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14
Functorial Factorization, Wellpointedness and Separability
"... A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and f ..."
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A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled 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: concordantdissonant and inseparableseparable.
Natural weak factorization systems
 Archivum Mathematicum
"... 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 categor ..."
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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.
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...
Algebraic model structures
"... Abstract. We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove “algebraic ” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that e ..."
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Cited by 7 (5 self)
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Abstract. We define a new notion of an algebraic model structure, in which the cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, and prove “algebraic ” analogs of classical results. Using a modified version of Quillen’s small object argument, we show that every cofibrantly generated model structure in the usual sense underlies a cofibrantly generated algebraic model structure. We show how to pass a cofibrantly generated algebraic model structure across an adjunction, and we characterize the algebraic Quillen adjunction that results. We prove that pointwise algebraic weak factorization systems on diagram categories are cofibrantly generated if the original ones are, and we give an algebraic generalization of the projective model structure. Finally, we prove that certain fundamental comparison maps present in any cofibrantly generated model category are cofibrations when the cofibrations are monomorphisms, a conclusion that does not seem to be provable in the classical, nonalgebraic, theory. Contents
Factorization Systems For Symmetric CatGroups
 THEORY AND APPLICATIONS OF CATEGORIES, PREPRINT
, 2000
"... This paper is a first step in the study of symmetric catgroups as the 2dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective fu ..."
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Cited by 6 (0 self)
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This paper is a first step in the study of symmetric catgroups as the 2dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective functor followed by a faithful one. Both these factorizations give rise to a factorization system, in a suitable 2categorical sense, in the 2category of symmetric catgroups. An application to exact sequences is given.
Free Quillen Factorization Systems
, 2002
"... 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 u ..."
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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.
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.
On the monad of proper factorisation systems in categories ( *)
, 2001
"... 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 constru ..."
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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.
ON THE FUNCTOR ℓ 2
"... and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous lin ..."
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and both categories are enriched over algebraic domains. The functor preserves daggers, monoidal structures, enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its direct image consists precisely of the partial isometries, but its essential image consists of all continuous linear maps between Hilbert spaces. 1.
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
, 2012
"... Restriction categories provide a convenient abstract formulation of partial functions. However, restriction categories can have a variety of structures such as finite partial products (cartesianess), joins, meets, and ranges which are of interest in computability theory, semigroup theory, topology, ..."
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Restriction categories provide a convenient abstract formulation of partial functions. However, restriction categories can have a variety of structures such as finite partial products (cartesianess), joins, meets, and ranges which are of interest in computability theory, semigroup theory, topology, and algebraic geometry. This thesis studies these structures. For finite partial products (cartesianess), a construction to add finite partial products to an arbitrary restriction category freely is provided. For joins, we introduce the notion of join restriction categories, describe a construction for the join completion of a restriction category, and show the completeness of join restriction categories in partial map categories using Madhesive categories and Mgaps. As the join completion for inverse semigroups is wellknown in semigroup theory, we show the relationships between the join completion for restriction categories and the join completion for inverse semigroups by providing adjunctions among restriction categories, join restriction categories, inverse categories, and join inverse categories.