Terminology for fiberwise maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:30:47Z http://mathoverflow.net/feeds/question/86473 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86473/terminology-for-fiberwise-maps Terminology for fiberwise maps Mike Shulman 2012-01-23T19:24:53Z 2012-02-07T09:33:06Z <p>I would like to know the standard terminology for the following two notions.</p> <p><strong>Notion 1:</strong> $E_1\to B$ 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.</p> <p><strong>Notion 2:</strong> $E_1\to B_1$ 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.</p> <p>Of course, notion 1 is the special case of notion 2 where $\phi$ is an identity map.</p> <p>Some phrases I can think of that might be used to describe either of the two notions are:</p> <ul> <li>map of fibrations</li> <li>parametrized map</li> <li>fiberwise map</li> <li>fiber-preserving map</li> </ul> <p>Is there a standard convention in algebraic topology regarding which phrase refers to which notion?</p> http://mathoverflow.net/questions/86473/terminology-for-fiberwise-maps/86481#86481 Answer by Ronnie Brown for Terminology for fiberwise maps Ronnie Brown 2012-01-23T20:38:00Z 2012-01-23T20:38:00Z <p>You could look at papers such as </p> <p>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. </p> <p>which study spaces of maps between maps and apply corresponding exponential laws. </p>