nLab
(infinity,n)-category with duals

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

• $C$ has duals for objects if its homotopy category is a rigid monoidal category.

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 aximatized inside opetopic type theory.

Examples

• (∞,n)-category of cobordisms

• (∞,n)-category of n-fold correspondences

Related concepts

• fundamental n-category?

• rigid monoidal category

• category with duals

• dualizable object, fully dualizable object

• blob n-category

References

• Jacob Lurie, On the Classification of Topological Field Theories

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