Multireflections and Multifactorization Systems (original) (raw)

1987, Demonstratio Mathematica

Recall from [8j that for a class of small categories D (containing the terminal one) and a category C, the objects of the category Pro(D,C) are the functors XsJD op-*• C , where © e D and its hom-sets are Pro(D,E)(X,Y) = lim(colim(C(X(d), _ de® eeE Y(e))). If D0 is the class of all small discrete categories (=all sets), then the set of Pro(DQ, O-morphisms from X:I-»• C to Y:J-» C ia TT (11 (C(X, ,Y-))) (where we write jej \iel 1 1 / X^ for X(i)). In other words, an object of this category is just a set of objects of C and a morphism can be seen as a set (possibly empty) of disjoint sources (in this paper, source means set-indexed source unless otherwise specified). We then call a morphism in Pro(DQ,C) a multisource in e. Sources and multisources in C will be denoted by f,g, etc... If we have to be more precise, we will write (X,fi)j or (X,f.) for sources and (X. ,f^) J or (X,f*') for multisources. J If f = jf*'} = { f i(j)| is a multi-source, f*' will be referred as a source of f and ^(jj as a (C-) morphism of f^. Composition of multisources is made in the most naive way through composition of t-morphisms. (X,<p) and (X,<p) will denote the empty source and multisouroe respectively. In what follows, all subcategories are assumed to be full and isomorphism-closed. Any subcategory >ft of C gives rise to an obvious inclusion functor Pro(D0,U) sPro(DQ,ift) Pro(DQ,e). Theorem 2.4 of [8] shows that is miiltireflective in C if and only if Pro(DQ,U) has a left adjoint. In fact, one can prove that any reflective subcategory of Pro(D0,e) is of the form Pro(D0,A) for some reflective subcategory <ft of C. This can be shown directly but it will follow from our results. 2.1. Definition. A multifactorization system in a category C consists of two classes of sources E and M such that: (i) B and M are both closed for composition with isomorphisms (in Pro(DQ,C)).