nLab
(infinity,1)-Kan extension

under construction

Context

(,1)(\infty,1)-Category theory

Limits and colimits

Contents

Idea

The notion if (,1)(\infty,1)-Kan extension is the generalization of the notion of Kan extension from category theory to (∞,1)-category theory.

Definition

General

Independent of any models or concrete realizations chosen, the notion of (,1)(\infty,1)-Kan extension is intrinsically determined from just the notions of

In terms of these, for f:CCf : C \to C' any (∞,1)-functor and any (∞,1)-category AA, there is an induced (,1)(\infty,1)-functor f *:Func (,1)(C,A)Func (,1)(C,A)f^* : Func_{(\infty,1)}(C',A) \to Func_{(\infty,1)}(C,A).

The left (,1)(\infty,1)-Kan extension functor is the left adjoint (∞,1)-functor to f *f^*.

The right (,1)(\infty,1)-Kan extension functor is the right adjoint (∞,1)-functor to f *f^*.

Given different incarnations of or models for the notion of (∞,1)-category, there are accordingly different incarnations and models of this general abstract prescription.

In terms of quasi-categories

(LurieHTT, def. 4.3.2.2, 4.3.3.2)

In terms of Kan-complex enriched categories

see homotopy Kan extension

In terms of simplicial model categories

see homotopy Kan extension

Properties

Pointwise (strong)

\infty-Kan extensions as above are pointwise/strong. That is in fact the very content of (LurieHTT, def. 4.3.2.2, 4.3.3.2).

As adjoints to pullbacks

left/right \infty-Kan extension is left/right adjoint (∞,1)-functor to restriction. (LurieHTT, prop. 4.3.3.7)

References

A general concept of (,1)(\infty,1)-Kan extensions in terms of quasi-categories are discussed in section 4.3 of

For simplicially enriched categories and model categories a discussion is in section A.3.3 there.

Coinciding left/righ (ambidextrous) \infty-Kan extensions along maps of ∞-groupoids are discussed in

Pointwise homotopy Kan extensions are discussed in

See also

Revised on February 3, 2014 11:59:20 by Urs Schreiber (89.204.135.219)