The weak equivalences in the covariant model structure

Question

Let $S$ be a simplicial set. Recall that there is a model structure, called the covariant model structure (see HTT ch. 2 and this question), on $\mathbf{SSet}/S$ such that:

1. The cofibrations are the monomorphisms.
2. A map $X \to Y$ of simplicial sets over $S$ is a weak equivalence if $X^\vartriangleleft \sqcup_X S \to Y^\vartriangleleft \sqcup_Y S$ is a categorical equivalence (i.e., a weak equivalence in the Joyal model structure -- that is, one such that when applying the simplicial category functor $\mathfrak{C}$ gives an equivalence of simplicial categories). Here the triangle denotes the left cone.
3. The fibrations are determined; the fibrant objects are the left fibrations $Y \to S$.

I think I understand the motivation for most of this: as Lurie explains, left fibrations are the $\infty$-categorical version of categories cofibered in groupoids (so the fibrant objects model a reasonable concept), and the cofibrations are as nice as can be. But I fail to understand the motivation for the weak equivalences -- not least because I don't have a particularly good picture of what these "left cones" are supposed to model. Why should the weak equivalences be what they are?

Answer 1

Maybe it would be helpful to think about the analogous situation in ordinary category theory. Suppose you are given a category $\mathcal{E}$ and a functor $F$ from $\mathcal{E}$ to the category of sets. There are several ways to encode this functor:

$(a)$: Via the Grothendieck construction, $F$ determines a category $\mathcal{C}$ cofibered in sets over $\mathcal{E}$, so that for each object $E \in \mathcal{E}$ you can identify $F(E)$ with the fiber $\mathcal{C}_E$ of the map $\mathcal{C} \rightarrow \mathcal{E}$ over $E$.

$(b)$: Using the functor $F$, you can construct an enlargement $\mathcal{E}_F$ of the category $\mathcal{E}$, adding a single object $v$ with $$Hom(E,v) = \emptyset \quad \quad Hom(v,E) = F(E) \quad \quad Hom(v,v) = \{ id \} $$

Now suppose we are given another functor $G$ from $\mathcal{E}$ to the category of sets, and a natural transformation $F \rightarrow G$. Then $G$ determines a category $\mathcal{D}$ cofibered in sets over $\mathcal{E}$, and an enlargement $\mathcal{E}_G$ of $\mathcal{E}$. The natural transformation $F \rightarrow G$ determines functors $$ \alpha: \mathcal{C} \rightarrow \mathcal{D} \quad \quad \beta: \mathcal{E}_F \rightarrow \mathcal{E}_G$$ In this situation, the following conditions are equivalent:

$(i)$: The natural transformation $F \rightarrow G$ is an isomorphism (that is, for each object $E \in \mathcal{E}$, the induced map $F(E) \rightarrow G(E)$ is bijective.

$(ii)$: The functor $\alpha$ is an equivalence of categories.

$(iii)$: The functor $\beta$ is an equivalence of categories.

Now observe that the category $\mathcal{E}_F$ can be described as the pushout (and also homotopy pushout) of the diagram $$\mathcal{E} \leftarrow \mathcal{C} \rightarrow \mathcal{C}^{\triangleleft},$$ where $\mathcal{C}^{\triangleleft}$ is the category obtained from $\mathcal{E}$ by adjoining a new initial object.

Let's now forget the original functors $F$ and $G$, and think only about the categories $\mathcal{C}$ and $\mathcal{D}$ cofibered in sets over $\mathcal{E}$. The equivalence of conditions $(ii)$ and $(iii)$ shows that functor $\alpha: \mathcal{C} \rightarrow \mathcal{D}$ of categories cofibered over $\mathcal{E}$ is an equivalence of categories if and only if the induced map $$ \mathcal{E} \amalg_{ \mathcal{C} } \mathcal{C}^{\triangleleft} \rightarrow \mathcal{E} \amalg_{ \mathcal{D} } \mathcal{D}^{\triangleleft}$$ is an equivalence of categories.

Now go to the setting of quasi-categories. Assume for simplicity that $S$ is a quasi-category, and let $f: X \rightarrow Y$ be a map of simplicial sets over $S$. If $X$ and $Y$ are left-fibered over $S$, then we would like to say that $f$ is a covariant equivalence if and only if it an equivalence of quasi-categories. However, we would like to formulate this condition in a way that will behave well also when $X$ and $Y$ are not fibrant. Motivated by the discussion above, we declare that $f$ is a covariant equivalence if and only if it induces a categorical equivalence $$ S \amalg_{X} X^{\triangleleft} \rightarrow S \amalg_{Y} Y^{\triangleleft}.$$ You can then prove that this is a good definition (it gives you a model structure with the cofibrations and fibrant objects that you described, and when $X$ and $Y$ are fibrant a map $f: X \rightarrow Y$ is a covariant equivalence if and only if it induces a homotopy equivalence of fibers $X_s \rightarrow Y_s$ for each vertex $s \in S$).