I have a concrete problem with the homotopy fiber and I am getting lost with the
literature. I state my question and, to avoid confusions, I state downwards the
definitions I am using.


Let $C$ be a pointed model category. To make things simpler, assume every object
is fibrant. Recall that the definition of the homotopy fiber of a morphism $f\colon
Y\to X$ is the homotopy pullback of the diagram
$$\begin{matrix}& & Y \\
&&\downarrow^f \\ \*&\to & X\end{matrix}
$$
That is to say, we replace $\*\to X$ and $f$ by fibrations (in the category of diagrams) and take the classic
pullback.


Now, if I have a map $Z\to X$ that fits in a commutative square
$$
\begin{matrix}
Z&\to &Y\\
\downarrow &&\downarrow^f\\
\*&\to & X.
\end{matrix}\quad (1)
$$
then it factors through the fiber. More concretely, there is a map $Z\to
\mathrm{fib}$ such that the diagram
$$
\begin{matrix}
Z&\to&\mathrm{fib}&\to &Y\\
&&\downarrow &&\downarrow^f\\
&&\*&\to & X.
\end{matrix}
$$
commutes in $C$ and the composed maps $Z\to \*$ and $Z\to X$ are those of (1). 


My question: **is there an analogue map to the homotopy
fiber?**, i. e., is there a commutative square in $C$
$$
\begin{matrix}
Z&\to&\mathrm{hofib}&\to &Y'\\
&&\downarrow &&\downarrow^{Rf}\\
&&\*'&\to & X'.
\end{matrix}\quad (2)
$$
where $\*'\to X'$ and $Rf\colon Y'\to X'$ are the fibrant replacements?




---


Let me recall the general definition of the homotopy limit. Denote $D$ the above
diagram. Then $C^D$ has a model structure. Note $\mathrm{lim}\colon C^D\to C$
the functor *taking pullback*. The homotopy pullback is the right derived
functor of $\mathrm{lim}$ in the sense of Quillen in "Homotopical algebra".
Therefore, in order to have such a map I should have a functor $F$ such that for
*every* diagram $d\colon D\to C$ I have an object $F(d)$ with maps as in (1) and
that in my concrete diagram gives $Z$. Then I would have a diagram like (2). Right?


I have read in n-category lab that this definition is called *global* and there is another definition of the homotopy fiber called *local* and that, under certein hypothesis, they seem coincide. Do this terminology refer to this problem? I am unable to understand the notations in the literature.



Fiber vs homotopy fiber in model categories: simple question
I work in a general pointed model category. The homotopy fibre of a morphism $f : Y \to X$ can be defined as follows: first, choose a fibrant replacement $w\_X : X \to \hat{X}$ and then factor $w\_X \circ f : Y \to \hat{X}$ as a weak equivalence $w\_Y : Y \to \hat{Y}$ followed by a fibration $\hat{f} : \hat{Y} \to \hat{X}$; then the homotopy fibre (with respect to these choices) is the ordinary fibre of $\hat{f} : \hat{Y} \to \hat{X}$.


Thus, we have the following commutative diagram,
$$\require{AMScd}
\begin{CD}
\operatorname{fib} f @>>> Y @>{w\_Y}>> \hat{Y} @<<< \operatorname{hfib} f\\
@VVV \mathrm{pb} @VV{f}V @V{\hat{f}}VV \mathrm{pb} @VVV \\
0 @>>> X @>>{w\_X}> \hat{X} @<<< 0
\end{CD}$$
and in particular, we get a comparison $\operatorname{fib} f \to \operatorname{hfib} f$ making the following diagram commute:
$$\begin{CD}
\operatorname{fib} f @>>> \operatorname{hfib} f \\
@VVV @VVV \\
Y @>>{w\_Y}> \hat{Y}
\end{CD}$$


If you can choose $(\hat{X}, w\_X, \hat{Y}, w\_Y, p)$ functorially, then you get a homotopy fibre functor, and it can be shown to be a right homotopical approximation of the fibre functor. This is the "global" universal property. The "local" universal property is more or less built into the definition I give above. One way of getting a feel for it is to think about what happens at the level of the homotopy category: in this case, it says that, for any morphism $g : Z \to Y$ such that $f \circ g = 0$ *in the homotopy category*, there is a (not necessarily unique) morphism $Z \to \operatorname{hfib} f$ making the evident diagram commute (in the homotopy category). But, of course, this is not a complete characterisation of $\operatorname{hfib} f$.


