### Context

#### Higher category theory

higher category theory

# Contents

## Idea

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

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

## References

The notion appears first in section 2.3 of

A model for $\left(\infty ,n\right)$-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 $\le n$-manifolds (in preparation)

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

previewed in

Revised on October 31, 2012 22:49:49 by Urs Schreiber (82.169.65.155)