In HTT.5.5.8.18 Lurie defines a projective object $P$ in a quasicategory $\bf C$ as an object such that its corepresented functor ${\rm Map}(P,-)$ "commutes with geometric realizations". I can catch the general idea that something like $\hom(P,|S|_{X^*}) \simeq |S|_{\hom(P,X^*)}$ should happen, if

- $X_*$ is a cosimplicial object in $\bf C$;
- $|-|_{\hom(P,X^*)}$ is the "realization" obtained via familiar abstraction from the cosimplicial object $X^*$: since $\hom(P,X^*)\in [\Delta, {\bf Set}]$ it can be Kan-extended to a functor ${\bf sSet}\to \bf C$, and this functor admits a right adjoint (the "nerve").
- $|-|_{X^*}$ is a realization
**similarly**induced as a ($\infty$?)-functor $N(\boldsymbol \Delta)\to \bf C$.

But this "similarly" is kind of too vague, I strive for a more precise description. Unfortunately HTT seems to skip a precise definition of the paradigm "geometric realization"-"nerve".

Does it transport to the $(\infty,1)$-categorical case? Do I recover the *precise* form of that statement (i.e.: give a cosimplicial object in $\bf C\in QCat$, then there is a pair of $(\infty,1)$-adjoint functors obtained as Yoneda extension / "nerve")?

On the same vein, Remark 5.5.8.5 is totally obscure to me:

The formation of the geometric realizations of simplicial objects should be thought of as the $\infty$-categorical analogue of the formation of reflexive coequalizers.

Can you help me?

Homotopical Algebra). $\endgroup$ – Mauro Porta Nov 6 '13 at 11:54