Grothendieck's relative point of view

Scheme theory | Category theory

Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object. It is named after Alexander Grothendieck, who made extensive use of it in treating foundational aspects of algebraic geometry. Outside that field, it has been influential particularly on category theory and categorical logic. In the usual formulation, the language of category theory is applied, to describe the point of view as treating, not objects X of a given category C as such, but morphisms f: X → S where S is a fixed object. This idea is made formal in the idea of the slice category of objects of C 'above' S. To move from one slice to another requires a base change; from a technical point of view base change becomes a major issue for the whole approach (see for example ). A base change 'along' a given morphism g: T → S is typically given by the fiber product, producing an object over T from one over S. The 'fiber' terminology is significant: the underlying heuristic is that X over S is a family of fibers, one for each 'point' of S; the fiber product is then the family on T, which described by fibers is for each point of T the fiber at its image in S. This set-theoretic language is too naïve to fit the required context, certainly, from algebraic geometry. It combines, though, with the use of the Yoneda lemma to replace the 'point' idea with that of treating an object, such as S, as 'as good as' the representable functor it sets up. The Grothendieck–Riemann–Roch theorem from about 1956 is usually cited as the key moment for the introduction of this circle of ideas. The more classical types of Riemann–Roch theorem are recovered in the case where S is a single point (i.e. the final object in the working category C). Using other S is a way to have versions of theorems 'with parameters', i.e. allowing for continuous variation, for which the 'frozen' version reduces the parameters to constants. In other applications, this way of thinking has been used in topos theory, to clarify the role of set theory in foundational matters. Assuming that we don't have a commitment to one 'set theory' (all toposes are in some sense equally set theories for some intuitionistic logic) it is possible to state everything relative to some given set theory that acts as a base topos. (Wikipedia).

Hegel versus Marx

Another re-upload from the previous channel. This short clip comes from the 1987 interview of Peter Singer with Bryan Magee on the life and philosophical work of Hegel and Marx. I put it together quite a long time ago, but I still think it provides a relatively decent summary of some of th

From playlist Hegel

Hegel's Absolute Idealism

A few introductory clips about Hegel's thought. #Philosophy #Hegel

From playlist Hegel

An introduction to the Gromov-Hausdorff distance

Title: An introduction to the Gromov-Hausdorff distance Abstract: We give a brief introduction to the Hausdorff and Gromov-Hausdorff distances between metric spaces. The Hausdorff distance is defined on two subsets of a common metric space. The Gromov-Hausdorff distance is defined on any

From playlist Tutorials

Hegel and his Heirs

Robert Harrison and Adrian Daub discuss Georg Wilhelm Friedrich Hegel and his heirs a few years back in an episode of Entitled Opinions, a KZSU Stanford University program. http://french-italian.stanford.edu/op... Hegel was one of the most important and influential 19th century German phi

From playlist Hegel

What is the difference between Special and General Relativity?

Subscribe to our YouTube Channel for all the latest from World Science U. Visit our Website: http://www.worldscienceu.com/ Like us on Facebook: https://www.facebook.com/worldscienceu Follow us on Twitter: https://twitter.com/worldscienceu

From playlist Science Unplugged: General Relativity

Schopenhauer's Thought

Professor Christopher Janaway discusses some of the ideas of Arthur Schopenhauer. This comes from an episode on Schopenhauer and Wagner. More Short Clips: https://www.youtube.com/playlist?list=PLhP9EhPApKE8v8UVlc7JuuNHwvhkaOvzc #Philosophy #Schopenhauer

From playlist Shorter Clips & Videos - Philosophy Overdose

Yves André - Grothendieck et les équations différentielles

From playlist Séminaire Grothendieck 30 mars 2016

Hegel & Marx - Bryan Magee & Peter Singer (1987)

Peter Singer discusses the thought of Hegel and Marx with Bryan Magee in this 1987 television series. Hegel was an important and influential 19th century German philosopher, best known for his dialectic, absolute idealism, and historicism, among various other things. The Hegelian dialectic

From playlist Hegel

General discussion

With Olivia Caramello, André Joyal, Laurent Lafforgue et Alain Connes

From playlist Topos à l'IHES

Signature de la convention de création du "Laboratoire Alexander Grothendieck"

Signature d'Emmanuel Ullmo (IHÉS) et Christoph Sorger (CNRS)

From playlist Journée inaugurale du Laboratoire Alexander Grothendieck

Gadamer on Schopenhauer (1991)

A very short clip of Gadamer discussing Schopenhauer. More details will be added at a later date. More Short Clips: https://www.youtube.com/playlist?list=PLhP9EhPApKE8v8UVlc7JuuNHwvhkaOvzc More Gadamer: https://www.youtube.com/playlist?list=PLhP9EhPApKE_MbPzGOh27qfdqiCaf7Uot #Philosophy

From playlist Gadamer

Jean-Pierre Serre & Alain Connes - Alexandre Grothendieck

Entretien enregistré à la Fondation Hugot du Collège de France le 27 novembre 2018 entre les mathématiciens Jean-Pierre Serre et Alain Connes à propos de la correspondance Serre / Grothendieck (Correspondance Grothendieck-Serre, Société mathématique de France, 2001 ; Grothendieck-Serre Cor

From playlist Math History

Céline Pessis - L'engagement d'Alexandre Grothendieck durant la première moitié des années 1970

Militant singulier ou porte-parole ? Retour sur l'engagement d'Alexandre Grothendieck durant la première moitié des années 1970 Le 27 janvier 1972, au Centre Européen de Recherches Nucléaires (CERN), citadelle d'une recherche de pointe, des centaines de technicien.

From playlist Séminaire Grothendieck 30 mars 2016

Markus Spitzweck: A Grothendieck Witt space for stable infinity categories with duality

The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: "Workshop: Hermitian K-theory and trace methods" In the talk we will construct a Grothendieck-Witt space for any stable infinity category with duality. We will show that if we apply our constru

From playlist HIM Lectures: Junior Trimester Program "Topology"

Huawei Young Talents Programme - Laurent Lafforgue

The online ceremony celebrating the official launch of the Huawei Young Talents Program at the Institut des Hautes Etudes Scientifiques was held on 6 November 2020. This program aims to support the work of talented researchers in mathematics and theoretical physics at the beginning of thei

From playlist Huawei Young Talents Program - November 2020

Bertrand Russell on Hegel (1957)

A few clips of Bertrand Russell discussing Hegel and his journey away from the Hegelian commitments of his early years. #Philosophy #BertrandRussell #Hegel

From playlist Shorter Clips & Videos - Philosophy Overdose

Entretien avec Luc Illusie, mené par Fabrice Orgogozo

Un entretien avec Luc Illusie, mené par Fabrice Orgogozo à l'IHÉS, le 12 juin 2021. Musique composée par Robert Schumann et interprétée par Paavali Jumppanen, utilisée sous licence CC BY-NC-ND 3.0. Une version de cette vidéo avec des sous-titres en anglais est disponible à l'adresse suiv

From playlist Les entretiens de l'IHES

An interview with Luc Illusie, conducted by Fabrice Orgogozo

An interview with Luc Illusie, conducted by Fabrice Orgogozo at the IHES on June 12, 2021. Subtitles: traductions [dot] mathematiques [at] gmail.com Music: Composed by Robert Schumann. Performed by Paavali Jumppanen and used under a CC BY-NC-ND 3.0 licence. A version of this video withou

From playlist Les entretiens de l'IHES

The Relative Frobenius

Here we give a definition of the relative Frobenius. We also give a good notation that helps you forget about the annoying tensor product that nobody can remember.

From playlist Cartier Operator

Olivia Caramello - 3/4 Introduction to Grothendieck toposes

This course provides an introduction to the theory of Grothendieck toposes from a meta-mathematical point of view. It presents the main classical approaches to the subject (namely, toposes as generalized spaces, toposes as mathematical universes and toposes as classifiers of models of firs

From playlist Olivia Caramello - Introduction to Grothendieck toposes

Grothendieck's relative point of view

Related pages