Categorical homotopy theory / Emily Riehl, Harvard University.

Format:

Book

Author/Creator:

Riehl, Emily, author.

Series:

New mathematical monographs ; 24.

New mathematical monographs ; 24

Language:

English

Subjects (All):

Homotopy theory.

Algebra, Homological.

Physical Description:

1 online resource (xviii, 352 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 2014.

Language Note:

English

Summary:

This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.

Contents:

Cover; Half title; Series; Title; Copyright; Dedication; Epigraph; Contents; Preface; Prerequisites; Notational Conventions; Acknowledgments; Part I Derived functors and homotopy (co)limits; 1 All concepts are Kan extensions; 1.1 Kan extensions; 1.2 A formula; 1.3 Pointwise Kan extensions; 1.4 All concepts; 1.5 Adjunctions involving simplicial sets; 2 Derived functors via deformations; 2.1 Homotopical categories and derived functors; 2.2 Derived functors via deformations; 2.3 Classical derived functors between abelian categories; 2.4 Preview of homotopy limits and colimits

3 Basic concepts of enriched category theory3.1 A first example; 3.2 The base for enrichment; 3.3 Enriched categories; 3.4 Underlying categories of enriched categories; 3.5 Enriched functors and enriched natural transformations; 3.6 Simplicial categories; 3.7 Tensors and cotensors; 3.8 Simplicial homotopy and simplicial model categories; 4 The unreasonably effective (co)bar construction; 4.1 Functor tensor products; 4.2 The bar construction; 4.3 The cobar construction; 4.4 Simplicial replacements and colimits; 4.5 Augmented simplicial objects and extra degeneracies

5 Homotopy limits and colimits: The theory5.1 The homotopy limit and colimit functors; 5.2 Homotopical aspects of the bar construction; 6 Homotopy limits and colimits: The practice; 6.1 Convenient categories of spaces; 6.2 Simplicial model categories of spaces; 6.3 Warnings and simplifications; 6.4 Sample homotopy colimits; 6.5 Sample homotopy limits; 6.6 Homotopy colimits as weighted colimits; Part II Enriched homotopy theory; 7 Weighted limits and colimits; 7.1 Weighted limits in unenriched category theory; 7.2 Weighted colimits in unenriched category theory

7.3 Enriched natural transformations and enriched ends7.4 Weighted limits and colimits; 7.5 Conical limits and colimits; 7.6 Enriched completeness and cocompleteness; 7.7 Homotopy (co)limits as weighted (co)limits; 7.8 Balancing bar and cobar constructions; 8 Categorical tools for homotopy (co)limit computations; 8.1 Preservation of weighted limits and colimits; 8.2 Change of base for homotopy limits and colimits; 8.3 Final functors in unenriched category theory; 8.4 Final functors in enriched category theory; 8.5 Homotopy final functors; 9 Weighted homotopy limits and colimits

9.1 The enriched bar and cobar construction9.2 Weighted homotopy limits and colimits; 10 Derived enrichment; 10.1 Enrichments encoded as module structures; 10.2 Derived structures for enrichment; 10.3 Weighted homotopy limits and colimits, revisited; 10.4 Homotopical structure via enrichment; 10.5 Homotopy equivalences versus weak equivalences; Part III Model categories and weak factorization systems; 11 Weak factorization systems in model categories; 11.1 Lifting problems and lifting properties; 11.2 Weak factorization systems; 11.3 Model categories and Quillen functors

11.4 Simplicial model categories

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-139-95051-7

1-139-96220-5

1-139-94946-2

1-139-96113-6

1-107-26145-7

1-139-95689-2

1-139-96008-3

1-139-95902-6

1-139-95796-1

OCLC:

881162803