# Is a pullback along a Dold fibration a homotopy pullback?

Let $$\begin{array}{ccc} A & \to & B \cr\downarrow&&\downarrow \cr A'& \to &B' \end{array}$$ be a pullback square in the category of all topological spaces (not just in a convenient one) and the arrow $B\rightarrow B'$ a Dold fibration.

Is the square always a homotopy pullback square

1. in the Quillen model structure?
2. in the Strom model structure?

In less abstract terms, is the canonical map from the pullback of the Hurewicz fibration associated to $B\rightarrow B'$ with $A'\rightarrow B'$ to $A$ a

1. weak homotopy equivalence?
2. homotopy equivalence?

Note that a Dold fibration may not be a Serre fibration.

• It seems potentially likely in the Strøm model structure. Note you should say a Serre fibration might not be a Dold fibration (not so much the other way around), but a Hurewicz fibration is a Dold fibration (see mathoverflow.net/questions/20442/…) Aug 3, 2015 at 15:18

In this paper of Charles Rezk he proves, among other things, that in any right proper model category, all base changes of a map $f:B \rightarrow B'$ give homotopy Cartesian squares if and only if the map is sharp.
Definition. A map $f:X \rightarrow S$ is sharp if, after any base change to $X'\rightarrow S'$ and weak equivalence $S''\rightarrow S'$ , the map $X'\times_{S'}S''\rightarrow X'$ is also a weak equivalence.