A functorial treatment of factorization structures is presented, under extensive use of well-pointed endofunctors. Actually, so-called weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordant-dissonant and inseparable-separable.

: The notion of closure operator on a category is explored, utilizing the approach of Dikranjan and Giuli. Conditions on the underlying factorization structure are given, which allow the construction of closure operators satisfying a variety of extra conditions. KEY WORDS: closure operator, factorization structure, separated object, sheaf, closure commuting with pullbacks CLASSIFICATION: 18A32, 18B99, 18D30 0 INTRODUCTION The basic idea for a closure operator on a category X is to have for each object X an extensive, isotone and idempotent operation on the partially ordered class of its subobjects. For these operations to be compatible with the structure of X , one would like the X - morphisms to be "continuous" in some sense with respect to them. If X has pullbacks (to be thought of as inverse images) of monos, this quite literally means that inverse images of closed subobjects are closed. It turns out that this notion of closure operator may be generalized in two ways. Often parti...

: Basic results are obtained concerning Galois connections between collections of closure operators (of various types) and collections consisting of subclasses of (pairs of) morphisms in M for an hE; Mi-category X . In effect, the "lattice" of closure operators on M is shown to be equivalent to the fixed point lattice of the polarity induced by the orthogonality relation between composable pairs of morphisms in M . KEY WORDS: Galois connection, polarity, closure operator, closure operator, composable pair of morphisms, factorization structure for sinks. CLASSIFICATION: Primary: 18A32 Secondary: 06A15, 54B30 0. INTRODUCTION To study closure operators we introduce two new concepts that are reasonably simple, yet apparently quite powerful. The first one is the relation ? , which naturally extends earlier notions of "diagonalizability" or "orthogonality". The second one is the subcollection M \Pi M of M \Theta M for M ` Mor (X ) that consists of the composable pairs in M . In Section...

: In an hE; M i - category X for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms in M to factor through the "lattice" of all closure operators on M , and to factor through certain sublattices. This leads to the notion of regular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplun--Rohrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class of X - objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators ...

It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the "squaring monad" on CAT. This structure largely facilitates natural choices for left or right liftings once certain basic natural choices have been made. The use of homomorphisms of such lax algebras is also discussed, with focus on "lax freeness". Mathematics Subject Classification: 18A32, 18C20, 18D05, 55P05. Key words: weak factorization system, cofibrantly generated system, (symmetric) lax factorization algebra, lax homomorphism. Supported by the Ministry of Education of the Czech Republic under project MSM 143100009. y Partial financial assistance by NSERC is acknowledged. 1 1. Introduction Weak factorization systems appear prominently in the definition of Quillen model category: for C, W, F the classes of cofibrations, weak equiva...

: The Galois connection given in 1985 by Pumplun and Rohrl between the classes of objects and the classes of morphisms in any category is shown (under ordinary circumstances) to have a "natural" factorization through the system of all idempotent closure operators over the category. Furthermore, each "component" of the factorization is a Galois connection in its own right. The first factor is obtained by using a generalization of the process, given by Salbany in 1975, that yields a closure operator for any class of topological spaces, while the second factor can be used to form the weakly hereditary core of an idempotent closure operator. KEY WORDS: Galois connection, closure operator, separated object, dense morphism CLASSIFICATION: Primary: 06A15, 18A20, 18A32, 18A40 Secondary: 18B30, 18E40, 20K40 0. INTRODUCTION In [10], Pumplun and Rohrl presented for any category X an important Galois connection between the collection S (X ) of all classes of X - objects, ordered by containmen...

: Methods developed for the study of general closure operators are used to construct various topological hulls for concrete categories. In particular, the notion of concretely cartesian closed topological hull for a concrete category over a cartesian closed base is generalized to arbitrary base categories. We clarify why this concept coincides with the one of universally topological hull for concrete categories over the terminal category, i.e., for pre-ordered classes, but not in general. The new notion is characterized in terms of injectivity in a suitable quasi-category. KEY WORDS: concrete category, (universally) topological category, final completion, closure operator, closure commuting with pullbacks, (concretely) cartesian closed category. CLASSIFICATION: 18D15, 18B25 0 INTRODUCTION In the following we will be concerned with concrete categories over a base category X , i.e., pairs hA; Ui consisting of a category A and a faithful functor A // U X . For convenience, and to si...