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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!!

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Isn't the proper setting for all of this model category theory (or at least some sort of setup where you have categories with weak equivalences? –  Harry Gindi Jun 3 '10 at 16:03
Well, you are probably right, but my supervisor doesn't want me to use model category theory because we have not covered model categories in a lectures and thus it would take quite long to develop the language and cover everything that is needed. –  user6499 Jun 3 '10 at 16:34
I'm guessing that by "$\nabla^n$", you mean the topological $n$-simplex (which people usually write $\Delta^n$). And that "$X_f$" and "$X_g$" are really just copies of $X$. Is that right? –  Charles Rezk Jun 3 '10 at 16:35
Yeah, you're right! Thank you! –  user6499 Jun 3 '10 at 20:17
I wouldn't be bothered by Harry; while model categories is one of the more general settings in which to understand homotopy colimits, the theory of homotopy colimits in model categories is inspired by the theory for topological spaces. So you are starting in historically the first place anyway. And an example (even a big example like this) is often a good way to get a handle on a general theory. –  Allison Smith Jun 7 '10 at 19:50
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2 Answers

up vote 6 down vote accepted

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".

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Re homotopy pushouts: These are also analysed in the chapter on Cofibrations in my book Topology and Groupoids (2006).

These ideas of homotopy colimits are also useful in group theory. For example you can replace the trefoil group T which has presentation with generators $x,y$ and relation $x^2y^{-3}$ by the trefoil groupoid $T'$ which has two objects $0,1$, a generator $x$ at $0$, a generator $y$ at $1$, an arrow $\iota: 0 \to 1 $ and a relation $y^3\iota = \iota x^2$. This corresponds to the fundamental groupoid on 2 base points of a double mapping cylinder of maps $S^1 \to S^1$, which is `better' than the ordinary pushout of these maps, as that is not Hausdorff.

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A further point about the use of homotopy colimits in the extension from group to groupoid theory is that in the example given above of the trefoil groupoid, $T'$, the map $\{x,y\} \to T'$ is injective, just as in the double mapping cylinder topological model there are two 1-cells corresponding to the generators. This type of use of groupoids was pointed out to me a long time ago by Eldon Dyer. –  Ronnie Brown May 3 '12 at 10:09
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