Factorizations, M-separation, and extremal-epireflective subcategories
Let C be a complete, well-powered, concrete category with forgetful functor \(T: C\to Set\) such that the class \(M_ 0\) of T-embeddings is the second factor of a factorization structure for single morphisms in c. Every (E,M)-factorization structure for single morphisms with \(M\subset M_ 0\) determines the full subcategory \(C_ M\) whose objects are the C- objects X such that the diagonal \(\Delta_ X\subset X\times X\) is in M (M-separated objects in a paper of \textit{D. Pumplün} and \textit{H. Röhrl} [Manuscr. Math. 50, 145-183 (1985; Zbl 0594.46064)]). For every subcategory A of C denote by \(M^ A\) the class of A-regular morphisms (i.e., equalizers of pairs of C-morphisms with codomain in A). Main results of the paper are: (a) \(C_ M\) is extremal epireflective in C and it is the largest \(B\subset C\) such that \(E\subset \{B\)-epis\(\}\) ; (b) If C is a connected topological category and A is extremal epireflective in C and such that \(M^ A\) is closed under composition then there exists a factorization structure (E,M) such tat \(C_ M=A\); (c) There is no (E,M)- factorization structure in Top such that \(Top_ M=FH\), where FH is the category of functionally Hausdorff spaces. It should be noted that it is implicitly shown in a paper of \textit{D. Dikranjan} and the reviewer [Topology Appl. 27, 129-143 (1987; Zbl 0634.54008)] for more general \(C: C_ M=A\) for some factorization structure (E,M) in C iff \(M^ A\) is closed under composition. This result depends on a general ``diagonal theorem'' shown by the reviewer, \textit{S. Mantovani} and \textit{W. Tholen} [J. Pure Appl. Algebra 51, 129-140 (1988)] (see also: the reviewer and \textit{M. Hušek} [Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006)] for the case \(C=Top)\).
scientific article; zbMATH DE number 4055850
scientific article