# A few questions while reading Higher Topos Theory

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?

• For the first question, do you mean: "no idea how Lemma 2.2.3.4 can be used here"? – Reid Barton Apr 3 '19 at 15:25
• Yes, I cannot see how it can be used here. – Frank Kong Apr 4 '19 at 1:59

$$\newcommand{\SSet}{\mathsf{SSet}}\DeclareMathOperator{\Map}{Map}$$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).