Results 11  20
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24
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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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...
Functorial calculus in monoidal bicategories
 Applied Categorial Structures
, 2002
"... The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomp ..."
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The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory VMod whose objects are categories enriched in a cocomplete symmetric monoidal
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.
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
Yoneda structures from 2toposes
"... Abstract. A 2categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples ..."
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Abstract. A 2categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2category [Str74b] and provides a selfcontained development of the necessary background material on yoneda structures.
Categorical Combinators for Charity
, 1996
"... A 2categorical notion of parametric combinator is introduced. These combinators are operations for constructing arrows in a category and satisfy a 2categorical abstraction of parametricity for strictly covariant functors. Inductive and coinductive datatypes are compatible with these combinators a ..."
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A 2categorical notion of parametric combinator is introduced. These combinators are operations for constructing arrows in a category and satisfy a 2categorical abstraction of parametricity for strictly covariant functors. Inductive and coinductive datatypes are compatible with these combinators and their universal properties are closely related to parametricity. The Charity programming language is based on strong datatypes which are a specialization of 2categorical datatypes. Stong combinators (which are an appropriate specialization of 2categorical combinators) provide a model of polymorphic functions in Charity. ii Contents Abstract ii Contents iii 1 Introduction 1 2 Charity 4 2.1 The Term Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Strong Datatypes . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Strong Combinators . . . . . . . . . . . . . . . . . . . . . . . . 17 3 2cate...
FROM TAMSAMANI’S WEAK 2CATEGORIES TO BICATEGORIES
"... Abstract. We give a conceptual description of the bicategory associated to a Tamsamani’s weak 2category, and we associate functorially to a morphism of Tamsamani’s weak 2categories a homomorphism of the corresponding bicategories. 1. Introduction. The goal of this paper is to construct a functor f ..."
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Abstract. We give a conceptual description of the bicategory associated to a Tamsamani’s weak 2category, and we associate functorially to a morphism of Tamsamani’s weak 2categories a homomorphism of the corresponding bicategories. 1. Introduction. The goal of this paper is to construct a functor from the category N2 of Tamsamani’s
Strict 2toposes
, 2006
"... Abstract. A 2categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some e ..."
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Abstract. A 2categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2category [Str74b] and provides a selfcontained development of the necessary background material on yoneda structures.
BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal
BIMONADS AND HOPF MONADS ON CATEGORIES BACHUKI MESABLISHVILI, TBILISI AND
, 710
"... Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal c ..."
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Abstract. The purpose of this paper is to develop a theory of bimonads and Hopf monads on arbitrary categories thus providing the possibility to transfer the essentials of the theory of Hopf algebras in vector spaces to more general settings. There are several extensions of this theory to monoidal categories which in a certain sense follow the classical trace. Here we do not pose any conditions on our base category but we do refer to the monoidal