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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

You could look at papers such as

Booth, Peter I.; Heath, Philip R.; Piccinini, Renzo A. Fibre preserving maps and functional spaces. Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), pp. 158–167, Lecture Notes in Math., 673, Springer, Berlin, 1978.

which study spaces of maps between maps and apply corresponding exponential laws.

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Thanks. Are you saying you think their use of terminology (whatever it is) is the standard one? – Mike Shulman Jan 23 '12 at 22:24
Can't say! There may not be "the" standard. Getting that can be like herding cats. But the work of Peter Booth on fibrewise topology was the foundation of work of Ioan James in this area, see his book "Fibrewise topology" CUP, which might be of help. – Ronnie Brown Jan 24 '12 at 10:21

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