nLab
symmetric monoidal (infinity,n)-category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Monoidal categories

Contents

Idea

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

Properties

Dualizable objects

Definition

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

Claim

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

C fdC C^{fd} \to C

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

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

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

This appears as (Lurie, claim 2.3.19).

Remark

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

Definition

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

Examples

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)