Research
In the past I worked on -Isovariant Homotopy Theory. I'm currently interested in cartesian fibrations in higher category theory, model structures over enriched virtual double categories and derived categories for real spaces. Below you can find more details about my research interests, current projects, publications, collaborations and more.
Research Areas
I developped a simplicial approach to -isovariant homotopy theory. This framework is meant to study spaces with an involution (action of the group ) where isotropy groups are preserved strictly. I'm currently working on extending this theory to arbitrary profinite groups and exploring connections with real algebraic geometry by means of isovariant Betti realization functors.
In collaboration with Pierre Martinez we are currently exploring the construction of a derived category of sheaves with transfers over real spaces (in the sense of Huisman, Martinez and Gleuher). This theory is meant to be an analogue of Voevodsky's derived category of motives but for real analytic spaces and it should enhance the existing cohomology theory of real spaces developed by Martinez and Gleuher.
I'm interested in the theory of cartesian fibrations in the setting of 2-categories and higher categories. In particular, I'm working in a collaborative project with M. Sarazola, P. Verdugo, J. Nickel, D. Teixeira and C. Bardomiano on the construction of a model category structure for 2-cartesian fibrations over a fixed base 2-category.
I'm also exploring the homotopy theory of enriched virtual double categories (or fc-multicategories). In particular, I'm currently investigating the construction of model category structures for these objects and their applications to higher category theory.
Publications and Preprints
Simplicial -Isovariant Homotopy Theory
Author(s): Santiago Toro
Submitted, 2024
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 , where morphisms are considered to be isovariant. Our methodology centers on simplicial techniques. We replace the conventional simplex category with a modified category and then delve into the study of presheaves of sets on . 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. Our approach distinguishes itself from prior work, such as Yeakel's which primarily focuses on a more topological context and, brings a new perspective to the study of isovariant homotopy theory for -spaces.
A model structure for 2 cartesian fibrations
Author(s): C. Bardomiano, J. Nickel, M. Sarazola, P. Verdugo, D. Teixeira, S. Toro Oquendo
In preparation
Derived categories for real spaces
Author(s): P. Martinez, S. Toro Oquendo
In preparation
Thesis
Simplicial -Isovariant Homotopy Theory
Author(s): Santiago Toro Oquendo
Advisor(s): Johannes Huisman
PhD Thesis, Université de Bretagne Occidentale, 2024
Applications of motivic homotopy theory to arithmetic counts of lines
Author(s): Santiago Toro Oquendo
Advisor(s): Johannes Huisman & Jean Fasel
Master's Thesis, Université Grenoble Alpes, 2019
The Riemann-Roch theorem for curves (in Spanish)
Author(s): Santiago Toro Oquendo
Advisor(s): Margarita Toro
Bachelor's Thesis, Universidad Nacional de Colombia, 2015