Does the reduced Mapping cylinder have the same homotopy type of unreduced Mapping cylinder?

Question

Let $ f:X\to Y $ be a map in the pointed category of topological spaces $ Top_* $. And let $ U:Top_*\to Top $ be the "forgetful" functor (which "forgets" the basepoint). We can look at the reduced mapping cylinder $ M_f $ and at the unreduced mapping cylinder $ M_{U(f)} $ in the category $ Top $. Until yesterday, I thought $ M_f $ has the same homotopy type of $ M_{U(f)} $. But, in May's Concise Course, he says "If $X, Y$ are well-pointed, then $ M_f $ has the same homotopy type of $ M_{U(f)} $."

Is this hypothesis necessary?

If it is truly necessary, I want to know where I'm wrong. I thought this: Since $ M_{U(f)} $ is a pushout of a trivial cofibration, we have that $ Y\to M_{U(f)} $ is a trivial cofibration. The same way (or only using a explicit homotopy), we have that $ Y\to M_f $ is a homotopy equivalence. So we have that $ M_{U(f)}\equiv Y\equiv M_f $.

I know that the first statement is right. If there is something wrong, it is in the second statement. I believed that we can factor any function in $ Top _* $ in the same way as in $ Top $, id est, $ f= R\circ j $, where $ j: X\to M_f $ is a cofibration and $ R: M_f\to Y $ is a strong retract. Is it wrong?

Answer 1

I don't trust anything with degenerate basepoints, but I do think you are right about the homotopy type via the parenthetical "(or only using an explicit homotopy)''. However, I was working in compactly generated spaces, denoted $\mathcal{U}$, including the weak Hausdorff property. I don't think you can prove that $Mf$, constructed as usual, is in $\mathcal{U}$ without assuming nondegenerate basepoints. Also, without nondegenerate base points, you cannot be sure that $j$ is a cofibration (at least not in the unbased sense; see p. 56 of Concise). In More Concise, Ponto and I take $\mathcal{T}$ to mean nondegenerately based spaces and $\mathcal{U}_*$ to mean based spaces in $\mathcal{U}$. This is justified model theoretically by noting that $\mathcal{T}$ is the full subcategory of $h$-cofibrant objects in $\mathcal{U}_{*}$ (that is, cofibrant in the based Hurewicz model structure on $\mathcal{U}_{*}$). Incidentally, one reason to start work in $\mathcal{U}_{*}$ and not $Top_{\ast}$ is that the smash product in $Top_{\ast}$ is not associative (a published source for a very old counterexample is Parametrized homotopy theory, by Sigurdsson and myself).

Answer 2

I'm lousy at point-set topology, always was, but I don't see that $X$ is closed in $X\wedge I_+$. This is the sort of question a working algebraic topologist does not want to think about. Cofibrant approximation in the $h$-model structure on $\mathcal{U}_{*}$ takes $X$ to the whiskered space $X\vee I$ with new basepoint at $1$ if the basepoint of $I$ is taken to be $0$. If $X$ has a nasty degenerate basepoint, take the nasty thing away: I do not want to think about it.