nLab
(infinity,n)-category with duals

Idea

An (∞,n)-category with adjoints and a 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 $1<k<n$ 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.

• $C$ has adjoints if it has adjoints for k-morphisms with $0<k<n$.

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).

Related concepts

• rigid monoidal category

• category with duals

• dualizable object, fully dualizable object

• blob n-category

References

• Jacob Lurie, On the Classification of Topological Field Theories