By Andre Joyal, Myles Tierney

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A continuous map f: X + Y is open iff for any subspace S0-—> Y, we have o _1 f^CS) = f (s) . Proof: Suppose f is open. For any open subspace Uc—> X we have 38 A. JOYAL CT M. TIERNEY U <_ f (S) £(U) < £(U) < o s s U £ f" X ( S ) U < f X (S) o showing that X f / -1 ° -\ (S) = f (S). be open, Conversely, suppose this is true, and le We have f(U) < S u< f" l(S) u< f" u< f" o 1 ° (S) f(U) < S showing that 4. f(U) is open. Open Surj ections, pullbacks Let f: X -*• Y be an open mapping. 3 f (f"(u)) = 3 f ( f ' ( u ) A l ) = U A 3 an epimorphism, and we call We say surjection.

An easy calculation shows that a point of Q is nothing but a function from IN to X, so sheaves(Q) classifies these GALOIS THEORY 37 functions. Now, to form the space of epimorphisms from IN to X, we simply extend the covers by specifying that, in addition, Vx e X{ [xQ,. . ,x n]. Again, it is immediate that this is a covering system, and a point of the new Q is now an epimorphism from ]N to X. ,x ] corresponding to x. Returning to characterizations of openness, we have Proposition 3. A continuous map open subspace U*"-—> X is open.

F: A -> B is an effective descent morphism for modules iff is pure as a morphism of A-modules. f We make a few remarks now on descent of structure, since we will need them in the following chapters. One basic fact underlies all the discussion. Namely, let g: C •> D be a morphism of commutative monoids. (M®N), c c so g! preserves 8-product. c As a consequence, consider 20 A. JOYAL $ M. (N). Moreover, suppose A A is, say, a B-algebra (a monoid in Mod(B)). Thus, we have an associative multiplication m: M0M B -* M M l~- f!