Kan Extensions in Enriched Category Theory

The original purpose of this paper was to provide suitable enriched completions of small enriched categories.

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Author: Eduardo J. Dubuc

Publisher: Springer

ISBN: 9783540363071

Category: Mathematics

Page: 174

View: 546

The original purpose of this paper was to provide suitable enriched completions of small enriched categories.

Basic Concepts of Enriched Category Theory

In theory , therefore , Kan extensions could replace indexed limits as the basic " limit " notion ; we could use ( 4.8 ) to define Kan extensions without explicit mention of indexed limits , merely using ordinary limits in V O to define ...

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Author: G. M. Kelly

Publisher: CUP Archive

ISBN: 0521287022

Category: Mathematics

Page: 245

View: 578

Category Theory in Context

On account of this, in his much cited book on enriched category theory, Kelly reserves the name “Kan extension” for pairs satisfying the condition to be introduced shortly, calling those of our Definition 6.1.1 “weak” and writing: Our ...

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Author: Emily Riehl

Publisher: Courier Dover Publications

ISBN: 9780486820804

Category: Mathematics

Page: 272

View: 566

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

Category Theory

Thus theorem 10 also illustrates that not every C-left adjoint is the extension of a B-left adjoint. References l E. Dubuc, Kan extensions in Enriched Category Theory, Springer Lecture Notes in Math. , 145 (1970).

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Author: K. H. Kamps

Publisher: Springer

ISBN: 9783540395508

Category: Mathematics

Page: 326

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Category Seminar

The problem amounts to a well-known one in the theory of V-enriched categories concerning the relationship between comma objects and pointwise kan extensions. If we naively take K to be the 2-category V-Cat then the pointwise left kan ...

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Author: G.M. Kelly

Publisher: Springer

ISBN: 9783540372707

Category: Mathematics

Page: 382

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Higher Topos Theory AM 170

Fix an excellent model category S and a combinatorial S-enriched model category A. Let f : C → C be a functor between small ... In §A.2.8, we defined homotopy right Kan extensions in the setting of the diagram categories Fun(C,A), ...

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Author: Jacob Lurie

Publisher: Princeton University Press

ISBN: 9781400830558

Category: Mathematics

Page: 944

View: 660

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Categorical Homotopy Theory

Six model structures for dg-modules over dgas: Model category theory in homological action. ... On the construction of functorial factorizations for model categories. Algebr. Geom. ... Kan extensions in enriched category theory.

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Author: Emily Riehl

Publisher: Cambridge University Press

ISBN: 9781107048454

Category: Mathematics

Page: 320

View: 706

This categorical perspective on homotopy theory helps consolidate and simplify one's understanding of derived functors, homotopy limits and colimits, and model categories, among others.

Formal Category Theory Adjointness for 2 Categories

Day, B.J., and Kelly, G. M., Enriched Functor Categories, Rep. Midw. Cat. Sem. III, Lecture Notes in Mathematics, vol. 106, (1969), Springer-Verlag, New York, p. 178-191. Dubuc, E. , Kan Extensions in Enriched Category Theory, ...

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Author: J.W. Gray

Publisher: Springer

ISBN: 9783540377689

Category: Mathematics

Page: 284

View: 669

Big Data Integration Theory

Theory and Methods of Database Mappings, Programming Languages, and Semantics Zoran Majkić. by the, equivalent to it, “computation” morphism in DB ... E. Dubuc, Kan extensions in enriched category theory, in Lecture Notes in Math., vol.

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Author: Zoran Majkić

Publisher: Springer Science & Business Media

ISBN: 9783319041568

Category: Computers

Page: 516

View: 461

This book presents a novel approach to database concepts, describing a categorical logic for database schema mapping based on views, within a framework for database integration/exchange and peer-to-peer. Database mappings, database programming languages, and denotational and operational semantics are discussed in depth. An analysis method is also developed that combines techniques from second order logic, data modeling, co-algebras and functorial categorial semantics. Features: provides an introduction to logics, co-algebras, databases, schema mappings and category theory; describes the core concepts of big data integration theory, with examples; examines the properties of the DB category; defines the categorial RDB machine; presents full operational semantics for database mappings; discusses matching and merging operators for databases, universal algebra considerations and algebraic lattices of the databases; explores the relationship of the database weak monoidal topos w.r.t. intuitionistic logic.

Handbook of Categorical Algebra Volume 2 Categories and Structures

Soc . , 67 , 1970 , 553-558 B. Day and G.M. Kelly , Enriched functor categories , Springer LNM , 137 , 1970 , 1-38 E. Dubuc , Kan extensions in enriched category theory , Springer LNM , 145 , 1970 J. Duskin , Variations on Beck's ...

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Author: Francis Borceux

Publisher: Cambridge University Press

ISBN: 052144179X

Category: Mathematics

Page: 464

View: 741

The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence. The second, which assumes familiarity with the material in the first, introduces important classes of categories that have played a fundamental role in the subject's development and applications. In addition, after several chapters discussing specific categories, the book develops all the major concepts concerning Benabou's ideas of fibred categories. There is ample material here for a graduate course in category theory, and the book should also serve as a reference for users.