# Section 2.4.4 Higher Topos Theory

In section 2.4.4 of Lurie's Higher Topos Theory, it is said multiple times that using

Proposition: Let $$p : \mathcal{C} \rightarrow \mathcal{D}$$ be an inner fibration of $$\infty$$-categories. Let $$x,y$$ be vertices of $$\mathcal{C}$$, let $$\tilde{e} : p(x) \rightarrow p(y)$$ be an edge of $$\mathcal{D}$$ and let $$e: x'\rightarrow y$$ be a locally $$p$$-Cartesian edge of $$\mathcal{C}$$ lifting $$\tilde{f}$$. Then in the homotopy category $$\mathcal{H}$$ of spaces there is a fiber sequence $$Map_{\mathcal{C}_{p(x)}}(x,x') \rightarrow Map_{\mathcal{C}}(x,y) \rightarrow Map_{\mathcal{D}}(p(x),p(y))$$ where the fiber is taken over $$\tilde{e}$$.

we have that any cartesian fibration of simplicial set $$p :\mathcal{C} \rightarrow \mathcal{D}$$ is fully faithful map in the sense that for every vertices $$x,y \in \mathcal{C}$$, the induced map $$Map_{\mathcal{C}}(x,y) \rightarrow Map_{\mathcal{D}}(p(x),p(y))$$ is a Kan weak equivalence of simplicial sets.

I see that from the fact that $$p$$ is a Cartesian fibration we have that $$p$$ satisfies the hypothesis of the above proposition. However I do not see how we can say that $$p$$ is fully faithful from it.

• He never makes that claim, and it’s not true (map a big simplicial set to a point). In each instance there are further properties assumed of p to ensure that fiber from 2.4.4.2 is zero, or the proposition is used in a different way. For example, in 2.4.4.4, we have a map between two loc Cartesian fibrations and it is assumed that the map preserves loc Cartesian edges and is a fiberwise equivalence- that tells you the map on fibers in 2.4.4.4 is an equivalence, and the map on bases is the identity, so the map in the middle is an equivalence. – Dylan Wilson Jan 2 at 14:54
• Is there a specific use of 2.4.4.2 that you’re interested in? – Dylan Wilson Jan 2 at 14:54
• I am mostly interested of its use in 2.4.4.5. I maybe should have written $\infty$-categories instead of simplicial sets. – Oscar P. Jan 2 at 15:04
• But it’s still false then- for example the map from Delta^1 to a point is a Cartesian fibration that is not fully faithful. In 2.4.4.5, the square with C’,C,D, and D’ induces a map of fiber sequences from 2.4.4.2. The map on bases is an equivalence bc D’—>D is assumed to be a categorical equivalence. The map on fibers is an isomorphism since C’ is the literal pullback. So the map in the middle is a weak equivalence, whence the claim about C’—>C being fully faithful. – Dylan Wilson Jan 2 at 15:20
• That’s exactly the content of the two sentences I wrote preceding “So the map in the middle is a weak equivalence.” – Dylan Wilson Jan 2 at 16:29