Homotopy theory of modules over diagrams of rings

Greenlees, J. P. C.; Shipley, Brooke

doi:10.1090/s2330-1511-2014-00012-2

Cited by 17 publications

(36 citation statements)

References 17 publications

(34 reference statements)

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“…It was later generalized by [1] to an arbitrary category C under suitable assumptions on F(c) for c ∈ C. See also [28,9,3]. This topic will not be addressed in this paper.…”

Section: Relation To Other Workmentioning

confidence: 99%

The Grothendieck construction for model categories

Harpaz

Prasma²

2015

Advances in Mathematics

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The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As a main result, we establish an equivalence between suitable diagrams of model categories indexed by M and a new notion of model fibrations over M. When M is a model category, our construction endows the Grothendieck construction with a model structure which gives a presentation of Lurie's ∞-categorical Grothendieck construction and enjoys several good formal properties. We apply our construction to various examples, yielding model structures on strict and weak group actions and on modules over algebra objects in suitable monoidal model categories.

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“…It was later generalized by [1] to an arbitrary category C under suitable assumptions on F(c) for c ∈ C. See also [28,9,3]. This topic will not be addressed in this paper.…”

Section: Relation To Other Workmentioning

confidence: 99%

The Grothendieck construction for model categories

Harpaz

Prasma²

2015

Advances in Mathematics

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“…We would like to assemble the category of all SO.2/-spectra from the category of F -complete objects and objects localized away from F. The way we do it here is to take suitable model categories of F -complete spectra, of spectra away from F , and Tate spectra, and then construct a model structure on the category S -mod of P -diagrams of such objects: a cellularization (K top -cell-S -mod) of this model category of P -diagrams is then shown to be Quillen equivalent to the original category of SO.2/-spectra essentially using the fact that the Tate square is a homotopy pullback. The machinery of Greenlees and Shipley [15] was built for this purpose.…”

Section: Contribution Of This Papermentioning

confidence: 99%

“…This idea has been studied in some detail in [15]. In this section we introduce the relevant structures and leave most of the proofs to the reference.…”

Section: Diagrams Of Model Categoriesmentioning

confidence: 99%

Rational SO(2)–equivariant spectra

Barnes

Greenlees

Kedziorek

et al. 2017

Algebr. Geom. Topol.

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“…There is a category Mod(P • ) whose objects are the P • -modules, with morphisms as defined above. By recent work of Greenlees and Shipley [14], the category Mod(P • ) can be given a strict projective model structure arising from the projective model structures on the categories Mod(P L ), in which a morphism f is a weak equivalence (or a fibration) if and only if each map f L is a weak equivalence (or, respectively, a fibration) of P L -modules.…”

Section: Pro-operads and Their Modulesmentioning

confidence: 99%

Cross-effects and the classification of Taylor towers

Arone

Ching

2016

Geom. Topol.

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Abstract. Let F be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of F have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of F are a right module over a certain 'pro-operad'. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivial operad. We show that the Taylor tower of the functor F can be reconstructed from this structure on the derivatives.Let C and D each be either the category of based topological spaces, or the category of spectra. Let F : C → D be a homotopy functor. Goodwillie's homotopy calculus provides a systematic way to decompose F into homogeneous pieces which are classified by certain spectra with Σ n actions, denoted ∂ n F and, by analogy with ordinary calculus, called the derivatives or Taylor coefficients of F .A key problem in the homotopy calculus is to describe all the relevant structure on the symmetric sequence ∂ * F , and to reconstruct the original functor F (or at least its Taylor tower) from this structure. In [2] we gave a general description of this structure. In this paper we give an alternative description of the structure on ∂ * F in cases when F takes values in the category of spectra. We interpret this structure as that of a 'module over a pro-operad'. The structure arises from actions of certain operads on the partially stablized cross-effects of F . Thus we connect the module structure on ∂ * F with Goodwillie's presentation of ∂ * F as stablized cross-effects of F .Let us review briefly our previous results on the subject. Let I Top * be the identity functor on the category of based spaces. It was shown by the second author in [5] that ∂ * I Top * has an operad structure. In fact, ∂ * I Top * is a topological realization of the classical Lie operad, and we refer to ∂ * I Top * informally as the topological Lie operad. In [1] we showed that for a functor F : Top f * → Sp, the symmetric sequence ∂ * F is a right module over ∂ * I Top * . We also remarked in [1] that the module structure was not sufficient for recovering the Taylor tower of F , and therefore there had to be additional structure, that still had to be described. In [2] we gave a description of this extra information. Specifically, we constructed a comonad C, defined on the category of right ∂ * I Top * -modules, which acts on the derivatives of a homotopy functor F . We showed that the Taylor tower of F can be recovered from this action.The methods of [2] were quite general (applying to functors between any combinations of the categories of based spaces and spectra). The comonad C has the form ∂ * Φ, where ∂ * is Goodwillie differentiation, and Φ is a functor right adjoint to ∂ * . In this paper we provide a more concrete description of t...

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Homotopy theory of modules over diagrams of rings

Cited by 17 publications

References 17 publications

The Grothendieck construction for model categories

The Grothendieck construction for model categories

Rational SO(2)–equivariant spectra

Cross-effects and the classification of Taylor towers

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