A few questions while reading Higher Topos Theory

Question

I am now reading Higher Topos Theory. I have recently met the following questions that I am not able to figure out and I am here to look for some answer or help.

First, in Lemma 2.2.3.6, while proving $(a)\implies (c)$, Lurie constructed a class $\scr U$ of simplicial sets which contains all simplicial sets $A$ satisfying $(c)$. The author claimed that using Lemma 2.2.3.4, it can be shown that the condition $(e)$ of Lemma 2.2.3.5 is satisfied. But I have no idea how Lemma 2.2.3.4 can be used here, nor can I figure out any other proof. How should I use Lemma 2.2.3.4 to prove this?

Second, in Corollary 2.2.3.12, Lurie claimed that for the simplicial equivalence $f:X\to Y$ in ${\sf SSet}_{/S}$ constructed there exists a map $g:Y\to X$ in ${\sf SSet}_{/S}$ and a homotopy $h:X\times\Delta[1]\to X$ from $\mathbb1_X$ to $g\circ f$. Such claim is true in a topological category, but in my thoughts two maps $f_1,f_2:A\to B$ in a simplicial category coincides in the homotopy category does not mean that there exists a homotopy $h:A\times\Delta[1]\to B$ connecting $f_1,f_2$. Is my thought incorrect, or the addition assumptions to $X,Y$ ($X\to S$ is a right fibration and $Y$ is contravariantly fibrant) may ensure that such $h$ does exist?

Answer 1

$\newcommand{\SSet}{\mathsf{SSet}}\DeclareMathOperator{\Map}{Map}$First, let's record the fact that for any $A$ in $\SSet_{/S}$ and any right fibration $p : X \to S$, the simplicial set $\Map_{\SSet_{/S}}(A, X)$ is a Kan complex. This follows from Lemma 2.2.3.4 by applying it to the inclusion $\emptyset \subset S$. As far as I can tell, this is the only way we'll actually use Lemma 2.2.3.4.

Here is another lemma, which is an exercise in diagram chasing: suppose given morphisms $A \to B$ and $X \to S$ and equip $B$ with some map $B \to S$. Then the induced map $\Map_{\SSet_{/S}}(B, X) \to \Map_{\SSet_{/S}}(A, X)$ is a pullback of the induced map $X^B \to X^A \times_{S^A} S^B$.

When $A \to B$ is a monomorphism and $X \to S$ is a right fibration, the latter map is again a right fibration by Corollary 2.1.2.9, so $\Map_{\SSet_{/S}}(B, X) \to \Map_{\SSet_{/S}}(A, X)$ is also a right fibration. But we know $\Map_{\SSet_{/S}}(A, X)$ is a Kan complex, so by Lemma 2.1.3.3 this map is actually a Kan fibration.

For your first question, suppose $A_0 \to A_1 \to A_2 \to \cdots$ is a sequence of monomorphisms between objects belonging to $\mathcal{U}$ and let $A$ be the colimit of the $A_i$ and suppose $A$ is equipped with some map to $S$. Then the $A_i$ inherit maps to $S$ by composition and $A$ is also the colimit of the $A_i$ in $\SSet_{/S}$. Then the mapping space $\Map_{\SSet_{/S}}(A, X)$ is the inverse limit of the spaces $\Map_{\SSet_{/S}}(A_i, X)$. We showed the that all the objects involved are Kan complexes and that the transition maps are fibrations. Therefore taking the inverse limit is a homotopy-invariant thing to do, so the induced map $\Map_{\SSet_{/S}}(A, X) \to \Map_{\SSet_{/S}}(A, Y)$ is a weak homotopy equivalence in the Kan model structure, hence a homotopy equivalence since both objects are Kan complexes. A similar argument applies to condition (iv).

For your second question, the existence of a genuine homotopy inverse follows from the fact that $\Map_{\SSet_{/S}}(A, X)$ and $\Map_{\SSet_{/S}}(A, Y)$ are Kan complexes for each $A$. For $X$ this follows from our first observation, and for $Y$ it follows from the fact that $Y$ is assumed to be fibrant in the simplicial model category $\SSet_{/S}$ (and every object is cofibrant).