# Which notion of equivalence is being used here?

I'm reading Homotopy associativiy of H-spaces I by Stasheff and in the proof of Proposition 2 he says that any map is equivalent to a fibring (in modern day language I think this is either a fibration or a fiber bundle). He refers to Lemma 13 of this paper by Nomura, where he states that any map $$f:X\to Y$$ is equivalent to a fibring. The fibring is of the form $$p:Z_f\to Y$$ where $$Z_f$$ is constructed in the proof.

I would like to know in what sense these maps are equivalents. The map $$f$$ need not be a fibring so this is not an equivalence of fibrings. A homotopy equivalence wouldn't make sense either because the domains are different. I haven't found in any of the linked papers any definition of this equivalence, so does anyone knows what they're talking about?

According to Stashef there should be a map $$X\to Z_f$$ commuting with $$f$$ and $$p$$, so maybe they mean that $$f$$ is homotopy equivalent to the composition of $$p$$ with this map.

By fibring Stasheff means Hurewicz fibration. That is a map $$p:E\rightarrow B$$ which has the homotopy lifting property with respect to all spaces.
By equivalent to Stasheff means the following. Given any map $$f:X\rightarrow Y$$ there is a Hurewicz fibration $$p_f:W_f\rightarrow Y$$ and inverse homotopy equivalences $$X\xrightarrow{i} W_f\xrightarrow{q} X$$ such that $$p_f\circ i=f,\qquad \qquad f\circ q\simeq p_f.$$ It is in the sense of the first equation that $$f$$ is equivalent to $$p_f$$.
As a model for the space it is standard to take $$W_f=\{(x,l)\in X\times Y^I\mid f(x)=l(0)\}$$ with $$p_f(x,l)=l(1).$$ Here $$Y^I$$ is the space of all maps $$I\rightarrow Y$$ in the compact-open topology. The map $$i$$ sends $$x$$ to the pair $$(x,c_{f(x)})$$, where $$c_{f(x)}$$ is the constant path at $$f(x)$$. The map $$q$$ is the projection $$(x,l)\mapsto x$$.
See, eg. M. Arkowitz's book Introduction to Homotopy Theory $$\S$$3.3 for details.