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Map** Spaces and Bousfield Localizations
Recall from Section 8.3 that a left Bousfield localization of a model category ε is a different model structure on the same category with more weak...
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An Overview of Rationalization Theories of Non-simply Connected Spaces and Non-nilpotent Groups
We give an overview of five rationalization theories for spaces (Bousfield-Kan’s ℚ-completion; Sullivan’s rationalization; Bousfield’s homology...
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K-flatness and orthogonality in homotopy categories
K-flatness for unbounded complexes of modules over a ring R was introduced by Spaltenstein [27], as an analogue of the classical notion of flatness...
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The Bousfield-Kuhn functor and topological André-Quillen cohomology
We construct a natural transformation from the Bousfield-Kuhn functor evaluated on a space to the Topological André-Quillen cohomology of the K ( n )-loc...
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On functorial (co)localization of algebras and modules over operads
Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are...
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Smashing localizations in equivariant stable homotopy
We study how smashing Bousfield localizations behave under various equivariant functors. We show that the analogs of the smash product and chromatic...
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Homotopy theory of monoids and derived localization
We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology, both known and...
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Model Categories and Bousfield Localization
For a systematic study of 2-Segal spaces it is convenient to work in the more general framework of model categories.
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Simplicial and Dendroidal Homotopy Theory
This open access book offers a self-contained introduction to the homotopy theory of simplicial and dendroidal sets and spaces. These are essential...
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Cohomological Localizations and Set-Theoretical Reflection
Homological localizations of spaces and spectra have been a fundamental tool in algebraic topology since the 1970s, especially in the setting of...
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Model structures on finite total orders
We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics . In the case of a...
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Admissibility of Localizations of Crossed Modules
The correspondence between the concept of conditional flatness and admissibility in the sense of Galois appears in the context of localization...
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Adjunctions and Adjoint Functor Theorems
In this section, we will discuss adjunctions between ∞-categories. We will define them in the language of fibrations and show that they may...
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S-Colocalization and Adams Cocompletion
A relationship between the S -colocalization of an object and the Adams cocompletion of the same object in a complete small 𝒰 -category ( 𝒰 is a...
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Right exact localizations of groups
We introduce several classes of localizations (idempotent monads) on the category of groups and study their properties and relations. The most...
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Group schemes and motivic spectra
By a theorem of Mandell, May, Schwede and Shipley [21] the stable homotopy theory of classical S 1 -spectra is recovered from orthogonal spectra. In...
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On the cokernel of the Baumslag rationalization
We prove that for the free group of rank two F the cokernel of the homomorphism to its Baumslag rationalization F → Bau( F ) is not abelian. Moreover,...
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The cube axiom and resolutions in homotopy theory
We show that a version of the cube axiom holds in cosimplicial unstable coalgebras and cosimplicial spaces equipped with a resolution model...
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Left Fibrations and the Covariant Model
In Section 9.5 we introduced the covariant model structure on a slice category dSets/B, for a fixed dendroidal set B. This model category represents...