I am following the explicit construction of homotopy colimits as described by Dugger in the paper: "Primer on homotopy colimits", which can be found here: http://www.uoregon.edu/~ddugger/hocolim.pdf
As described in the appendix of Topological hypercovers and A1-realizations, Mathematische Zeitschrift 246 (2004) in the category of topological spaces no cofibrant-replacement functor is needed when computing the homotopy colimit of a small diagram in $\mathcal{T}op$.
For the index category $\mathcal{I} = \cdot \rightrightarrows \cdot$ and a small diagram $D: \mathcal{I} \rightarrow \mathcal{T}op$ with the image $X \rightrightarrows Y$ where $f, g: X \rightarrow Y$ this yields the space $T := (X \times \nabla^0 \amalg Y \times \nabla^0 \amalg X_g \times \nabla^1 \amalg X_f \times \nabla^1) / \sim$ where $\sim$ is given by:
$(x, 1) \sim (x, (0,1)) \in X_f \times \nabla^n, X_g\times \nabla^n$, $(f(x), 1) \sim (x, (1,0)) \in X_f \times \nabla^n$ and $(g(x), 1) \sim (x, (1,0)) \in X_g \times \nabla^n$ for all $x \in X$.
Notation: $\nabla^n$ is the topogical n-simplex, $X_f$ and $X_g$ are just copies of X indexed by a map in the diagram to keep track of all the identifications
1) Are any requirements necessary for $T$ to be homotopy equivalent or weakly homotopy equivalent to $colim_{\mathcal{I}}{D}$?
2) What are the requirements for a homotopy pushout to be homotopy or weakly homotopy equivalent to the ordinary pushout?
3) What are the requirements for a homotopy colimit of a small diagram from the category obtained from the preorder $(\mathbb{N}, \leqslant)$ to $\mathcal{T}op$ to be be homotopy or weakly homotopy equivalent to the infinite mapping telescope as described in Section 3F (page 312) in the book about algebraic topology by Hatcher?
Since I barely know any model category theory, I would appreciate any elementary answers to this! Thank you very much!!
Best Answer
Here's an answer to question 2: A sufficient condition for $\mathrm{colimit}(X \leftarrow A\rightarrow Y)$ to be weakly equivalent to the homotopy colimit, is (a) for one of the maps (say $A\to X$) to have the homotopy extension property. Another sufficient condition is that the diagram $(X\leftarrow A\rightarrow Y)$ is (b) an "excisive triad". Take a look at Chapter 10, section 7 of May's "Consise Course", where (b) is proved to be such a sufficient condition. You can show that (a) is a sufficient condition by showing directly that it is homotopy equivalent to the double mapping cylinder $X\cup A\times \Delta^1\cup Y$, which can be re-analyzed as an "excisive triad".