Categorical Homotopy Theory

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Cambridge University Press, May 26, 2014 - Mathematics - 372 pages
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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. 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.
 

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Contents

All concepts are Kan extensions
3
Derived functors via deformations
17
Basic concepts of enriched category theory
32
The unreasonably effective cobar construction
58
The theory
69
The practice
76
Weighted limits and colimits
99
Categorical tools for homotopy colimit computations
121
Algebraic perspectives on the small object argument
190
Enriched factorizations and enriched lifting properties
222
A brief tour of Reedy category theory
240
Preliminaries on quasicategories
263
Simplicial categories and homotopy coherence
282
Isomorphisms in quasicategories
298
A sampling of 2categorical aspects of quasicategory theory
318
Bibliography
337

Weighted homotopy limits and colimits
136
Derived enrichment
145
FACTORIZATION SYSTEMS
167
Glossary of Notation
343
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About the author (2014)

Emily Riehl is a Benjamin Peirce Fellow in the Department of Mathematics at Harvard University, Massachusetts and a National Science Foundation Mathematical Sciences Postdoctoral Research Fellow.

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