Homotopy Theory of Diagrams
Title | Homotopy Theory of Diagrams PDF eBook |
Author | Wojciech Chachólski |
Publisher | American Mathematical Soc. |
Pages | 106 |
Release | 2002 |
Genre | Mathematics |
ISBN | 0821827596 |
In this paper the authors develop homotopy theoretical methods for studying diagrams. In particular they explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept introduced is that of a model approximation. A model approximation of a category $\mathcal{C}$ with a given class of weak equivalences is a model category $\mathcal{M}$ together with a pair of adjoint functors $\mathcal{M} \rightleftarrows \mathcal{C}$ which satisfy certain properties. The key result says that if $\mathcal{C}$ admits a model approximation then so does the functor category $Fun(I, \mathcal{C})$.
Categorical Homotopy Theory
Title | Categorical Homotopy Theory PDF eBook |
Author | Emily Riehl |
Publisher | Cambridge University Press |
Pages | 371 |
Release | 2014-05-26 |
Genre | Mathematics |
ISBN | 1139952633 |
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.
Homotopy Theory of Diagrams
Title | Homotopy Theory of Diagrams PDF eBook |
Author | Wojciech Chachólski |
Publisher | |
Pages | 90 |
Release | 2014-09-11 |
Genre | Categories |
ISBN | 9781470403294 |
In this paper the authors develop homotopy theoretical methods for studying diagrams. In particular they explain how to construct homotopy colimits and limits in an arbitrary model category. The key concept introduced is that of a model approximation. A model approximation of a category $\mathcal{C}$ with a given class of weak equivalences is a model category $\mathcal{M}$ together with a pair of adjoint functors $\mathcal{M} \rightleftarrows \mathcal{C}$ which satisfy certain properties. The key result says that if $\mathcal{C}$ admits a model approximation then so does the functor category $Fun(I, \mathcal{C})$.
Diagram Cohomology and Isovariant Homotopy Theory
Title | Diagram Cohomology and Isovariant Homotopy Theory PDF eBook |
Author | Giora Dula |
Publisher | American Mathematical Soc. |
Pages | 97 |
Release | 1994 |
Genre | Mathematics |
ISBN | 0821825895 |
Obstruction theoretic methods are introduced into isovariant homotopy theory for a class of spaces with group actions; the latter includes all smooth actions of cyclic groups of prime power order. The central technical result is an equivalence between isovariant homotopy and specific equivariant homotopy theories for diagrams under suitable conditions. This leads to isovariant Whitehead theorems, an obstruction-theoretic approach to isovariant homotopy theory with obstructions in cohomology groups of ordinary and equivalent diagrams, and qualitative computations for rational homotopy groups of certain spaces of isovariant self maps of linear spheres. The computations show that these homotopy groups are often far more complicated than the rational homotopy groups for the corresponding spaces of equivariant self maps. Subsequent work will use these computations to construct new families of smooth actions on spheres that are topologically linear but differentiably nonlinear.
Cubical Homotopy Theory
Title | Cubical Homotopy Theory PDF eBook |
Author | Brian A. Munson |
Publisher | Cambridge University Press |
Pages | 649 |
Release | 2015-10-06 |
Genre | Mathematics |
ISBN | 1107030250 |
A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.
Combinatorial Foundation of Homology and Homotopy
Title | Combinatorial Foundation of Homology and Homotopy PDF eBook |
Author | Hans-Joachim Baues |
Publisher | Springer Science & Business Media |
Pages | 379 |
Release | 2013-03-09 |
Genre | Mathematics |
ISBN | 3662113384 |
A new combinatorial foundation of the two concepts, based on a consideration of deep and classical results of homotopy theory, and an axiomatic characterization of the assumptions under which results in this field hold. Includes numerous explicit examples and applications in various fields of topology and algebra.
Homotopy Theory: An Introduction to Algebraic Topology
Title | Homotopy Theory: An Introduction to Algebraic Topology PDF eBook |
Author | |
Publisher | Academic Press |
Pages | 383 |
Release | 1975-11-12 |
Genre | Mathematics |
ISBN | 0080873804 |
Homotopy Theory: An Introduction to Algebraic Topology