Badajoz, Spain English Notes

A classifying space of a topological group \(G\) is a quotient of a contractible space by a proper free action of \(G\). There are a variety of functorial classifying space models, such as the Milgram model \(B\) based on the bar construction, or the classifying complex \(W\) for simplicial groups. The Milgram model can be generalized to topological group-like monoids, but the same does not happen with the classifying complex. In this talk, we present a simplicial model of a classifying space for topological group-like monoids by means of the homotopy coherent nerve. As an application, we describe a model of the fundamental \(\infty\)-groupoid of a topological space as a topologically enriched category using Moore paths.

Barcelona, Spain English Notes

In these notes, we will present an introduction to hypercommutative algebras. This structure was discovered by Dijkgraaf, Verlinde and Verlinde, and can be found with different names and small variations, like Witten-Dijkgraaf-Verlinde-Verlinde algebras, formal Frobenius manifolds, genus zero reduction of Gromov-Witten theories or genus zero cohomological field theories. The hypercommutative algebra structure plays a crucial role in mathematical physics because it describes the algebraic structure of quantum cohomology of varieties.

Barcelona, Spain English Notes

In higher category theory, \(\infty\)-groupoids are \(\infty\)-categories whose morphisms are weakly invertible at all orders. Every topological space has an associated \(\infty\)-groupoid, named its fundamental \(\infty\)-groupoid, which encodes the information of higher paths over the space. The statement that every space can be recovered up to homotopy from its fundamental \(\infty\)-groupoid is known as Grothendieck’s homotopy hypothesis. In this presentation, we choose a model of \(\infty\)-categories based on topologically enriched categories, and discuss the homotopy hypothesis in this context.

Barcelona, Spain English Notes

In higher category theory, \(\infty\)-groupoids are \(\infty\)-categories whose morphisms are weakly invertible at all orders. Every topological space has an associated \(\infty\)-groupoid, named its fundamental \(\infty\)-groupoid, which encodes the information of higher paths over the space. The statement that every space can be recovered up to homotopy from its fundamental \(\infty\)-groupoid is known as Grothendieck’s homotopy hypothesis. In this talk we present the model of \(\infty\)-categories based on topologically enriched categories, and discuss the homotopy hypothesis in this context.

Sevilla, Spain Spanish Notes

En teoría de categorías de orden superior, los \(\infty\)-grupoides son \(\infty\)-categorías con morfismos débilmente invertibles en todos los órdenes. A cada espacio topológico podemos asociarle un \(\infty\)-grupoide, llamado su \(\infty\)-grupoide fundamental, que codifica la información de todas las homotopías de orden superior sobre el espacio. La hipótesis de homotopía de Grothendieck postula que un espacio topológico puede ser recuperado a partir de su \(\infty\)-grupoide fundamental salvo homotopía. En esta charla se explicará un modelo de \(\infty\)-categorías basado en categorías enriquecidas en espacios topológicos, se discutirá la hipótesis de homotopía en ese contexto y se definirá el \(\infty\)-grupoide fundamental mediante una construcción basada en las categorías de caminos de Moore. Este punto de vista se sustenta en una demostración de la equivalencia entre el nervio coherente de un \(\infty\)-grupoide y el nervio de Segal de su categoría topológica asociada.

Barcelona, Spain English Notes

Left Bousfield localization of model categories is one of the possible ways to create new model structures from existing ones. Under certain assumptions on the original model category, a left Bousfield localization always exists. This construction could be sum up informally as taking the same class of cofibrations but adding new weak equivalences. The first appearance of this construction was in the context of localizing simplicial sets with respect to a generalized homology theory, by Bousfield. When we localize a model structure over simplicial sets with respect to a homology theory \(E_*\), we “lose” homotopical information for every simplicial set, but retaining the \(E_*\)-accessible parts. The main examples of this type of constructions are the rational model structure and the \(p\)-adic model structure over simplicial sets.

Online Spanish

Recientemente se ha incorporado un nuevo lenguaje en los estándares de la web: WebAssembly. Este nuevo lenguaje no solo destaca por su rol dentro de la infraestructura web, sino también por el que puede tener en los ambientes nativos en un futuro cercano. En esta charla, exploraremos algunas de sus características y veremos que implicaciones puede tener en cuanto a seguridad y portabilidad.

Barcelona, Spain Catalan

Barcelona, Spain Catalan Notes

Barcelona, Spain Catalan Notes

Sevilla, Spain Spanish Notes

La teoría homotópica de tipos es una rama de las matemáticas originada en la década de 2010, que relaciona la teoría de tipos de Martin-Löf con el estudio de los \(\infty\)-grupoides. Sus elementos fundamentales son el axioma de univalencia de Voevodsky y los tipos inductivos de orden superior. Cualquier resultado en teoría homotópica de tipos puede ser formalizado mediante un asistente de demostraciones. Los tipos inductivos de orden superior permiten definir tipos generados libremente por estructuras inspiradas en CW-complejos como la circunferencia, el toro o la botella de Klein. En este póster se definen los recubridores de tipos inductivos y se esboza una demostración de que el tipo de la botella de Klein admite un recubridor de dos hojas equivalente al toro.