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Higher Vector Bundles
Category Theory · Submitted 2 Sep 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Concretely, we set an equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. Concretely, we set an equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage. Our result is an incarnation of the Higher Grothendieck Correspondence, it can be seen as a relative Dold-Kan Theorem and extends earlier work of Gracia-Saz and Mehta on VB-groupoids.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. | contribution 1.00 | 0.93 | 0.07 | asserted 1.00 |
| s1 | We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. | contribution 0.84 | 0.93 | 0.11 | asserted 1.00 |
| s2 | Concretely, we set an equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage. | result 0.90 | 0.93 | 0.62 | asserted 1.00 |
| s3 | Our result is an incarnation of the Higher Grothendieck Correspondence, it can be seen as a relative Dold-Kan Theorem and extends earlier work of Gracia-Saz and Mehta on VB-groupoids. | interpretation 0.99 | 0.79 | 0.60 | asserted 0.95 |
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The paper
Original abstract
We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. Concretely, we set an equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage. Our result is an incarnation of the Higher Grothendieck Correspondence, it can be seen as a relative Dold-Kan Theorem and extends earlier work of Gracia-Saz and Mehta on VB-groupoids.
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How the literature uses it
3 arXiv papers cite this in 34 sentences
Papers citing it · 23 sentences without a confident label
Builds on it 1 paper
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To achieve that, we are going to extend the splitting construction of [ref] for higher vector bundles equipped with cleavages to morphisms between them.
The Adjoint Representation of a Higher Lie Groupoid · Introduction · 0.98
Uses its method or tool 2 papers
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The key ingredient is the semi-direct product $G⋉_R E$ for general representations up to homotopy $GE=⊕_n≥0E_n$, recently developed [ref] , which produces a higher vector bundle $V→G$, a simplicial vector bundle over the nerve, generalizing VB-groupoids.
On the cohomology of differentiable stacks · Introduction · 0.80
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Most of what we are going to say in the next two sections is drawn out from [ref] .
The Adjoint Representation of a Higher Lie Groupoid · The splitting construction for higher vector bundles · 0.80
Cites it as support 1 paper
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It turns out that $$ admits a canonical homotopy inverse [ref] and, therefore, becomes an isomorphism within the homotopy category $^N_+(G)$ of $^N_+(G)$; in particular, down to the homotopy category, the functor $E ↦E$ becomes an essentially surjective one.
The Adjoint Representation of a Higher Lie Groupoid · Further applications · 0.97
Cites it as background 3 papers
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As to the remaining sections, [ref] and [ref] review a selection of topics from [ref] which constitute the indispensable conceptual background for the subject matter of section [ref].
The Adjoint Representation of a Higher Lie Groupoid · Introduction · 1.00
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See [ref] for further details.
On the cohomology of differentiable stacks · Representations up to homotopy · 0.87
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We conclude the appendix recalling that there is an equivalence between the category of RUTHs of a fixed Lie groupoid $G$, concentrated in non-positive degrees, and the category of higher VB groupoids on $G$, i.e. simplicial vector bundles over $G^(∙)$ (up to some technical aspects, see [ref] ).
Shifted Contact Structures on Differentiable Stacks · Representations up to Homotopy · 0.91
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