nLab
(infinity,n)-category with adjoints

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

An (∞,n)-category 𝒞\mathcal{C} is said to have 1-adjoints if in its homotopy 2-category Ho 2(𝒞)Ho_2(\mathcal{C}) every 1-morphism is part of an adjunction. By recursion, for n3n \geq 3 and k2k \geq 2 an (∞,n)-category has kk-adjoints if for every pair X,YX, Y of objects the hom (∞,n-1)-category 𝒞(X,Y)\mathcal{C}(X,Y) has adjoints for (k1)(k-1)-morphisms.

An (,n)(\infty,n)-category has all adjoints (or just has adjoints, for short) if it has adjoints for kk-morphisms for 0<k<n0 \lt k \lt n.

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

Examples

References

The notion appears first in section 2.3 of

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

previewed in

Revised on October 13, 2014 12:19:09 by Urs Schreiber (185.26.182.37)