In a recent paper (TAC, Vol 4, 208-248) On Generic Separable Objects, by Robbie Gates, the author mentioned a well known "boolean algebraic structure of the summands of an object in an extensive category". This reminded me a paper I posted to my homepage last year (8/30/98, see the abstract below), in which the same boolean structure was reconstructed (at that time this was not "well known" to me), and was applied to define the Pierce topology for any extensive category, extending some results of Diers. The paper Pierce Topologies of Extensive Categories is available at Categorical Geometry Homepage at the following new (permanent, hopefully) address http://www.geometry.net (or http://www.azd.com) (The new service is a little bit slow, but offers more functions than the old one, so please be patient.) Best wishes, Zhaohua Luo ----------------------------------------------------------------------------------------- Pierce Topologies of Extensive Categories by Zhaohua Luo Abstraction An extensive category is a category with finite stable disjoint sums. In this note we show that each extensive category carries a natural subcanonical coherent Grothendieck topology defined by injections of sums. This Grothendieck topology is induced by a strict metric topology, which is a functor to the category of Stone spaces. We call this metric topology the Pierce topology of the category, as it generalizes the classical Pierce spectrums of commutative rings. Recall that the Pierce spectrum of a commutative ring R is the spectrum of the Boolean algebra of idempotents of R, which is a Stone space. A theorem of R. S. Pierce states that R can be represented as the ring of global sections of a sheaf of commutative rings on its Pierce spectrum (called the Pierce sheaf or representation), whose stalks are indecomposable rings (with respect to product decomposations). Diers showed that Pierce's theorem can be extended to any object in a locally finitely presentable category such that the opposite of the subcategory of finitely presentable objects is lextensive (called a locally indecomposable category). We shall see that a weak form of Pierce representation exists for any object in an extensive category.