Mathematical Structuralism à la Grothendieck

The few people who have a wide enough background and perspective enabling them to feel the main questions, are devoting their energies to things which seem more directly rewarding. Maybe even a wind of disrepute for any foundational matters whatever is blowing nowadays.

Pursuing Stacks, Alexander Grothendieck

This is an open-ended course aiming at all mathematics and computer science students, covering some parts of modern mathematics from a logical, synthetic and structural point of view. It starts with some category theory as the language of mathematical structuralism. Then, it moves to the main organizational concept of the course, i.e., the space as the world of finitely verifiable constructions. We start with the propositional part, namely the theory of locales and then move to the higher-order versions of Grothendieck toposes and higher toposes. We finally use the ultimate notion of space to touch some parts of synthetic functorial geometry as the theory of locally simple spaces.

Recordings:

They can be found on Youtube and Aparat (Parts I and II).

Lecture Notes:

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

Exercises:

See the exercises here. It gets updated regularly.

Expository Papers:

Synthetic Differential Geometry, Michael Shulman.
Quantum Quandaries A Category Theoretic Perspective, John Baez.
Structuralism, Invariance, and Univalence, Steve Awodey.
Five Stages of Accepting Constructive Mathematics, Andrej Bauer.
Calculus is Algebra, William S. Hatcher.
What is Applied Category Theory? Tai-Danae Bradley.

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.
Lecture 22: Profinite sets, Profinite groups and some philosophical notes on pro-objects, their topological representations and the structuralism behind the scene.
Lecture 23: Flat modules, functoriality of (co)limits and (co)limit-preserving functors.
Lecture 24: (Co)limits in functor categories and non-(co)limit preserving functors.
Lecture 25: Computability of tensors, van Kampen theorem and the limit-preserving nature of corepresentables.
Lecture 26: “The projective line”, its non-representability and reflected (co)limits.
Lecture 27: Created (co)limits, commutation of (co)limits with (co)limits and some philosophical discussions.
Lecture 28: Some examples of the commutation of (co)limits with (co)limits, Some examples and non-examples of the commutation of limits with colimits, a computation rule for the filtered colimits and some philosophical discussions.
Lecture 29: Commutation of finite limits and filtered colimits in Set and the category of elements.
Lecture 30: The examples of the category of elements, action groupoid and the priority of the identification to the identity.
Lecture 31: Strong yoneda lemma, discrete fibrations and their examples.
Lecture 32: Examples of discrete fibrations and covering spaces.
Lecture 33: Lifiting Properties of Covering Spaces..
Lecture 34: The equivalence between discrete fibrations and variable sets, Grothendieck Galois theory.
Lecture 35: Density theorem, free co-completion.

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Mathematical Structuralism à la Grothendieck

The few people who have a wide enough background and perspective enabling them to feel the main questions, are devoting their energies to things which seem more directly rewarding. Maybe even a wind of disrepute for any foundational matters whatever is blowing nowadays.

Pursuing Stacks, Alexander Grothendieck

Recordings:

They can be found on Youtube and Aparat (Parts I and II).

Lecture Notes:

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

Exercises:

See the exercises here. It gets updated regularly.

Expository Papers:

Synthetic Differential Geometry, Michael Shulman.

Quantum Quandaries A Category Theoretic Perspective, John Baez.

Structuralism, Invariance, and Univalence, Steve Awodey.

Five Stages of Accepting Constructive Mathematics, Andrej Bauer.

Calculus is Algebra, William S. Hatcher.

What is Applied Category Theory? Tai-Danae Bradley.

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.

Lecture 22: Profinite sets, Profinite groups and some philosophical notes on pro-objects, their topological representations and the structuralism behind the scene.

Lecture 23: Flat modules, functoriality of (co)limits and (co)limit-preserving functors.

Lecture 24: (Co)limits in functor categories and non-(co)limit preserving functors.

Lecture 25: Computability of tensors, van Kampen theorem and the limit-preserving nature of corepresentables.

Lecture 26: “The projective line”, its non-representability and reflected (co)limits.

Lecture 27: Created (co)limits, commutation of (co)limits with (co)limits and some philosophical discussions.

Lecture 28: Some examples of the commutation of (co)limits with (co)limits, Some examples and non-examples of the commutation of limits with colimits, a computation rule for the filtered colimits and some philosophical discussions.

Lecture 29: Commutation of finite limits and filtered colimits in Set and the category of elements.

Lecture 30: The examples of the category of elements, action groupoid and the priority of the identification to the identity.

Lecture 31: Strong yoneda lemma, discrete fibrations and their examples.

Lecture 32: Examples of discrete fibrations and covering spaces.

Lecture 33: Lifiting Properties of Covering Spaces..

Lecture 34: The equivalence between discrete fibrations and variable sets, Grothendieck Galois theory.

Lecture 35: Density theorem, free co-completion.