Weighted limits in simplicial homotopy theory

Gambino, Nicola

doi:10.1016/j.jpaa.2009.10.006

Cited by 24 publications

(20 citation statements)

References 18 publications

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“…The following proposition shows that the limit of any diagram of quasi-categories weighted by a projective cofibrant functor is again a quasi-category. The proof is very simple and indeed related conclusions have been drawn elsewhere; see for instance [8]. For the duration of this section and the next we shall assume that A is a small simplicial category.…”

Section: 2mentioning

confidence: 54%

Homotopy coherent adjunctions and the formal theory of monads

Riehl

Verity

2016

Advances in Mathematics

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Abstract. In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes.We extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"-much of it independent of the quasi-categorical context.

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Section: 2mentioning

confidence: 54%

Homotopy coherent adjunctions and the formal theory of monads

Riehl

Verity

2016

Advances in Mathematics

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“…Also, one can easily check that for each inclusion of a square-shaped diagram the map from the pushout of the upper left horn to the terminal vertex is a cofibration by iterated use of the pushout product axiom (cf Lemma 3.4.6 where essentially the same result is proven in more detail). The fact that the canonical map from |srpFq| Ñ colim F is a weak equivalence for projectively cofibrant A is Theorem B.1.2, which is a consequence of Theorems 3.2 and 3.3 in [21].…”

Section: Whitehead Towersmentioning

confidence: 83%

“…Definition 3.4.2 If I ‚ is a cofibrant decreasingly filtered commutative monoid in C, X ‚ is a simplicial finite set, and E˚is a connective generalized homology theory on C, then by the topological Hochschild-May spectral sequence for X ‚b I ‚ we mean the spectral sequence obtained by applying E˚to the tower of cofiber sequences in C (21) . .…”

Section: 4mentioning

confidence: 99%

A May-type spectral sequence for higher topological Hochschild homology

Angelini-Knoll

Salch

2018

Algebr. Geom. Topol.

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Given a filtration of a commutative monoid A in a symmetric monoidal stable model category C, we construct a spectral sequence analogous to the May spectral sequence whose input is the higher order topological Hochschild homology of the associated graded commutative monoid of A, and whose output is the higher order topological Hochschild homology of A. We then construct examples of such filtrations and derive some consequences: for example, given a connective commutative graded ring R, we get an upper bound on the size of the THH-groups of E 8 -ring spectra A such that π˚pAq -R. 55P42; 55T05

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“…Our main sources are [12], [28], [30], [33], [34], [36], [52]. For simplicity, we restrict our discussion to pointed simplicial model categories, which is sufficient for our purposes.…”

Section: Coreflective Localizing Subcategoriesmentioning

confidence: 99%

Are all localizing subcategories of stable homotopy categories coreflective?

Casacuberta

Gutiérrez

Rosický

2014

Advances in Mathematics

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Abstract. We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopěnka's principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. The existence of such a bijection was left as an open problem by Hovey, Palmieri and Strickland in their axiomatic study of stable homotopy categories and also by Neeman in the context of well-generated triangulated categories. IntroductionThe main purpose of this article is to address a question asked in [37, p. 35] of whether every localizing subcategory (i.e., a full triangulated subcategory closed under coproducts) of a stable homotopy category T is the kernel of a localization on T (or, equivalently, the image of a colocalization). We prove that the answer is affirmative if T arises from a combinatorial model category, assuming the truth of a large-cardinal axiom from set theory called Vopěnka' More precisely, we show that, if K is a stable combinatorial model category, then every semilocalizing subcategory C of the homotopy category Ho(K) is coreflective under Vopěnka's principle, and the coreflection is exact if C is localizing. We call C semilocalizing if it is closed under coproducts, cofibres and extensions, but not necessarily under fibres. Examples include kernels of nullifications in the sense of [11] or [19] on the homotopy category of spectra.We also prove that, under the same hypotheses, every semilocalizing subcategory C is singly generated ; that is, there is an object A such that C is the smallest semilocalizing subcategory containing A. The same result is true for localizing subcategories. The

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Weighted limits in simplicial homotopy theory

Cited by 24 publications

References 18 publications

Homotopy coherent adjunctions and the formal theory of monads

Homotopy coherent adjunctions and the formal theory of monads

A May-type spectral sequence for higher topological Hochschild homology

Are all localizing subcategories of stable homotopy categories coreflective?

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