nLab (infinity,1)-Kan extension

under construction

Idea

The notion of $(\infty,1)$-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 $(\infty,1)$-Kan extension is intrinsically determined from just the notions of

• (∞,1)-functor,

• (∞,1)-category of (∞,1)-functors

• adjoint (∞,1)-functor.

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

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

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

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)

Related concepts

• Kan extension

• $(\infty,1)$-Kan extension

References

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

• Jacob Lurie, Higher Topos Theory .

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

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

• Michael Hopkins, Jacob Lurie, section 4 of Ambidexterity in K(n)-Local Stable Homotopy Theory

Pointwise homotopy Kan extensions are discussed in

• Andrei Radulescu-Banu, Cofibrations in Homotopy Theory (arXiv:0610009)

• Denis-Charles Cisinski, Locally constant functors, Math. Proc. Camb. Phil. Soc. (2009), 147, 593 (pdf)

• Beatriz Rodriguez Gonzalez, section 4 of Realizable homotopy colimits (arXiv:1104.0646)

See also

• Samuel Isaacson, A note on unenriched homotopy coends (pdf)