Luciano Stramaccia

Characterizing shape theories by Kan extensions

Abstract Every pair ( C , K ) of categories, where K is a proreflective subcategory of C , generates a shape theory. As a main result in this paper we give a characterization of such pairs, showing that these are exactly those having the property that every functor F : K → A has a Kan extension Ran F : C → A , which is preserved by all functors commuting with inverse limits.

A NOTE ON REGULAR AND EXTREYAL SUBOBJECTS

ABSTRACT In the general setting of a complete, well-powered category A, we define and study two universal closure operations: regularization and extremalization, by means of regular and extremal subobjects of A. respectively. A general theorem of characterization of epimorphisms in A is given. When A is an epireflective subcategory of TOP, such operations are shown to coincide with A-closure [11] and epiclosure [2]. respectively. In the topological contest, regularization and extremalization are studied in detail and compared with r-closure, defined in [13].