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Consider the category $\mathbf{Top}$ of topological spaces as a cofibrantly generated model category, which is generated by $I=\{S^{n-1}\to D^n; n\geq 0\}$ (boundary inclusions) and $J=\{D^n\to D^n\times[0,1]; n\geq 0\}$ (inclusions $x\mapsto(x,0)$). For a continuous map $f\colon X\to Y$ between CW-complexes $X$ and $Y$ (they are I-cell complexes in model categorical sense), its mapping cylinder $M_f=D\times [0,1]\cup_f Y$ could be seen as the pushout of $X\times[0,1]\xleftarrow{(1_X,1)}X\xrightarrow{f}Y$.

My question is I do not know whether the following inclusion is a cofibration in this model categorical sense. $$X\to M_f\quad,\quad x\mapsto(x,0)$$

Or a conclusion of whether this inclusion is a Serre cofibration is also helpful for me! These two questions are equivalent.

(To avoid the confusion on marks 0 and 1, I get the mapping cylinder $M_p$ by attaching $X\times\{1\}$ of $X\times[0,1]$ and $Y$, and the inclusion is to the other side $X\times \{0\}$ in cylinder $M_p$).

Frank
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1 Answers1

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There is a trick. The point is that $X \amalg X \to X \times [0, 1]$ is a cofibration (because $X$ is cofibrant), and we have the following pushout diagram, $$\require{AMScd} \begin{CD} X \amalg X @>{\textrm{id}_X \amalg f}>> X \amalg Y \\ @VVV @VVV \\ X \times [0, 1] @>>> M_f \end{CD}$$ so $X \amalg Y \to M_f$ is also a cofibration. Now use the fact that $X \to X \amalg Y$ is a cofibration (because $Y$ is cofibrant).

Zhen Lin
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