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I want to understand the notion of "fiberwise cone". The only "definition" I found is the following (Ichirō Satake, University of Toronto Press, 1991 ):

There is a more general construction. If $f:Y\rightarrow X$ a map of spaces, think of $Y$ as a space over $X$, think of the mapping cylinder of $f$ as the fiberwise cone of $Y$ over $X$ (another space over $X$), and denote it $C_XY$.

I understand that the mapping cylinder of a $f:Y\rightarrow X$ is the space

$$M_f = ((Y\times [0,1])\sqcup X)\,/_\sim$$

where the union is disjoint, and $\sim$ is the equivalence relation generated by $(y,0)\sim f(y)\quad\text{for each }y\in Y$. That is, the mapping cylinder $M_f$ is obtained by gluing one end $Y\times\{0\}$ of $Y\times[0,1]$ to $X$ via the map $f$. But I don't understand how to link this with the construction of "fiberwise cone" as it is described above, thank you for your help!

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Mapping cylinder retracts to $X$, so $M_f$ is a space over $X$. The preimage of a point $x \in X$ is produced by taking the preimage of $x$ in a map $Y \to X$ and connecting all points there with $x$, so it's just a cone. –  Dmitry Feb 24 '14 at 21:28

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