The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan extensions (it's not hard to show that the theory of homotopy limits reduces to this case (the same holds for left Kan extensions/colimits)), we say that given a functor $f:C\to C'$ between small categories, and a functor $F:C\to A$ where $A$ is a cofibrantly-generated model category, that a natural transformation of functors $\alpha: H\to Ran_f F$ exhibits $H$ as the homotopy right Kan extension of $F$ if there exists an injectively fibrant replacement $G$ of $F$ such that the composite $H\to Ran_fF \to Ran_fG$ is a weak equivalence.
When $A$ is combinatorial, we can also simply define a homotopy right Kan extension functor along $f$ to be $Ran_f (Q (-))$, where $Q$ is a functorial injectively fibrant replacement functor.
This is easy enough to define, but why is this the definition? Why would we want to take fibrant/cofibrant replacements and consider their ordinary Kan extensions/limits/etc? I suspect that it has to do with the fact that homotopy is an honest equivalence relation on arrows from a cofibrant object into a fibrant object, but I would appreciate an actual explanation.
Best Answer
There is a notion of homotopical Kan extension defined for "homotopical categories" (cats with a class of weak equivalences satisfying the 2-out-of-6 property). I cannot go into much detail without making a bazillion definitions, but the reference is Dwyer, Kan, Hirschorn, and Smith's "Homotopy Limit Functors on Model Categories and Homotopical Categories". Homotopical kan extensions and homotopy limit functors are defined, and model categories are demonstrated to be homotopically complete and cocomplete, and of course, fibrant and cofibrant replacement play a pretty big role in the proof. I think of the book as MacLane's chapter on Kan extensions generalized to the homotopical case.