nLab (infinity,n)-category with adjoints

Idea

An (∞,n)-category $\mathcal{C}$ is said to have 1-adjoints if in its homotopy 2-category $Ho_2(\mathcal{C})$ every 1-morphism is part of an adjunction. By recursion, for $n \geq 3$ and $k \geq 2$ an (∞,n)-category has $k$-adjoints if for every pair $X, Y$ of objects the hom (∞,n-1)-category $\mathcal{C}(X,Y)$ has adjoints for $(k-1)$-morphisms.

An $(\infty,n)$-category has all adjoints (or just has adjoints, for short) if it has adjoints for $k$-morphisms for $0 \lt k \lt n$.

If in addtition every object in $\mathcal{C}$ is a fully dualizable object, then $\mathcal{C}$ is called an (∞,n)-category with duals.

Properties

Internal language

The internal language of $(\infty,n)$-categories with duals seems plausibly to be axiomatizable in opetopic type theory.

Examples

• An (∞,n)-category of cobordisms has all adjoints.

Related concepts

• (∞,n)-category with duals

• fully dualizable object

References

The notion appears first in section 2.3 of

• Jacob Lurie, On the Classification of Topological Field Theories

A model for $(\infty,n)$-categories with all adjoints in terms of (∞,1)-sheaves on a site of a variant of $n$-dimensional manifolds with embeddings between them is discussed in

• David Ayala, Nick Rozenblyum, Weak $n$-categories are sheaves on iterated submersions of $\leq n$-manifolds (in preparation)

• David Ayala, Nick Rozenblyum, Weak $n$-categories with adjoints are sheaves on $n$-manifolds (in preparation)

previewed in

• David Ayala, Higher categories are sheaves on manifolds, talk at FRG Conference on Topology and Field Theories, U. Notre Dame (2012) (video)

  Abstract Chiral/factorization homology gives a procedure for constructing a topological field theory from the data of an En-algebra. I’ll explain a multi-object version of this construction which produces a topological field theory from the data of an $n$-category with adjoints. This construction is a consequence of a more primitive result which asserts an equivalence between n-categories with adjoints and “transversality sheaves” on framed $n$-manifolds - of which there is an abundance of examples.

• Nick Rozenblyum, Manifolds, Higher Categories and Topological Field Theories, talk Northwestern University (2012) (pdf slides)