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(infinity,n)-category with adjoints

Context

Higher category theory

Basic concepts

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Basic theorems

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Applications

Models

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Morphisms

Functors

Universal constructions

Extra properties and structure

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1-categorical presentations

Contents

Idea

An (โˆž,n)-category ๐’ž\mathcal{C} is said to have 1-adjoints if in its homotopy 2-category Ho 2(๐’ž)Ho_2(\mathcal{C}) every 1-morphism is part of an adjunction. By recursion, for nโ‰ฅ3n \geq 3 and kโ‰ฅ2k \geq 2 an (โˆž,n)-category has kk-adjoints if for every pair X,YX, Y of objects the hom (โˆž,n-1)-category ๐’ž(X,Y)\mathcal{C}(X,Y) has adjoints for (kโˆ’1)(k-1)-morphisms.

An (โˆž,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 ๐’ž\mathcal{C} is a fully dualizable object, then ๐’ž\mathcal{C} is called an (โˆž,n)-category with duals.

Properties

Internal language

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

Examples

References

The notion appears first in section 2.3 of

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

previewed in

Last revised on November 6, 2014 at 18:27:56. See the history of this page for a list of all contributions to it.