,n)-category with adjoints.

Last revised on November 17, 2022 at 12:40:57. See the history of this page for a list of all contributions to it.