arXiv subjects
Publications and source records attributed to Sofía Martínez Alberga.
This article takes steps to show that an equivariant version of the well known phenomena of Koszul Duality can be formulated in a manner not considered prior: via dg (co)algebra valued functors on the orbit category. As such techniques used in this article mix traditional and modern categorical methods in homotopy theory.
We provide a complete description of the model category structures on the nonmodular lattice $N_5$. Furthermore we explain how these model category structures are related to each other via Bousfield localization. This work heavily relies on the use of combinatorical objects from equivariant homotopy theory known as \emph{transfer systems}, and it results in a wealth of interesting interactions between combinatorial and homotopical methods.
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.
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.
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.
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.