When I learn model category, the important compute tool is the lifting property between $(\operatorname{Cof}, \operatorname{Fib} \cap W)$ and $(\operatorname{Cof} \cap W, \operatorname{Fib}) $, where $W$ is class of weak equivalence. This means for commutative diagram whose 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 $f$, $g$.
- 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.
- 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 don't know mean what adjoint. But I know there are some redicular. because $N(C)$ is infinity category. So just is a question of $\operatorname{sk_1}(\Delta^3) \rightarrow X$ where X is just simplicial set. in this morphism, this two extension have what relevant?
- in my view $\Delta^3$ like association, maybe $\Delta^n$ like higher association. this intuition is true? if it is true what is special description.
