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Left and right Bousfield localization on lattices
Authors:
Andrés Carnero Bravo,
Shuchita Goyal,
Sofía Martínez Alberga,
Cherry Ng,
Constanze Roitzheim,
Daniel Tolosa
Abstract:
The key information of a model category structure on a poset is encoded in a transfer system, which is a combinatorial gadget, originally introduced to investigate homotopy coherence structures in equivariant homotopy theory. We describe how a transfer system associated with in a model structure on a lattice is affected by left and right Bousfield localization and provide a minimal generating syst…
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The key information of a model category structure on a poset is encoded in a transfer system, which is a combinatorial gadget, originally introduced to investigate homotopy coherence structures in equivariant homotopy theory. We describe how a transfer system associated with in a model structure on a lattice is affected by left and right Bousfield localization and provide a minimal generating system of morphisms which are responsible for the change in model structure. This leads to new concrete insights into the behavior of model categories on posets in general.
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Submitted 11 November, 2025;
originally announced November 2025.
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Equivariant Homotopy Theory via Simplicial Coalgebras
Authors:
Sofía Martínez Alberga,
Manuel Rivera
Abstract:
Given a commutative ring $R$, a $π_1$-$R$-equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers. When $R$ is an algebraically closed field, Raptis and Rivera described a full and faithful model for the homotopy theory of spaces up to $π_1$-$R$-equivalence by means of simplicial coalgebras considered up to a…
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Given a commutative ring $R$, a $π_1$-$R$-equivalence is a continuous map of spaces inducing an isomorphism on fundamental groups and an $R$-homology equivalence between universal covers. When $R$ is an algebraically closed field, Raptis and Rivera described a full and faithful model for the homotopy theory of spaces up to $π_1$-$R$-equivalence by means of simplicial coalgebras considered up to a notion of weak equivalence created by a localized version of the Cobar functor. In this article, we prove a $G$-equivariant analog of this statement using a generalization of a celebrated theorem of Elmendorf. We also prove a more general result about modeling $G$-simplicial sets considered under a linearized version of quasi-categorical equivalence in terms of simplicial coalgebras.
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Submitted 7 April, 2025; v1 submitted 6 October, 2024;
originally announced October 2024.
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Transplanting Trees: Chromatic Symmetric Function Results through the Group Algebra of $S_n$
Authors:
Angèle M. Foley,
Joshua Kazdan,
Larissa Kröll,
Sofía Martínez Alberga,
Oleksii Melnyk,
Alexander Tenenbaum
Abstract:
One of the major outstanding conjectures in the study of chromatic symmetric functions (CSF's) states that trees are uniquely determined by their CSF's. Though verified on graphs of order up to twenty-nine, this result has been proved only for certain subclasses of trees. Using the definition of the CSF that emerges via the Frobenius character map applied to $\mathbb{C}[S_n]$, we offer new algebra…
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One of the major outstanding conjectures in the study of chromatic symmetric functions (CSF's) states that trees are uniquely determined by their CSF's. Though verified on graphs of order up to twenty-nine, this result has been proved only for certain subclasses of trees. Using the definition of the CSF that emerges via the Frobenius character map applied to $\mathbb{C}[S_n]$, we offer new algebraic proofs of several results about the CSF's of trees. Additionally, we prove that a "parent function" of the CSF defined in the group ring of $S_n$ can uniquely determine trees, providing further support for Stanley's conjecture.
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Submitted 20 January, 2022; v1 submitted 18 December, 2021;
originally announced December 2021.
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Spiders and their Kin: An Investigation of Stanley's Chromatic Symmetric Function for Spiders and Related Graphs
Authors:
Angèle M. Foley,
Joshua Kazdan,
Larissa Kröll,
Sofía Martínez Alberga,
Oleksii Melnyk,
Alexander Tenenbaum
Abstract:
We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish that a subclass of generalized spiders, which we call…
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We study the chromatic symmetric functions of graph classes related to spiders, namely generalized spider graphs (line graphs of spiders), and what we call horseshoe crab graphs. We show that no two generalized spiders have the same chromatic symmetric function, thereby extending the work of Martin, Morin and Wagner. Additionally, we establish that a subclass of generalized spiders, which we call generalized nets, has no e-positive members, providing a more general counterexample to the necessity of the claw-free condition. We use yet another class of generalized spiders to construct a counterexample to a problem involving the $e$-positivity of claw-free, P4-sparse graphs, showing that Tsujie's result on the e-positivity of claw-free, P4-free graphs cannot be extended to graphs in this set. Finally, we investigate the e-positivity of another type of graphs, the horseshoe crab graphs (a class of unit interval graphs), and prove the positivity of all but one of the coefficients. This has close connections to the work of Gebhard and Sagan and Cho and Huh.
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Submitted 28 June, 2022; v1 submitted 9 December, 2018;
originally announced December 2018.