# Terminology for fiberwise maps

I would like to know the standard terminology for the following two notions.

Notion 1: $E_1\to B$ and $E_2\to B$ are fibrations over the same base space, and $f\colon E_1\to E_2$ is a map making the evident triangle commute.

Notion 2: $E_1\to B_1$ and $E_2\to B_2$ are fibrations over possibly different base spaces, and $f\colon E_1\to E_2$ and $\phi\colon B_1\to B_2$ are maps making the evident square commute.

Of course, notion 1 is the special case of notion 2 where $\phi$ is an identity map.

Some phrases I can think of that might be used to describe either of the two notions are:

• map of fibrations
• parametrized map
• fiberwise map
• fiber-preserving map

Is there a standard convention in algebraic topology regarding which phrase refers to which notion?

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I think the first one (map of fibrations) is the simplest and also most common one. – Martin Brandenburg Jan 23 '12 at 19:54
Martin, is that an answer? Which phrase are you saying applies to which notion? – Mike Shulman Jan 23 '12 at 22:23
Hatcher calls Notion 1 a fiber-preserving map (p. 406). You might also consult Ioan James' books on fiberwise topology. – Dan Ramras Jan 23 '12 at 22:41
In "notion 1," why not "map of spaces over $B$?" In fact fibrations over $B$ are fibrant objects in a Quillen model structure on the comma category $\text{TOP}/B$. What you have is a morphism of fibrant objects in that model category. – John Klein Jan 23 '12 at 23:30