I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure?


I'm sorry for not providing motivation, but as you can see, I'm unable to do so. 


Here's a link to the ArXiv version: <http://arxiv.org/pdf/math/0608040v4>



Motivation for the covariant model structure on SSet/S
If you have a category $C$, then you can consider the category of functors $Func(C,Sets)$ from $C$ to sets. This comes with the Yoneda functor, $C^{op}\to Func(C,Sets)$, and is a generally useful thing to think about.


In $(\infty,1)$-category land, you want to start with an $(\infty,1)$-category $S$, and build the $(\infty,1)$-category of functors $Func(S,Spaces)$ from $S$ to spaces. 


Lurie is describing one way to do this. Given a quasicategory $S$, he produces a closed model category $(sSet/S, covariant)$ on the slice category $sSet/S$ which "models" this functor category.


The analogy is to the "point category" construction. Given a functor $F:C\to Sets$, you can build a category $P\_F$, whose objects are pairs $(c,x)$ where $c$ is an object of $C$, and $x\in F(c)$, and maps $(c,x)\to (c',x')$ are $f:c\to c'$ in $C$ such that $Ffx=x'$. Such a thing comes with a forgetful functor $U:P\_F\to C$, and one can produce an equivalence of categories
$$
Func(C,Sets) \qquad \Leftrightarrow\qquad (\text{certain full subcategory of $Cat/C$}).
$$ 
The subcategory is that of functors $U:D\to C$ such that given $f:c\to c'$ in $C$ and $d\in D$ such that $U(d)=c$, there is a unique $g: d\to d'$ in $D$ such that $U(g)=f$. 


So, fibrant objects in Lurie's covariant model category for $sSet/S$ are supposed to look like left Grothendieck fibrations, and this model category should model "$Func(S,Spaces)$".


**Edit:** originally, I said that the certain full subcategory was that of "left Grothendieck fibrations", but I don't think that's right. You can play the above story a bit more generally, with $Sets$ replaced by $Groupoids$; the functor $P\_F\to C$ corresponding to a (pseudo)functor $F:C\to Groupoids$ is a typical example of what's called a "left Grothendieck fibration" (or "fibered groupoid"), consult nLab for details. 


