arXiv subjects

Santiago Toro Oquendo

Publications and source records attributed to Santiago Toro Oquendo.

2 recordsLinked to original sources

arXiv2026

A model structure for cartesian 2-fibrations

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical ``straightening'' construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

math.CT↗

arXiv2023

Simplicial $C_2$-Isovariant Homotopy

This article presents a novel approach to construct a model category structure designed to model the homotopy theory of spaces equipped with an action by the group $C_2$, where morphisms are considered to be isovariant. Our methodology centers on simplicial techniques. We replace the conventional simplex category $Δ$ with a modified category $C_2 Δ$ and then delve into the study of presheaves of sets on $C_2 Δ$. To establish the model category structure we employ Cisinski's methods for model structures in categories of presheaves. In particular, we use an analogous idea to the one employed by Cisinski and Moerdjik in the construction of a model category structure for Dendroidal Sets (arXiv:0902.1954). Our approach distinguishes itself from prior work, such as Yeakel's (arXiv:1907.12135) which primarily focuses on a more topological context and, brings a new perspective to the study of isovariant homotopy theory for $C_2$-spaces.

math.AT↗