TY - GEN

T1 - Weak products and Hausdorff locales

AU - Johnstone, Peter

AU - Sun, Shuhao

PY - 1988

Y1 - 1988

N2 - It is well known that one of the salient differences between the category of locales and that of spaces is that the notions of product in the two categories do not coincide: that is, if X and Y are spaces, with open-set lattices Ω(X) and Ω(Y), the locale product Ω(X) xl Ω(Y) is in general different from Ω(X x Y). (Note: here and throughout the paper, our notation relating to locales is taken from [5].) In this paper, we introduce a construction which goes some way towards "reconciling" this difference: specifically, we show that the assignment (Ω(X), Ω(Y)) → Ω(X x Y) is the restriction to spatial locales of a symmetric monoidal structure defined on the whole category of locales (which we denote by (A,B) → A⊗B, and call the weak product structure), such that A⊗B is (naturally) a dense sublocale of A x1 B for any A and B. (There is also an infinitary version of ⊗, which we shall define although we shall not investigate its properties in any great detail.) Consideration of the extent to which ⊗ differs from the categorical product leads us to introduce anew class of "weakly spatial" locales, which forms the largest (coreflective) subcategory of Loc on which the restriction of ⊗ yields the categorical product; thanks to a recent example of Kriz and Pultr, we know that it is strictly larger than the class of spatial locales.

AB - It is well known that one of the salient differences between the category of locales and that of spaces is that the notions of product in the two categories do not coincide: that is, if X and Y are spaces, with open-set lattices Ω(X) and Ω(Y), the locale product Ω(X) xl Ω(Y) is in general different from Ω(X x Y). (Note: here and throughout the paper, our notation relating to locales is taken from [5].) In this paper, we introduce a construction which goes some way towards "reconciling" this difference: specifically, we show that the assignment (Ω(X), Ω(Y)) → Ω(X x Y) is the restriction to spatial locales of a symmetric monoidal structure defined on the whole category of locales (which we denote by (A,B) → A⊗B, and call the weak product structure), such that A⊗B is (naturally) a dense sublocale of A x1 B for any A and B. (There is also an infinitary version of ⊗, which we shall define although we shall not investigate its properties in any great detail.) Consideration of the extent to which ⊗ differs from the categorical product leads us to introduce anew class of "weakly spatial" locales, which forms the largest (coreflective) subcategory of Loc on which the restriction of ⊗ yields the categorical product; thanks to a recent example of Kriz and Pultr, we know that it is strictly larger than the class of spatial locales.

M3 - Conference Paper

SN - 9783540503620

VL - 1348

T3 - Lecture Notes in Mathematics

SP - 173

EP - 193

BT - Categorical Algebra and its Applications

A2 - Borceux, Francis

PB - Springer

CY - Germany

ER -