Note on a theorem of Bousfield and Friedlander

关于经检验的Bousfield-Friedlander局部化

On tested Bousfield-Friedlander localizations

该研究将经检验的Bousfield-Friedlander局部化刻画为左Bousfield局部化，建立了相关齐性模型结构，证明离散演算与Goodwillie演算的齐性函子同伦等价，并指出Weiss演算的多项式模型结构是其特例。

When Bousfield localizations and homotopy idempotent functors meet again

We adopt semimodel categories to extend fundamental results related to Bousfield localizations of model categories. More specifically, we generalize Bousfield-Friedlander Theorem and Hirschhorn Localization Theorem of cellular model categories to settings where their classical formulation does not apply. We use such results to answer, in the world of semimodel categories, an open problem posed by May-Ponto about the existence of Bousfield localizations for Hurewicz and mixed type model structures.

Truncated resolution model structures

Using the dual of Bousfield-Friedlander localization we colocalize resolution model structures on cosimplicial objects over a left proper model category to get truncated resolution model structures. These are useful to study realization and moduli problems in algebraic topology.

Substitudes, Bousfield localization, higher braided operads, and Baez-Dolan stabilization

This short note reports on joint work with Michael Batanin towards a general machine for proving Baez-Dolan Stabilization Theorems for various models of higher categories, based on substitudes, Bousfield localization, and homotopical Beck-Chevalley squares. I provide a road map to our recent papers, and include new results proving Baez-Dolan Stabilization Theorems for Tamsamani weak $n$-categories, higher Segal categories, Ara's $n$-quasi-categories, and cartesian models of Segal and complete Segal objects due to Bergner and Rezk. I also attempt to clarify the connection to higher braided oper…

Morita theory in enriched context

We develop a homotopy theoretical version of classical Morita theory using the notion of a strong monad. It was Anders Kock who proved that a monad T in a monoidal category E is strong if and only if T is enriched in E. We prove that this correspondence between strength and enrichment follows from a 2-isomorphism of 2-categories. Under certain conditions on T, we prove that the category of T-algebras is Quillen equivalent to the category of modules over the endomorphism monoid of the T-algebra T(I) freely generated by the unit I of E. In the special case where E is the category of Gamma-spaces…

Localisation and Completion with an addendum on the use of Brown-Peterson homology in stable homotopy

These are notes, by Z. Fiedorowicz, from lectures given by J. Frank Adams at the University of Chicago in spring of 1973. They give an elegant axiomatic presentation of localization and completion in algebraic topology. The construction of localization and completion functors with respect to an arbitrary generalized homology theory is derived from the axioms by using the Brown representability theorem. These notes were never formally published, due to an apparent flaw in the proof. The relevant representable functors could not be shown to be set-valued, as opposed to class-valued. Subsequent w…