Date: Wed, 2 Dec 92 08:13:34 GMT-0500 From: jds@rademacher.math.upenn.edu
what in the world is the Beck-Chevalley condition?
The Chevalley-Beck condition actually deals with bi-fibrations. Let p : E ---> B be a bi-fibration, and consider a pullback square in the base category B, say f A ----------> B | _| | | | h | | g | | V V C ----------> D k Probably the most intuitive case is the one of the codomain functor from the full subcategory of the arrow category Set^2 (where 2 denotes the two-element chain with 0 < 1) that has all monos as objects to Set. The fibres essentially are the power sets. It is a trivial exercise to show that it doesn't matter whether we first take the inverse image under h and then the direct image under f, or first the direct image under k and then the inverse image under g. The Chevalley-Beck condition generalizes this pleasant situation to arbitrary bi-fibrations. Consider a commutative square in E, say t X ----------> Y | | | | u | | v | | V V Z ----------> W w "over" the pullback above, such that u and v are cartesian (this corresponds to taking inverse images in the simple setting) and w is co-cartesian (this corresponds to to taking direct images). The Chevalley-Beck condition now states that t has to be co-cartesian as well. Other people on the net are better qualified to point out the significance of this condition. But I'd like to mention that even in the study of categorical closure operators this type of condition is the *right* one and arises naturally. This discovery was quite a pleasant surprize for me, after wrestling with unwieldy other conditions for a while. I hope this helps! -- J"urgen Koslowski | If I don't see you no more in this world Department of Mathematics | I meet you in the next world Kansas State University | and don't be late! koslowj@math.ksu.edu | Jimi Hendrix (Voodoo Chile) ==============================================================================