higher category theory - The Yoneda Lemma for (oo,1)-categories?

Here is a (tautological) proof in the setting of quasi-categories. Let $A$ be a quasi-category. In ordinary category theory, one can describe the category of presheaves of sets over small category $C$ as the full subcategory of $Cat/C$ spanned by Grothendieck fibrations wih discrete fibers. A quasi-category version would consist to say that a presheaf over a quasi-category $A$ is a Grothendieck fibration (cartesian fibration in Lurie's terminology), whose fibers are $infty$-groupoids (this the way of being discrete in the higher setting). Such fibrations are simply the right fibrations.

More precisely, a model for the theory of presheaves (or $infty$-stacks) over $A$ is the model category of simplicial sets over $A$, in which the fibrant objects are right fibrations $Xto A$, while the cofibrations are the monomorphisms. The weak equivalences between two right fibrations over $A$ are simply the fiberwise categorical equivalence (for the different models for the theory of stacks over a quasi-category, see §5.1.1 in Lurie's book). From this point of view, the representable stacks over $A$ are the right fibrations $Xto A$ such that $X$ has a terminal object. If $a$ is an object ($0$-cell) of $A$, there is a canonical right fibration $A/a to A$ (from the general theory of joins): this is the representable stack associated to $a$. You can also construct a model of $A/a$ by taking a fibrant replacement of the map $a:Delta[0]to A$ (seen as an object of $SSet/A$). This model category has the good taste of being a simplicial model category. In particular, you have a simplicially enriched Hom, which I will denote here by $Map_A$, and which can be described as follows. If $X$ and $Y$ are two simplicial sets over $A$, there is a simplicial set $Map_A(X,Y)$ of maps from $X$ to $Y$ over $A$: if $underline{Hom}$ is the internal Hom for simplicial sets, then $Map_A(X,Y)$ is simply the fiber of the obvious map $underline{Hom}(X,Y)tounderline{Hom}(X,A)$ over the $0$-cell corresponding to the structural map $Xto A$. If $Y$ is fibrant (i.e. $Yto A$ is a right fibration), then $Map_A(X,Y)$ is a Kan complex (because it is the fiber of a right fibration, hence of a conservative inner Kan fibration), which is the mapping space of maps from $X$ to $Y$ for this model structure on $Sset/A$. $Map_A$ is a Quillen functor in two variables with value in the usual model category of simplicial sets.

If $a$ is an object of $A$, seen as a map $a: Delta[0]to A$, i.e. as an object of $SSet/A$, then for any right fibration $Fto A$, we see that $Map_A(a,F)$ is isomorphic to $F_a$, that is the fiber of the map $Fto A$ at $a$ (which is also an homotopy fiber for the Joyal model structure). Considering the weak equivalence from $a: Delta[0]to A$ to $A/ato A$, we also have a weak equivalence of Kan complexes

$$Map_A(A/a,F)overset{sim}{to}Map_A(a,F)=F_a, .$$

This gives the full Yoneda lemma.

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