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We prove two representability theorems, up to homotopy, for presheaves taking values in a closed symmetric combinatorial model category \cat V. The first theorem resembles the Freyd representability theorem, the second theorem is closer to the Brown representability theorem. As an application we discuss a recognition principle for mapping spaces., Comment: V2: minor improvements suggested by the referee; 10 pages

We prove the equivalence of several hypotheses that have appeared recently in the literature for studying left Bousfield localization and algebras over a monad. We find conditions so that there is a model structure for local algebras, so that localization preserves algebras, and so that localization lifts to the level of algebras. We include examples coming from the theory of colored operads, and applications to spaces, spectra, and chain complexes.

Let $R$ be an associative ring with unit and denote by $K({\rm R \mbox{-}Proj})$ the homotopy category of complexes of projective left $R$-modules. Neeman proved the theorem that $K({\rm R \mbox{-}Proj})$ is $\aleph_1$-compactly generated, with the category $K^+ ({\rm R \mbox{-}proj})$ of left bounded complexes of finitely generated projective $R$-modules providing an essentially small class of such generators. Another proof of Neeman's theorem is explained, using recent ideas of Christensen and Holm, and Emmanouil. The strategy of the proof is to show that every complex in $K({\rm R \mbox{-}Proj})$ vanishes in the Bousfield localization $K({\rm R \mbox{-}Flat})/\langle K^+ ({\rm R \mbox{-}proj}) \rangle.$, Comment: 5 pages

We show that the complete Segal model structure extends to a model structure on bimplicial presheaves on a small site $\mathscr{C}$, for which the weak equivalences are local (or stalkwise) weak equivalences. This model structure can be realized as a left Bousfield localization of the Jardine model structure on the simplicial presheaves on a site $\mathscr{C}/ \Delta^{op}$. Furthermore, it is shown that this model structure is Quillen equivalent to the model structure of the author's previous preprint entitled 'the Local Joyal Model Structure'. This Quillen equivalence extends an equivalence between the complete Segal space and Joyal model structures, due to Joyal and Tierney., Comment: This is the pre copy edited, post peer review version of an article to appear in 'Applied Categorical Structures'

We provide conditions on a monoidal model category $\mathcal{M}$ so that the category of commutative monoids in $\mathcal{M}$ inherits a model structure from $\mathcal{M}$ in which a map is a weak equivalence or fibration if and only if it is so in $\mathcal{M}$. We then investigate properties of cofibrations of commutative monoids, rectification between $E_\infty$-algebras and commutative monoids, the relationship between commutative monoids and monoidal Bousfield localization functors, when the category of commutative monoids can be made left proper, and functoriality of the passage from a commutative monoid $R$ to the category of commutative $R$-algebras. In the final section we provide numerous examples of model categories satisfying our hypotheses., Version 2 adds material about rectification between strict commutative monoids and $E_\infty$-algebras, adds material about lifting Quillen equivalences to categories of commutative monoids, and adds several new examples: simplicial presheaves, diagram categories, and commutative Smith ideals of ring spectra

In this note, we give a brief overview of the telescope conjecture and the chromatic splitting conjecture in stable homotopy theory. In particular, we provide a proof of the folklore result that Ravenel's telescope conjecture for all heights combined is equivalent to the generalized telescope conjecture for the stable homotopy category, and explain some similarities with modular representation theory., Accepted for publication in Surveys around Ohkawa's theorem on Bousfield classes. All comments welcome. v2: Corrected some typos

The aim of this short paper is two-fold: (i) to construct a TQ-localization functor on algebras over a spectral operad O, in the case where no connectivity assumptions are made on the O-algebras, and (ii) more generally, to establish the associated TQ-local homotopy theory as a left Bousfield localization of the usual model structure on O-algebras, which itself is almost never left proper, in general. In the resulting TQ-local homotopy theory, the "weak equivalences" are the TQ-homology equivalences, where "TQ-homology" is short for topological Quillen homology, which is also weakly equivalent to stabilization of O-algebras. More generally, we establish these results for TQ-homology with coefficients in a spectral algebra A. A key observation, that goes back to the work of Goerss-Hopkins on moduli problems, is that the usual left properness assumption may be replaced with a strong cofibration condition in the desired subcell lifting arguments: Our main result is that the TQ-local homotopy theory can be established (e.g., a semi-model structure in the sense of Goerss-Hopkins and Spitzweck, that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are TQ-equivalences., Comment: 21 pages. We have made the TQ-local homotopy theory explicit (e.g., as a semi-model structure in the sense of Goerss-Hopkins and Spitzweck, that is both cofibrantly generated and simplicial) by localizing with respect to a set of strong cofibrations that are TQ-equivalences

For a systematic study of 2-Segal spaces it is convenient to work in the more general framework of model categories.

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products (pullbacks) of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For stable model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization., 20 pages. The paper "Bousfield localisations along Quillen bifunctors and applications" (arXiv:1411.0500v1) has been divided into two parts: "Towers and fibered products of model structures", which is this arXiv submission, and "Bousfield localisations along Quillen bifunctors" (arXiv:1411.0500v2)

In this paper we show how to modify cofibrations in a monoidal model category so that the tensor unit becomes cofibrant while keeping the same weak equivalences. We obtain aplications to enriched categories and coloured operads in stable homotopy theory., 13 pages. Comments welcome

We prove a general version of Quillen's Theorem B, for actions of simplicial categories, in an arbitrary left Bousfield localization of the homotopy theory of simplicial presheaves over a site. As special cases, we recover a version of the group completion theorem in this general context, as well a version of Puppe's theorem on the stability of homotopy colimits in an infinity-topos, due to Rezk., Comment: 16 pages

In this paper, we prove a conjecture of Blumberg and Hill regarding the existence of $N_\infty$-operads associated to given sequences $\mathcal{F} = (\mathcal{F}_n)_{n \in \mathbb{N}}$ of families of subgroups of $G\times \Sigma_n$. For every such sequence, we construct a model structure on the category of $G$-operads, and we use these model structures to define $E_\infty^{\mathcal{F}}$-operads, generalizing the notion of an $N_\infty$-operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these $E_\infty^{\mathcal{F}}$-operads, obtaining some new results as well for $N_\infty$-operads., Comment: This version has been accepted to Algebraic & Geometric Topology

In 1979, Bousfield defined an equivalence relation on the stable homotopy category. The set of Bousfield classes has some important subsets such as the distributive lattice DL of all classes 〈E〉 which are smash idempotent and the complete Boolean algebra cBA of closed classes. We provide examples of spectra that are in DL, but not in cBA; in particular, for every prime p, the Bousfield class of the Eilenberg–MacLane spectrum 〈HFp〉 is in DL∖cBA.

In this paper, we study the extent to which Bousfield and finite localizations relative to a thick subcategory of equivariant finite spectra preserve various kinds of highly structured multiplications. Along the way, we describe some basic, useful results for analyzing categories of acyclics in equivariant spectra, and we show that Bousfield localization with respect to an ordinary spectrum (viewed as an equivariant spectrum with trivial action) always preserves equivariant commutative ring spectra.

For a "genuine" equivariant commutative ring spectrum $R$, $\pi_0(R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ arising from the existence of multiplicative norm maps. Motivated by the surprising fact that Bousfield localization can destroy some of the norm maps, in previous work we studied equivariant commutative ring structures parametrized by $N_\infty$ operads. In a precise sense, these interpolate between "naive" and "genuine" equivariant ring structures. In this paper, we describe the algebraic analogue of $N_\infty$ ring structures. We introduce and study categories of incomplete Tambara functors, described in terms of certain categories of bispans. Incomplete Tambara functors arise as $\pi_0$ of $N_\infty$ algebras, and interpolate between Green functors and Tambara functors. We classify all incomplete Tambara functors in terms of a basic structural result about polynomial functors. This classification gives a conceptual justification for our prior description of $N_\infty$ operads and also allows us to easily describe the properties of the category of incomplete Tambara functors.

Let BP be the Brown-Peterson spectrum at an odd prime p, and L2 denote the Bousfield localization functor with respect to [Formula: see text]. The Ravenel spectrum T(1) is characterized by BP*(T(1)) = BP*[t1] on the primitive generator t1. In this paper, we determine the homotopy groups π*(L2M ∧ T(1)) for the mod p Moore spectrum M.

We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.

We exhibit a triangulated category T having both products and coproducts, and a triangulated subcategory S of T which is both localizing and colocalizing, for which neither a Bousfield localization nor a colocalization exists. It follows that neither the category S nor its dual satisfy Brown representability. Our example involves an abelian category whose derived category does not have small Hom-sets., 5 pages

The global structure of the unbounded derived category of a truncated polynomial ring on countably many generators is investigated, via its Bousfield lattice. The Bousfield lattice is shown to have cardinality larger than that of the real numbers, and objects with large tensor-nilpotence height are constructed.

This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be $\Sigma$-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter-Shipley about a zig-zag of Quillen equivalences between commutative $H\mathbb{Q}$-algebra spectra and commutative differential graded $\mathbb{Q}$-algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization., Comment: This is the final, journal version

We describe a proof of the Baez–Dolan Stabilization Hypothesis for Rezk’s model of weak n-categories. This proof proceeds via abstract homotopy theory, and en route we discuss a version of left Bousfield localization which does not require left properness. We also discuss conditions under which various categories of operads can be made left proper, but these conditions are difficult to be satisfied, as a counterexample in the context of simplicial sets demonstrates.

We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on N-infinity ring spectra, we construct categories of equivariant operadic modules over N-infinity rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an "incomplete Mackey functor in homotopical categories". In particular, we construct internal norms which satisfy the double coset formula. We regard the work of this paper as a first step towards equivariant derived algebraic geometry., Revised to include appendix on compact Lie groups, reflect referee comments

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E-localisation of this model category. We study the properties of this new construction and relate it to some well-known categories., Comment: 18 pages

Given a spectrum $X$, we construct a spectral sequence of $BP_{*}BP$-comodules that converges to $BP_{*}(L_{n}X)$, where $L_{n}X$ is the Bousfield localization of $X$ with respect to the Johnson?Wilson theory $E(n)_{*}$. The $E_{2}$-term of this spectral sequence consists of the derived functors of an algebraic version of $L_{n}$. We show how to calculate these derived functors, which are closely related to local cohomology of $BP_{*}$-modules with respect to the ideal $I_{n + 1}$.

In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a v_n self map of a finite S--module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) is independent of choices. Goodwillie's general theory says that to any homotopy functor F from S--modules to S--modules, there is an associated tower under F, {P_dF}, such that F --> P_dF is the universal arrow to a d--excisive functor. Our first theorem says that P_dF --> P_{d-1}F always admits a homotopy section after localization with respect to T(n) (and so also after localization with respect to Morava K--theory K(n)). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second theorem which is equivalent to the following: for any finite group G, the Tate spectrum t_G(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees--Sadofsky, Hovey--Sadofsky, and Mahowald--Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus., 25 pages. AmsLatex. Uses XYpics

Let L 2 L_2 denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum E ( 2 ) E(2) . A spectrum L 2 X L_2X is called invertible if there is a spectrum Y Y such that L 2 X ∧ Y = L 2 S 0 L_2X\wedge Y=L_2S^0 . Hovey and Sadofsky, Invertible spectra in the E ( n ) E(n) -local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E ( 2 ) E(2) -local sphere L 2 S 0 L_2S^0 at a prime p > 3 p>3 . At the prime 3 3 , it is shown, A relation between the Picard group of the E ( n ) E(n) -local homotopy category and E ( n ) E(n) -based Adams spectral sequence, that there exists an invertible spectrum X X that is not homotopy equivalent to a suspension of L 2 S 0 L_2S^0 . In this paper, we show the homotopy equivalence v 2 3 : Σ 48 L 2 V ( 1 ) ≃ V ( 1 ) ∧ X v_2^3\colon \Sigma ^{48}L_2V(1)\simeq V(1)\wedge X for the Smith-Toda spectrum V ( 1 ) V(1) . In the same manner as this, we also show the existence of the self-map β : Σ 144 L 2 V ( 1 ) → L 2 V ( 1 ) \beta \colon \Sigma ^{144}L_2V(1)\to L_2V(1) that induces v 2 9 v_2^9 on the E ( 2 ) ∗ E(2)_* -homology.

We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories.