n-category = (n,n)-category
n-groupoid = (n,0)-category
An (∞,n)-category is said to have 1-adjoints if in its homotopy 2-category every 1-morphism is part of an adjunction. By recursion, for and an (∞,n)-category has -adjoints if for every pair of objects the hom (∞,n-1)-category has adjoints for -morphisms.
An -category has all adjoints (or just has adjoints, for short) if it has adjoints for -morphisms for .
The notion appears first in section 2.3 of
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 -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 -manifolds - of which there is an abundance of examples.