Homotopy limits of model categories and more general homotopy theories

The Stable Homotopy Category Has a Unique Model at the Prime 2

Given an appropriate diagram of left Quillen functors between model categories, one can define a notion of homotopy fiber product, but one might ask if it is really the correct one. Here, we show that this homotopy pullback is well-behaved with respect to translating it into the setting of more general homotopy theories, given by complete Segal spaces, where we have well-defined homotopy pullbacks.

We begin the chapter by discussing homotopy pushouts and homotopy pullbacks in the first two sections. These are analogues in the homotopy category of pushouts and pullbacks. They have many applications in homotopy theory and are necessary for the proofs in Section 6.4. That section contains the statement and proof of many of the major theorems of classical homotopy theory.

Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left Bousfield localizations Existence of right Bousfield localizations Fiberwise localization Homotopy theory in model categories: Summary of part 2 Model categories Fibrant and cofibrant approximations Simplicial model categories Ordinals, cardinals, and transfinite composition Cofibrantly generated model categories Cellular model categories Proper model categories The classifying space of a small category The Reedy model category structure Cosimplicial and simplicial resolutions Homotopy function complexes Homotopy limits in simplicial model categories Homotopy limits in general model categories Index Bibliography.

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.

Three models for the homotopy theory of homotopy theories

This thesis consists of two independent parts. In the first part we ask how traces in monoidal categories behave under homotopical operations. In order to investigate this question we define traces in closedmonoidal derivators and establish some of their properties. In the stable setting we derive an explicit formula for the trace of the homotopy colimit over finite categories in which every endomorphism is invertible. In the second part, we study motives of algebraic varieties over a subfield of the complex numbers, as defined by Nori on the one hand and by Voevodsky, Levine, and Hanamura on the other. Ayoub attached to the latter theory a motivic Galois group using the Betti realization, based on a weak Tannakian formalism. Our main theorem states that Nori’s and Ayoub’s motivic Galois groups are isomorphic. In the process of proving this result we construct well-behaved functors relating the two theories which are of independent interest.

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.

Let [math] be a finite group. We define a suitable model-categorical framework for [math] –equivariant homotopy theory, which we call [math] –model categories. We show that the diagrams in a [math] –model category which are equipped with a certain equivariant structure admit a model structure. This model category of equivariant diagrams supports a well-behaved theory of equivariant homotopy limits and colimits. We then apply this theory to study equivariant excision of homotopy functors.

Chapter 2 - Homotopy Theories and Model Categories

The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

Spectra and symmetric spectra in general model categories

Handbook of Algebraic Topology

We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.

Function complexes in homotopical algebra