This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of ‘directed structures’, e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence of categories. Here we introduce past and future equivalences of categories—sort of symmetric versions of an adjunction—and use them and their combinations to get ‘directed models ’ of a category; in the simplest case, these are the join of the least full reflective and the least full coreflective subcategory.

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 Aurelio Carboni on the occasion of his sixtieth birthday

This article shows that the distributive laws of Beck in the bicategory of sets and matrices determine strict factorization systems on their composite monads (=categories). Conversely, it is shown that strict factorization systems on categories give rise to distributive laws. Moreover, these processes are shown to be mutually inverse in a precise sense. Further, an extension of the distributive law concept provides a correspondence with the classical orthogonal factorization systems.

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A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3–for–2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory. 1.

(communicated by George Janelidze) We show that the class of separable morphisms in the sense of G. Janelidze and W. Tholen in the case of Galois structure