The Yoneda Lemma for $(\infty,1)$-categories?

Question

According to this page on the nLab, it is currently unclear whether or not the entire Yoneda lemma generalizes to $(\infty ,1)$-categories rather than just the Yoneda embedding. Have there been counterexamples to the stronger statement? If not, what complications are there in generalizing the entire Yoneda lemma? The whole lemma is a very powerful tool in ordinary category theory, and if there is no definitive answer, are there any reasons to believe that the generalization should or shouldn't be true?

By the "entire" Yoneda lemma, I mean the isomorphism $F(X)\cong Nat(h_X,F)$

Answer 1

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 $X\to A$, 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 $X\to A$ 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 $\underline{Hom}(X,Y)\to\underline{Hom}(X,A)$ over the $0$-cell corresponding to the structural map $X\to A$. If $Y$ is fibrant (i.e. $Y\to A$ 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 $F\to A$, we see that $Map_A(a,F)$ is isomorphic to $F_a$, that is the fiber of the map $F\to A$ at $a$ (which is also an homotopy fiber for the Joyal model structure). Considering the weak equivalence from $a: \Delta[0]\to A$ to $A/a\to A$, 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.

Answer 2

The Yoneda lemma is certainly true for $(\infty,1)$-categories, and I can give you a proof: if you let me choose the model of $(\infty,1)$-categories I use!

I'll take topological categories (=categories enriched over spaces) as my model for an $(\infty,1)$-category. Let $C$ be a small topological category, and let $D_C$ denote the category of topological functors from $C^{\mathrm{op}}$ to spaces (and by "topological functor" I mean functor between categories enriched over spaces compatible with the enrichment).

The category $D_C$ is a closed model category using the projective model structure (weak equivalences and fibrations are maps which are so when evaluated at objects of $C$; this means in particular that all objects of $D_C$ are fibrant, and that representable functors are cofibrant in $D_C$).

Let $D_C'$ be the full subcategory $D_C'\subset D_C$ of fibrant-cofibrant objects in $D_C$, viewed as a topological category. I will take the topological category $D_C'$ as my model for the $(\infty,1)$-category of $\infty$-groupoid valued presheaves. Note that the Yoneda functor $C\to D_C$ (which is a topological functor), actually factors through $D_C'$, and I take this as my model for the $(\infty,1)$-categorical Yoneda functor.

If you buy all this, then the $(\infty,1)$-categorical Yoneda lemma is immediate (it follows from the usual Yoneda lemma for enriched categories).

Of course, you might have a different model of $(\infty,1)$-categories in mind. In which case, you'll want to know whether my model is really "equivalent" to yours. I'm sure the state of the art would allow you to prove this (if your model is one of the usual suspects), though I don't have a reference handy.

(I would be really surprised if one couldn't give a proof in terms of quasicategories; I'd guess that the "cartesian fibration" formalism for representable functors that Lurie uses in his book would do this easily, but since I haven't read that far in the book yet, I don't know.)

Answer 3

I realize it is a bit late to respond to the question... but this is precisely Lemma 5.1.5.2 of HTT (and the proof is very much along the same lines as the accepted one).