nLab
symmetric monoidal (infinity,n)-category

Idea

A symmetric monoidal $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },n)$-category is called dualizable if …

Claim

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

$$C^{\mathrm{fd}}\to C$$C^{fd} \to C

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

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

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

This appears as (Lurie, claim 2.3.19).

Remark

$C^{\mathrm{fd}}$ 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 $C^{\mathrm{fd}}\to C$.

Examples

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

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

Related concepts

• monoidal category, monoidal (∞,1)-category

  • symmetric monoidal category, symmetric monoidal (∞,1)-category, symmetric monoidal $(\mathrm{\infty },n)$-category

  • closed monoidal category , closed monoidal (∞,1)-category

• symmetric monoidal 2-category

References

A discussion of dualizable objects is in section 2.3 of

• Jacob Lurie, On the Classification of Topological Field Theories