Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.01350

A homotopical enhancement of Neisendorfer's algebraic models

Bruno Stonek
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give an $\infty$-categorical enhancement of Neisendorfer's equivalences between the homotopy categories of rational simplicial sets, commutative dg algebras (cdgas), cocommutative dg coalgebras (cdgcs) and dg Lie algebras (dgls), where these objects have respective hypotheses of connectedness, nilpotency and finite type. We do so by proving that linear duality is a partial Quillen equivalence between weak finite type connected cdgcs and cdgas, and that Quillen's Chevalley--Eilenberg adjunction is a partial crossed derivable equivalence between cdgcs and dgls, a concept we introduce in this paper to axiomatize precisely the behavior of this adjunction.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bruno Stonek. 2026-10-01. A homotopical enhancement of Neisendorfer's algebraic models. https://arxiv.org/abs/2610.01350

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the structure of abelian Hopf algebras

We study the structure of the category of graded, connected, countable-dimensional, commutative and cocommutative Hopf algebras over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct sum of explicitly given indecomposables. This gives rise to a similar classification of not necessarily $p$-torsion objects that are either free as commutative algebras or cofree as cocommutative coalgebras. We also completely classify those objects that are indecomposable modulo $p$.

math.AT↗
arXiv2026

Galois and separable extensions of Tambara functors

For a map of Tambara functors $R \rightarrow T$ we define when $T$ is separable over $R$ and if $T$ carries an action by some finite group $H$ that fixes $R$ we also define when $R \rightarrow T$ is an $H$-Galois extension of Tambara functors. We show that flat and separable extensions are (formally) étale in the sense of Hill and that Galois extensions are separable. We prove that the Burnside Tambara functor is separably closed and we express Nullstellensatzian Tambara functors as filtered colimits of Galois extensions.

math.AT↗
arXiv2026

From configuration spaces to graph complexes via FA-modules

Work of Gadish and Hainaut (after Petersen) models the compactly supported cohomology of a wedge of circles as a polynomial functor. We identify the coefficients of this functor, $Φ[n,m]$, via a cobar construction of $\mathbf{FA}$-modules. This identification formally implies that these coefficients will arise in computations of graph homology, and we use this result to give examples of graph complexes whose homology may be embedded in $H_c^\ast(F(S^1\vee S^1,n))$. This includes the Payne-Willwacher marked graph complex in genus 2, for which we give a new, explicit decomposition in terms of $\mathbf{FA}$-modules. This allows us to describe $\mathsf{gr}_{11}H_c^{\ast}(\mathcal{M}_{2,n})$ as the cohomology of a complex of decorated trees and to show, for example, $\mathsf{gr}_{11}H_c^{n+1}(\mathcal{M}_{2,n})=0$.

math.AT↗