under construction
equivalences in/of $(\infty,1)$-categories
The notion if $(\infty,1)$-Kan extension is the generalization of the notion of Kan extension from category theory to (∞,1)-category theory.
Independent of any models or concrete realizations chosen, the notion of $(\infty,1)$-Kan extension is intrinsically determined from just the notions of
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.
(LurieHTT, def. 4.3.2.2, 4.3.3.2)
$\infty$-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).
left/right $\infty$-Kan extension is left/right adjoint (∞,1)-functor to restriction. (LurieHTT, prop. 4.3.3.7)
$(\infty,1)$-Kan extension
A general concept of $(\infty,1)$-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
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