Intersections of reflective subcategories

It is shown that the class of full reflective subcategories of 5t (and of other concrete categories) is not closed under intersections. This answers a question raised by Herrlich in 1967. A natural example of nonreflective intersection is presented in the category of bitopological spaces. We investigate reflective subcategories (subcategories are always understood full and replete) of concrete categories, i.e., categories equipped with a faithful functor to 5 er. H. Herrlich asked whether each subcategory of 5? has a reflective hull (or, equivalently, whether reflective subcategories of gare closed under intersections) at the conference "Contributions to extension theory of topological structures" in Berlin, 1967; see also [H1]. The question has remained open in spite of considerable effort in the area; see [T] and references there. We present here a negative solution. Recall from [PT] that a concrete category X is universal if each concrete category has a full embedding into Z. Further, X is almost universal if either it is universal, or if (a) constant maps between underlying sets always carry X-morphisms, and (b) each concrete category ' has an almost full embedding E: ' Z. This means that E(hom(C,C')) = {f: EC -* EC'If is nonconstant} for all objects C,0' of W. An object is said to have rank n for some regular cardinal n if the hom-functor of that object preserves n-directed colimits of extremal monos. THEOREM. LetX% be a cocomplete, cowellpowered and almost universal concrete category in which each object has a rank. Then X has a collection of reflective subcategories with nonreflective intersection. PROOF. Morphisms in X have (epi, extremal mono-) factorizations (see [HS, 34.1]), and thus it follows from [Ke, 10.1] that for each morphism f: A -* B the orthogonal subcategory {f}1 is reflective. (Recall that {f}' consists of all K such that for each p: A -* K there is a unique q: B -* K with p = q.f.) It is sufficient to find a class Z of morphisms such that R" = nfc } is nonreflective. Define an auxiliary category W. Its objects are Ai, Bi, and C for all ordinals i. Its morphisms are freely generated by the following morphisms aij: Ai -* Aj (i, j E Ord, i < j), Itijk: Ai -* Bk (i, k E Ord), and -yi: C -* Ai (i E Ord), and the Received by the editors January 5, 1987 and, in revised form, April 8, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 18A40; Secondary 18B30, 54B30. ? 1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page