Tangent categories of algebras over operads

Harpaz, Yonatan; Nuiten, Joost; Prasma, Matan

doi:10.48550/arxiv.1612.02607

Cited by 5 publications

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“…Let M be a left proper combinatorial model category (so that M ∞ is presentable) and let (N × N) ⋆ be the category obtained by freely adjoining a zero object. The main result of [HNP18] shows that the tangent ∞-category TM ∞ is presented by a left Bousfield localisation of the Reedy model structure on functors (N × N) ⋆ → M. In [HNP16], this construction of the tangent ∞-category is used prove an equivalence between parametrised spectra in the ∞-category of algebras over an ∞-operad and operadic modules.…”

Section: Global Theorymentioning

confidence: 99%

Combinatorial parametrised spectra

Braunack-Mayer

2019

Preprint

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We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey's sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted cohomology groups.As an application of our results we prove a tangent prolongation of Simpson's theorem, characterising tangent ∞-categories of presentable ∞-categories as accessible localisations of ∞-categories of presheaves of parametrised spectra. Applying these results to the homotopy theory of smooth ∞-stacks produces well-behaved (symmetric monoidal) model categories of smooth parametrised spectra. These models provide a concrete foundation for studying twisted differential cohomology, and subsume previous work of Bunke and Nikolaus.

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Section: Global Theorymentioning

confidence: 99%

Combinatorial parametrised spectra

Braunack-Mayer

2019

Preprint

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“…Finally, in §2.4 we recall the notions of stabilization, abstract parameterized spectra and Quillen cohomology, whose specialization to the case of (∞, 2)-categories is our main interest in this paper. As in the previous papers, [HNP17a], [HNP17b] and [HNP17c] we adopt the formalism of tangent categories and tangent bundles, which follow Lurie's abstract cotangent complex formalism developed in [Lur14,§7.3].…”

Section: Recollectionsmentioning

confidence: 99%

Quillen cohomology of $(\infty,2)$-categories

Harpaz,

Nuiten,

Prasma

2018

Preprint

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In this paper we study the homotopy theory of parameterized spectrum objects in the ∞-category of (∞, 2)-categories, as well as the Quillen cohomology of an (∞, 2)-category with coefficients in such a parameterized spectrum. More precisely, we construct an analogue of the twisted arrow category for an (∞, 2)-category C, which we call its twisted 2-cell ∞-category. We then establish an equivalence between parameterized spectrum objects over C, and diagrams of spectra indexed by the twisted 2-cell ∞-category of C. Under this equivalence, the Quillen cohomology of C with values in such a diagram of spectra is identified with the two-fold suspension of its inverse limit spectrum. As an application, we provide an alternative, obstruction-theoretic proof of the fact that adjunctions between (∞, 1)-categories are uniquely determined at the level of the homotopy (3, 2)-category of Cat∞.

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“…This paper is part of an on going work concerning the abstract cotangent complex and Quillen cohomology. In [HNP17a] and [HNP17b], the authors study the tangent categories of algebras over an operad and of enriched categories, as well as their cotangent complex. The work in loc.…”

Section: Introductionmentioning

confidence: 99%

“…was carried out in a model-categorical framework. The subject of this note, which was splitted from [HNP17a], is the development of such a framework, which we believe is of independent interest.…”

Section: Introductionmentioning

confidence: 99%

“…whose fibers are the tangent ∞-categories of the ∞-category M ∞ underlying M. Our main results are that TM enjoys particularly favorable properties on the model categorical level: it exhibits TM as a relative model category over M, in the sense of [HP15], and forms a model fibration when restricted to the fibrant objects of M. Furthermore, when M is tensored over a suitable model category S, the tangent bundle TM inherits this structure, and thus becomes enriched in S. This enrichment plays a key role in the description of the tangent categories of algebras and enriched categories as in [HNP17a] and [HNP17b], and may be useful for other purposes as well.…”

Section: Introductionmentioning

confidence: 99%

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The tangent bundle of a model category

Harpaz,

Nuiten,

Prasma

2018

Preprint

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This paper studies the homotopy theory of parameterized spectrum objects in a model category from a global point of view. More precisely, for a model category M satisfying suitable conditions, we construct a relative model category TM → M, called the tangent bundle, whose fibers are models for spectra in the various over-categories of M, and which presents the ∞-categorical tangent bundle. Moreover, the tangent bundle TM inherits an enriched model structure when such a structure exists on M. This additional structure is used in subsequent work to identify the tangent bundles of algebras over an operad and of enriched categories. Contents 1. Introduction 1 2. Tangent model categories 2 2.1. Spectrum objects 3 2.2. Parameterized spectrum objects 6 2.3. Suspension spectra 8 2.4. Differentiable model categories and Ω-spectra 11 3. The tangent bundle 15 3.1. The tangent bundle as a relative model category 15 3.2. Tensor structures on the tangent bundle 23 3.3. Comparison with the ∞-categorical construction 24 References 27

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Tangent categories of algebras over operads

Cited by 5 publications

References 0 publications

Combinatorial parametrised spectra

Combinatorial parametrised spectra

Quillen cohomology of $(\infty,2)$-categories

The tangent bundle of a model category

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