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.

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...

Dedicated to Jiˇrí Rosick´y at the occasion of his sixtieth birthday Abstract. In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category K is introduced, as a pair (comonad, monad) overK 2. The link with existing notions in terms of morphism classes is given via the respective Eilenberg– Moore categories. 1.

ABSTRACT. For the 2-monad ((−) 2,I,C)onCAT, with unit I described by identities and multiplication C described by composition, we show that a functor F: K2 � � K satisfying FIK =1K admits a unique, normal, pseudo-algebra structure for (−) 2 if and only if there is a mere natural isomorphism F · F 2 � � � F · CK. We show that when this is the case the set of all natural transformations F · F 2 � � F · CK formsa commutative monoid isomorphic to the centre of K. 1. Preliminaries 1.1. When we speak of ‘the 2-monad (−) 2 on CAT ’ we understand the canonical monad that arises by exponentiation of the cocommutative comonoid structure

For the 2-monad ((-) 2 , I, C) on CAT, with unit I described by identities and multiplication C described by composition, we show that a functor F : K 2 ## K satisfying F I K = 1 K admits a unique, normal, pseudo-algebra structure for (-) 2 if and only if there is a mere natural isomorphism F F 2 # ## F CK . We show that when this is the case the set of all natural transformations F F 2 ## F CK forms a commutative monoid isomorphic to the centre of K.

The notion of Quillen factorization system is obtained by strengthening the left and right lifting properties in a Quillen model category to the unique diagonalization property. An equivalent description of this notion is given in terms of a double factorization system which decomposes each arrow uniquely into three factors. The free category with Quillen factorization system over a given category is described.

Abstract. It is known that factorisation systems in categories can be viewed as unitary pseudo algebras for the monad (–) 2, in Cat. We show in this note that an analogous fact holds for proper (i.e., epimono) factorisation systems and a suitable quotient of the former monad, deriving from a construct introduced by P. Freyd for stable homotopy. Structural similarities of the previous monad with the path endofunctor of topological spaces are considered.