Mathematical Structuralism a la Grothendieck

This is a course aiming at all mathematics and computer science students, covering some parts of modern mathematics from a logical, synthetic structural point of view. It starts with some category theory as the language of mathematical structuralism. Then, it moves to point-free topology and its higher-order versions, Grothendieck toposes and higher toposes to finally reach synthetic functorial geometry and abstract homotopy theory.

Recordings:

They can be found in the Youtube channel of the course.

Lecture Notes:

See the full lecture notes here. It gets updated regularly.

Exercises:

See the exercises here. It gets updated regularly.

All Lecture Notes:

Lecture 01: Introduction
Lecture 02: Categories and their examples
Lecture 03: Some examples of categories, Representation theorems, Baby Erlangen program
Lecture 04: Baby Erlangen program continued
Lecture 05: Construction of new categories, Functors
Lecture 06: Some examples of functors
Lecture 07: Fundamental sets and fundamental groupoids
Lecture 08: Brouwer’s fixed-point theorem and natural transformations
Lecture 09: Some examples of natural transformations, CAT as a 2-space
Lecture 10: Some examples of non-natural transformations including no-deleting and no-cloning theorems
Lecture 11: Functor categories, some examples from algebra, topology and logic including motivations for sheaves and quantum contexuality
Lecture 12: Some philosophical discussions on the unity of mathematics, its connection with physics and computer science and some points on intuitionism.
Lecture 13: Functors as ideal objects, representable functors
Lecture 14: The Yoneda lemma, the universal elements
Lecture 15: Morphisms as the generalised elements and the fibrations, terminal objects, products and pullbacks
Lecture 16: Initial objects, coproducts, pushouts, exponential objects and some points on the categorical way of thinking
Lecture 17: Some applications of Yoneda Lemma
Lecture 18: Equalizers and coequalizers, the relationship between pullbacks, terminals, products and equalizers
Lecture 19: Limits and their examples, completion of rings and solenoids
Lecture 20: Sheaves as limits, limits in Set, limits by product and equalizers, colimits and their examples
Lecture 21: Germs as colimits, colimits in Set, colimits by coproduct and coequalizers, completeness in posets, filtered colimits and cofiltered limits, stable linear groups