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On a Generalized SmallObject Argument for the Injective Subcategory Problem
 Cah. Topol. Géom. Différ. Catég
, 2000
"... For locally ranked categories A, which include all locally presentable categories and the category Top, we prove that, given any set ..."
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Cited by 21 (10 self)
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For locally ranked categories A, which include all locally presentable categories and the category Top, we prove that, given any set
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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Cited by 13 (4 self)
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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.
...Cat Is Locally Presentable Or Locally Bounded If ... Is So
, 2001
"... We show, for a monoidal closed category V = (V 0 , #, I), that the category V Cat of small V categories is locally #presentable if V 0 is so, and that it is locally #bounded if the closed category V is so, meaning that V 0 is locally #bounded and that a side condition involving the monoidal ..."
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Cited by 6 (2 self)
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We show, for a monoidal closed category V = (V 0 , #, I), that the category V Cat of small V categories is locally #presentable if V 0 is so, and that it is locally #bounded if the closed category V is so, meaning that V 0 is locally #bounded and that a side condition involving the monoidal structure is satisfied.
ALGEBRAICALLY CLOSED AND EXISTENTIALLY CLOSED SUBSTRUCTURES IN CATEGORICAL CONTEXT
 THEORY AND APPLICATIONS OF CATEGORIES
, 2004
"... We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir [Fa, 75], as well as some of his results, are revisited and extended. Related preservation the ..."
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Cited by 4 (1 self)
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We investigate categorical versions of algebraically closed (= pure) embeddings, existentially closed embeddings, and the like, in the context of locally presentable categories. The definitions of S. Fakir [Fa, 75], as well as some of his results, are revisited and extended. Related preservation theorems are obtained, and a new proof of the main result of Rosicky, Adamek and Borceux ([RAB, 02]), characterizing #injectivity classes in locally #presentable categories, is given.
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
Categories: A Free Tour
"... Category theory plays an important role as a unifying agent in a rapidly expanding universe of mathematics. In this paper, an introduction is given to the basic denitions of category theory, as well as to more advanced concepts such as adjointness, factorization systems and cartesian closedness. ..."
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Category theory plays an important role as a unifying agent in a rapidly expanding universe of mathematics. In this paper, an introduction is given to the basic denitions of category theory, as well as to more advanced concepts such as adjointness, factorization systems and cartesian closedness. In the past decades, the subject of mathematics has experienced an explosive increase both in diversity and in the sheer amount of published material. (E.g., the Mathematical Reviews volume of 1950 features 766 pages of reviews, compared to a total of 4550 pages in the six volumes for the rst half of 2000.) It has thus become inevitable that this growth, taking place in numerous and increasingly disconnected branches, be complemented by some form of unifying theory. There have been attempts at such unications in the past, such as Birkhostyle universal algebra or the encyclopedic work of Bourbaki. However, the most successful and universal approach so far is certainly the theory of cat...
DESCENT THEORY FOR SCHEMES
"... Abstract. We give a complete characterization of the class of quasicompact morphisms of schemes that are stable effective descent morphisms for the SCHEMESindexed category given by quasicoherent sheaves of modules. 1. ..."
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Abstract. We give a complete characterization of the class of quasicompact morphisms of schemes that are stable effective descent morphisms for the SCHEMESindexed category given by quasicoherent sheaves of modules. 1.
Extended Galois Theory And Dissonant Morphisms
"... For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of socalled weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various ..."
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For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of socalled weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two results of M. Barr and R. Diaconescu [BD]. We also give an elaborate list of examples and applications.
STRONGLY SEPARABLE MORPHISMS IN GENERAL CATEGORIES
"... Dedicated to Dominique Bourn on the occasion of his sixtieth birthday ..."
ON REFLECTIVECOREFLECTIVE EQUIVALENCE AND ASSOCIATED PAIRS
"... Abstract. We show that a reflective/coreflective pair of full subcategories satisfies a “maximalnormal”type equivalence if and only if it is an associated pair in the sense of Kelly and Lawvere. 1. ..."
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Abstract. We show that a reflective/coreflective pair of full subcategories satisfies a “maximalnormal”type equivalence if and only if it is an associated pair in the sense of Kelly and Lawvere. 1.