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I am a little bit confused by Serre and Hurewicz fibrations in the context of pointed spaces, i.e. in $Top_*$.

Serre and Hurewicz fibrations are defined in $Top$, i.e. for non-pointed spaces, as having the RLP for inclusions $i_0 \colon I^n \hookrightarrow I^n \times I$ and $i_0 \colon X \hookrightarrow X \times I$ respectively.

But if I consider now a pointed map $p \colon (E,e_0) \to (B,b_0)$ and a pointed homotopie $H \colon (I^n,\partial I^n) \times I \to (B,b_0)$ for Serre or $H \colon (X,x_0)\times I \to (B,b_0)$ for Hurewicz, and if $p$ is a Serre/Hurewicz fibration in Top, there is no garantee that the lift $\tilde{H}$ is also a pointed homotopy. All we can say is that $\tilde{H}(* \times I) \subset F$ where $F = p^{-1}(b_0)$ is the fiber of $p$. So a priori $p$ is not a Serre/Hurewicz fibration in $Top_*$

Thus it seems that a specific more restrictive notion of Serre/Hurewicz fibration should be used in the context of $Top_*$, as having the RLP in $Top_*$ for the mentionned inclusions, i.e. the lift is should also be a pointed homotopy.

But I have never seen explicitely such a definition in classic books or pdf, although in most theorems involving homotopy groups we are in fact in the $Top_*$ context. Why ?

Now it is true that Serre fibration also have the RLP for $i \colon J^n \hookrightarrow I^n \times I$ where $J^n = \partial I^n \times I \cup I^n \times 0$. If there is a commutative square as usual where the initialization is $J^n \mapsto e_0$, then the lift is such that $\tilde{H}(\partial I^n \times I) = e_0$ and $\tilde{H}$ is a pointed homotopy. In this way, a Serre fibration is also a "pointed" Serre fibration. But this does not work for Hurewicz.

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This all works as long as the basepoint $x_0$ of $X$ is nondegenerate. The general context for this is due to Arne Strom who showed that the category of topological spaces, with the classic (i.e. Hurewicz) definitions of cofibration and fibration, is a model category. The key theorem goes as follows: Suppose given a commutative square $$(i: A \rightarrow Y) \rightarrow (p: E \rightarrow B),$$ with $i$ a Hurewicz cofibration and $p$ a Hurewicz fibration. If either one is also a homotopy equivalence, there exists a map $Y \rightarrow E$ making the two evident triangles commute (i.e. extending $A\rightarrow E$ and lifting $Y \rightarrow B$).

Now apply this to the case when $Y=X \times I$ and $A = X \times 0 \cup x_0 \times I$. You can check that $A \hookrightarrow Y$ will be a cofibration and a homotopy equivalence if $x_0$ is nondegenerate.

Besides Strom's original papers from 1966 and 1968, a nice very short proof of this general theorem appears at the beginning of section 17 of the book by May and Ponto.

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