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The shape of a category up to directed homotopy
 Theory Appl. Categ
, 2004
"... 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 ..."
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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.
Lax Factorization Algebras
"... 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 facili ..."
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Cited by 9 (4 self)
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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...
Internal monotonelight factorization for categories via preorders
 Theory Appl. Categories
"... Dedicated to Aurelio Carboni on the occasion of his sixtieth birthday ..."
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Dedicated to Aurelio Carboni on the occasion of his sixtieth birthday
Factorization Systems And Distributive Laws
"... 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 process ..."
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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.
Journal of Homotopy and Related Structures, vol. 2(2), 2007, pp.295–314 FACTORIZATION, FIBRATION AND TORSION
"... 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 theo ..."
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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.
Journal of Homotopy and Related Structures, vol. 1(1), 2006, pp.1–5 SEPARABLE MORPHISMS OF SIMPLICIAL SETS
, 2006
"... (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 ..."
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(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
FACTORIZATION, FIBRATION AND TORSION
"... Abstract. 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 hom ..."
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Abstract. 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. Dedicated to the memory of Saunders Mac Lane 1.
LAWVERE COMPLETENESS AS A TOPOLOGICAL PROPERTY
"... Abstract. Lawvere’s notion of completeness for quantaleenriched categories has been extended to the theory of lax algebras under the name of Lcompleteness. In this paper we introduce the corresponding morphism concept and examine its properties. We explore some important relativized topological co ..."
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Abstract. Lawvere’s notion of completeness for quantaleenriched categories has been extended to the theory of lax algebras under the name of Lcompleteness. In this paper we introduce the corresponding morphism concept and examine its properties. We explore some important relativized topological concepts like separatedness, denseness, compactness and compactification with respect to Lcomplete morphisms. Moreover, we show that separated Lcomplete morphisms belong to a factorization system. 1.