Homotopy colimits/limits using model categories

Question

A homotopy (limits and) colimit of a diagram $D$ topological spaces can be explicitly described as a geometric realization of simplicial replacement for $D$.

However, a homotopy colimit can also be described as a derived functor of limit. A model category structure can be placed on the category $\mathrm{Top}^I$, where $I$ is a small index category, where weak equivalences and fibrations are objectwise, so that $\mathrm{colim} : \mathrm{Top}^I \leftrightarrow \mathrm{Top} : c$ form a Quillen pair, where $c$ is the diagonal functor taking an object $A$ to the constant diagram at $A$. Then the homotopy colimit can be described as a derived functor for $\mathrm{colim}$: take a cofibrant replacement $QD$ for a diagram $D$, then compute $\mathrm{colim}(QD) = \mathrm{hocolim}(D)$. It turns out that two cofibrant replacements will give weakly equivalent homotopy colimits. As such, you would suspect that this choice is not really important.

This leaves two questions: firstly, is it necessary in most cases to construct homotopy colimits explicitly, or are its properties as a homotopical functor enough? Secondly, do any problems arise from the fact that homotopy colimit is well-defined only up to weak equivalence (through the derived functor angle)? Do cases ever arise where a more canonical definition is required?

Context: I am reading through Goodwillie's "Calculus II: Analytic Functors." There the explicit simplicial construction is used, and in particular it is needed that certain maps from holim(D) are fibrations (Definition 1.1a, for example). However, being a fibration is not invariant under weak equivalence. Does this reflect that properties of this particular choice of holim are needed, or that the paper itself is too rigid? Can these arguments be made with a non-canonical choice of holim?

I apologize ahead of time for the vague question: I've been trying to read up in this subject area for a few months now, and this has been a stumbling block.

Answer 1

In the context of Goodwillie's paper, he's got an explicit natural transformation $f:holim_I(X)\to holim_J(X|_J)$, where $X:I\to Top$ is a functor to spaces, and $J\subset I$ is a subcategory of $I$. With the construction of holim he's using, this map is always a fibration.

What if you tried to use a different construction of holim? Then maybe you get a map $f'$ which is not a fibration anymore. In that case, you could still have taken the homotopy fiber of $f'$, and this would be a notion which is invariant under weak equivalence. That is, you could (functorially) replace $f'$ with a fibration via the path construction, and take the fiber of that.

Of course, the homotopy fiber is exactly the thing he wants here. In fact, he's manufactured the situation exactly so that the homotopy fiber he wants is just the fiber of this map.

(It's worthwhile to note that in his setting, the category $I$ (which is a cube) has an initial object $\varnothing$. This means that the evident map $holim_I(X)\to X(\varnothing)$ is a weak equivalence. In other words, $holim_I(X)$ is really just $X$ evaluated at $\varnothing$, but modified so that it maps to (and fibers over) $holim_J X|_J$.)