# nLab symmetric monoidal (infinity,n)-category

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Monoidal categories

monoidal categories

# Contents

## Idea

A symmetric monoidal $(\infty,1)$-category is the analog of a symmetric monoidal (∞,1)-category for (∞,n)-category theory.

## Properties

### Dualizable objects

###### Definition

An object $X$ in a symmetric monoidal $(\infty,n)$-category is called dualizable if

###### Claim

Let $C$ be a symmetric monoidal $(\infty,n)$-category. Then there exists another symmetric monoidal $(\infty,n)$-category $C^{fd}$ and a symmetric monoidal functor

$C^{fd} \to C$

such that $C^{fd}$ has duals and is universal with these properties:

for any symmetric monoidal $(\infty,n)$-category with duals $D$ and any symmetric monoidal functor $F : D \to C$ there exists a symmetric monoidal functor $f : D \to C^{fd}$, unique up to equivalence, and an equivalence

$\array{ && C^{fd} \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\simeq}& \searrow \\ D &&\stackrel{F}{\to}&& C } \,.$

This appears as (Lurie, claim 2.3.19).

###### Remark

$C^{fd}$ is obtained from $C$ by discarding all objects that do not have duals and all k-morphisms that do not admit right and left adjoints.

###### Definition

An object $X \in C$ is called fully dualizable if it is in the essential image of $C^{fd} \to C$.

## Examples

• For all $n \in \mathbb{N}$, the (∞,n)-category of cobordisms $Bord_n$ is symmetric monoidal. By the cobordism hypothesis this should be the free symmetric monoidal $(\infty,n)$-category on the point.

• For all $n$ and $C$ any symmetric monoidal $(\infty,n)$-category, there is a symmetric monoidal (∞,n)-category of spans of ∞-groupoids over $C$.

## References

A discussion of dualizable objects is in section 2.3 of

Revised on February 5, 2013 02:17:04 by Urs Schreiber (89.204.154.134)