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17
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 14 (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.
Free adjunction of morphisms
 APPL. CAT. STRUCT
"... We develop a general setting for the treatment of extensions of categories by means of freely adjoined morphisms. To this end, we study what we call composition graphs, i.e. large graphs with a partial binary operation on which we impose only rudimentary requirements. The quasicategory thus obtaine ..."
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Cited by 8 (6 self)
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We develop a general setting for the treatment of extensions of categories by means of freely adjoined morphisms. To this end, we study what we call composition graphs, i.e. large graphs with a partial binary operation on which we impose only rudimentary requirements. The quasicategory thus obtained contains the quasicategory of all categories as a full reflective subquasicategory; we characterize composition graphs for which this reflexion is of a particularly simple nature. This leads to the concept of semicategory; we apply semicategories to solve characterization problems concerning absolutely initial sources, absolute monosources
Tholen: A functional approach to general topology
 Categorical Foundations (Cambridge
, 2004
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A Note on Local Compactness
"... We propose a categorical definition of locallycompact Hausdorff object which gives the right notion both, for topological spaces and for locales. Stability properties follow from easy categorical arguments. The map version of the notion leads to an investigation of restrictions of perfect maps t ..."
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We propose a categorical definition of locallycompact Hausdorff object which gives the right notion both, for topological spaces and for locales. Stability properties follow from easy categorical arguments. The map version of the notion leads to an investigation of restrictions of perfect maps to open subspaces. AMS Subj. Class.: 18B30, 54B30, 54D30, 54D45, 54C10. Key words: closure operator, locally compact object, open map, perfect map. 1 Introduction Both, for topological spaces and for locales, locally compact Hausdorff spaces are characterized as the spaces which are openly embeddable into compact Hausdorff spaces. While in Top this is an obvious consequence of Alexandroff's onepointcompactification, in Loc one uses results of Vermeulen [10] to establish this result. In this note we show that, taking this characterization as the defining property for (Hausdorff) local compactness, one establishes practically all standard stability properties of local compactness  with ...
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...
On completeness in a nonArchimedean setting, via firm reflections
, 2008
"... We develop a completion theory for (general) nonArchimedean spaces based on the theory on ”a categorical concept of completion of objects” as introduced by G.C.L. Brümmer and E. Giuli in [7]. Our context is the construct NA0 of all Hausdorff nonArchimedean spaces and uniformly continuous maps and ..."
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We develop a completion theory for (general) nonArchimedean spaces based on the theory on ”a categorical concept of completion of objects” as introduced by G.C.L. Brümmer and E. Giuli in [7]. Our context is the construct NA0 of all Hausdorff nonArchimedean spaces and uniformly continuous maps and V is the class of all epimorphic embeddings in NA0. We determine the class Inj V of all Vinjective objects and we present an internal characterization as ”complete objects”. The basic tool for this characterization is a notion of small collections that in some sense preserve the inclusion order on the nonArchimedean structure. We prove that the full subconstruct CNA0 consisting of all complete objects forms a firmly Vreflective subcategory. This means that every object X in NA0 has a completion which is a Vreflection rX: X → RX into the full subconstruct CNA0 of ”complete spaces”. Moreover this completion is unique (up to isomorphism) in the sense that, considering L(CNA0), the class of all those morphisms u: X → Y for which Ru: RX → RY is an isomorphism, one has that V is contained in L(CNA0). In fact one even has V = L(CNA0). Finally we apply our constructions to the classical case of Hausdorff nonArchimedean uniform spaces, in that case our completion reduces to the standard one [21], [22].
Traces of Epimorphism Classes
"... Classes of morphisms that occur as preimages of the class of all epimorphisms, of all extremal epimorphisms, respectively of all retractions under some faithful functor are characterized by suitable closure properties. The corresponding question for regular or strict epimorphisms is presented as an ..."
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Classes of morphisms that occur as preimages of the class of all epimorphisms, of all extremal epimorphisms, respectively of all retractions under some faithful functor are characterized by suitable closure properties. The corresponding question for regular or strict epimorphisms is presented as an open problem. A further application of the methods developed in this context yields a similar characterization of classes of morphisms that are final w.r.t. some faithful functor.
PROTOLOCALISATIONS OF EXACT MAL’CEV CATEGORIES
, 2008
"... A protolocalisation of a regular category is a full reflective regular subcategory, whose reflection preserves pullbacks of regular epimorphisms along arbitrary morphisms. We devote special attention to the epireflective protolocalisations of an exact Mal’cev category; we characterise them in terms ..."
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A protolocalisation of a regular category is a full reflective regular subcategory, whose reflection preserves pullbacks of regular epimorphisms along arbitrary morphisms. We devote special attention to the epireflective protolocalisations of an exact Mal’cev category; we characterise them in terms of a corresponding closure operator on equivalence relations. We give some examples in algebra and in topos theory.