Abstract:Abstract. In this note we consider spatiality of directed inverse limits of spatial locales. We give an example which shows that directed inverse limits of compact spatial locales are not necessarily spatial. This answers a question posed by John Isbell. We also give a condition which, if satisfied by the maps of a directed inverse system, implies that taking limits preserves local compactness and hence produces spatial locales.The question "Which limits are preserved by the embedding of sober topological spac… Show more
Structures of products, coproducts, subobjects, extremal subobjects, quotient objects and extremal quotient objects in CL (the category of closed-set lattices) and CLPair (the category of closed-set lattice pairs) are given.
Abstract-The way-below relation defined on dcpos is a important concept, it is the basis for the continuity of dcpos. And a poset is also a category, so we want to generalize the concept of way-below to category and further define continous category. In this paper, we give the way-below relation on arbitrary small categories and even a topos and introduce the concept of continuous category and obtain some correspontive characterizations by mean of the diagram proof. This also shows diagram method can be used to reconstruct the classical order theory in an arbitrary elementary topos.
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