# Serre and Hurewicz fibrations definition for pointed spaces?

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.

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.