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Suppose we have a finite diagram in the category of topological spaces. What is the condition for homotopy equivalence of the natural map from homotopy direct limit of this diagram to it's direct limit? I'm also interested in good sufficient conditions if there is no simple necessary and sufficient condition.
For an explicit example, I'd like to know the answer for a diagram consisting of three space $A, B, C$ with two maps from $A$ to $B$ and $C$.
Thanks!
Here is the general condition: If $\mathcal D:\mathscr J\to\mathbf{Top}$ is a diagram of spaces and $\mathscr J$ is a very small category (meaning that for any object $a$ in $\mathscr J$ there is an upper bound for the length of compositions of arrows ending at $a$), then $D$ may be called cofibrant if for every $a\in \mathscr J$, the arrow $s^\mathcal D_a:\partial_a(\mathcal D)\to \mathcal D(a)$ is a cofibration, where $s^\mathcal D_a$ is defined as follows:
Let $\partial a$ be the subcategory of $(\mathscr J\downarrow a)$ missing the object $\mathbf 1_a$, and let $\pi_a:\partial a\to \mathscr J$ be the projection functor. If $\partial_a(\mathcal D)$ denotes the colimit of the functor $\mathcal D\pi_a$, then the maps $\mathcal D(m):D(e)\to D(a)$ at the object $m:e\to a$ of $\partial a$ induce an arrow $s^\mathcal D_a: \partial_a(\mathcal D)\to \mathcal D(a)$.
For the diagram $\mathcal D=B\leftarrow A\to C$ this amounts to both maps being cofibrations since $s^\mathcal D_B=(A\to B),\ s^\mathcal D_C=(A\to C),$ and $s^\mathcal D_A=(\emptyset\to A)$.
For cofibrant diagrams $\mathcal D$, the fiber projection $p_f:\text{hocolim}(\mathcal D)\to\text{colim}(\mathcal D)$ is a homotopy equvalence.
The reason behind this is that $\mathbf{Top}$ is a model category, and so is $\mathbf{Top}^\mathscr J$ when $\mathscr J$ is a very small category. The fibrations and weak equivalences are defined pointwise. The cofibrations, however, are a bit difficult to describe, but cofibrant diagrams are then just the diagrams satisfying the property above.
Now the homotopy colimit can be expressed as the colimit of a certain cofibrant diagram $Q\mathcal D$, which is pointwise homotopy equivalent by a canonical diagram map $Q\mathcal D\to D$, and the colimit of that map is just the fiber projection $p_f$. But between cofibrant diagrams the colimit preserves weak equivalences, being left adjoint to the constant diagram functor $\Delta:\mathbf{Top}\to\mathbf{Top}^\mathscr J$, which preserves fibrations and acyclic fibrations.
