This is a Soft question.When I learn model category, the important compute tool is Lifting property between $(Cof, Fib \cap W)$ and $(Cof \cap W, Fib) $,W is class of weak equivalence.This means for commutative diagram which vertical map are f,g, $f \in \operatorname{Cof}$,$g \in \operatorname{fib}\cap W$.So in localization of C by W,we must have this filler of f,g.

 1. so my first question is why we need filler of g,f in localization of C which must exist one in C, this property means what in the category special or in general.
 2. we can regard filler of f,g as every $\Delta^1 \times \Delta^1 \rightarrow N(C),$ which two vertical map are f,g, which is restriction of $\Delta^3 \rightarrow N(C)$. so we say commutative diagram has the structure of the 3-dimension. but in $\infty-$category. we proof many statements relative composition,homotopy using $\operatorname{sk}_1(\Delta^3) \rightarrow X$ in which $d_0(\sigma)$ ,$d_3(\sigma)$ is commute,then $d_1(\sigma)$ iff $d_2(\sigma)$,$\sigma: \Delta^3 \rightarrow X $ by definition of $\infty-$category.so this is also question existing the structure of 3-dimension.I know model category and $\infty-$category using extension in different way.  in model category means commutative diagram extension to $\Delta^3$. in $\infty-$category, i have two $x_1,x_2:\Delta^2 \rightarrow X$ which $d_0 (x_1) = d_2(x_2)$. when $d_0(x_2) \circ d_1(x_1) = d_1 \circ d_2(\sigma)$ this composition in meaning of $\infty-$category. we can extension $x_1 ,x_2$ in $\Delta^3 \rightarrow X$,then we have diagram commutative. this two way i feel some adjoint,but i dont know mean what adjoint. but i know there are some redicular. because $N(C)$ is infinity category.so just is a question of $\operatorname{sk_1}(\Delta^3) \rightarrow X$,here X is just simplicial set. in this morphism ,this two extension have what relevent?
 3. in my view $\Delta^3$ like association, maybe $\Delta^n$ like higher association. this intuition is true? if it is true what is special describtion.