There is a theorem of Dold (in Partitions of unity in the theory of fibrations) saying that if $X$ is a CW-complex and $Y\to X$, $Y'\to X$ are two fibrations connected by a map $f:Y\to Y'$ over $X$ then $f$ is a homotopy equivalence iff $f$ is a fibre homotopy equivalence (meaning that the homotopy inverse is also a map over $X$).

I wonder if this, for a connected $X$ and CW-complexes $Y$ and $Y'$, is an easy consequence of the modern language (of course, the difficulties are just hidden then) as follows:

The model structure on topological spaces $TOP$ induces a model category on the slice category $TOP/X$. Then $Y$ and $Y'$ are cofibrant (as they are CW-complexes) and fibrant(!) objects and $f$ is a morphism in $TOP/X$ which is a weak equivalence. Hence, the Whitehead theorem implies that $f$ is a homotopy equivalence in $TOP/X$ and therefore the inverse is also over $X$.


1 Answer 1


I think your argument is correct. I assume you are working with Quillen model structure, since you mention $Y, Y \prime$ being cofibrant since they are CW-complexes. However, due to this fact the result is not really as strong as we'd want it to be, since very important classes of fibrations do not have domain a CW-complex (say, the path fibration).

You could, of course, work with Strøm model structure to obtain the general theorem but proving the model category axioms in this case is known to be rather delicate. To quote nLab:

The theorem might have been a folklore at the time, but the actual paper has a number of subtleties.

Strøm’s proofs are not that well-known today and use techniques better known to the topologists of that time, and there is consequently a slight controversy among topologists now. One of these is that there are modern reproofs, but these modern techniques essentially use compactly generated spaces, while Strøm’s proofs succeeded in avoiding that assumption.

I've been thinking about this phenomena for a while when proving the "invariance of homotopy pullbacks" using basically only axioms of fibration categories, which would probably require a substantial homotopical argument if I was trying to prove it directly. In short, it seems that model structures on spaces - if we take them for granted - can lead to short and conceptual arguments, like yours.


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