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The shape of a category up to directed homotopy
 Dip. Mat. Univ. Genova
, 2004
"... Abstract. 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, wider than ordin ..."
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Cited by 14 (6 self)
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Abstract. 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, wider 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 larg ..."
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Cited by 12 (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
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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.
GALOIS THEORIES OF COMMUTATIVE SEMIGROUPS VIA SEMILATTICES
"... Abstract. The classes of stablyvertical, normal, separable, inseparable, purely inseparable and covering morphisms, defined in categorical Galois theory, are characterized for the reflection of the variety of commutative semigroups into its subvariety of semilattices. It is also shown that there ..."
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Abstract. The classes of stablyvertical, normal, separable, inseparable, purely inseparable and covering morphisms, defined in categorical Galois theory, are characterized for the reflection of the variety of commutative semigroups into its subvariety of semilattices. It is also shown that there is an inseparableseparable factorization, but there is no monotonelight factorization. 1.
STRONGLY SEPARABLE MORPHISMS IN GENERAL CATEGORIES
"... Dedicated to Dominique Bourn on the occasion of his sixtieth birthday ..."
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.
References
"... Starting from a ground structure consisting of an adjunction C → X and a prefactorization system (E,M) which factorizes the unit morphisms ϕC: C → GF (C), C ∈ C, through an epimorphism ηC, a full epireflection C → M is obtained with unit η: 1C → HI. A reflective factorization system is associated, ..."
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Starting from a ground structure consisting of an adjunction C → X and a prefactorization system (E,M) which factorizes the unit morphisms ϕC: C → GF (C), C ∈ C, through an epimorphism ηC, a full epireflection C → M is obtained with unit η: 1C → HI. A reflective factorization system is associated, and there may be a concordantdissonant (in a similar sense to the one referred in [3]) and also a monotonelight factorization. We will show that in the case of any adjunction SetA → SetB, given by right Kan extensions along a functor K: B → A, there is a monotonelight factorization which coincides with the concordantdissonant one, provided the objects in the image of the functor K are a cogenerating set for A. Remark that this condition is equivalent to demanding that the composition SetK ·y with the Yoneda embedding is a faithful functor. A generalization of the given results, to left adjoints from presheaves into a cocomplete category, is then possible.