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?
Let T : A^op --> Cat be a (pseudo) functor (such as obtained from a fibred category p : E --> A by taking T(a) to be the fibre of E over a in A). Suppose that A has pullbacks and that each T(f) : T(a) --> T(b) has an adjoint. Take a pullback square in A; apply T; now replace a pair of opposite sides in the square of functors by their adjoints. There is a canonical natural transformation in that square. We say T satisfies the Chevalley-Beck condition when this nat tran is invertible for all pullback squares. [See Benabou and Roubaud, Monades et descente, CR Acad Sc Paris 270 (1970) 96-98, and Lawvere, Equality in hyperdoctrines & comprehension schema as an adjoint functor, Proc. Symposia in Pure Math 17 (AMS 1970) 1-14.] The Chev-Beck cond is part of the requirement that T (as a category varying over A) should "have small coproducts" in the sense that every pointwise left kan extension into T, along a functor between small discrete variable categories, should exist. It is also required that each category T(a) should have small coproducts and each functor T(f) should preserve them. --Ross ==============================================================================