Lectures related to Cohomology: Recollection and Foliations

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Algebraic Kunneth Theorem

Covers the Algebraic Kunneth Theorem, explaining chain complexes and cohomology computations.

Cohomology: Cross Product

Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.

Derived functors: Identity and Homotopy Categories

Explores derived functors in model categories, focusing on identity and homotopy categories.

Acyclic Models: Cup Product and Cohomology

Covers the cup product on cohomology, acyclic models, and the universal coefficient theorem.

Homotopy theory of chain complexes

Explores the homotopy theory of chain complexes, focusing on retractions and model category structures.

Injective Modules: Ox-Modules and Injectives

Covers injective modules, Ox-modules, and their relevance in algebraic structures, emphasizing their importance in resolving acyclic resolutions and computing cohomology.

The Topological Künneth Theorem

Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in chain complexes.

Homotopy Categories: Model Structures

Explores homotopy categories in model structures, emphasizing weak equivalences and the Whitehead Lemma.

Homotopy Category and Derived Functors

Explores the homotopy category of chain complexes and the relation between quasi-isomorphisms and chain homotopy equivalences.

Derived functors: Two technical lemmas

Covers two technical lemmas essential for the Fundamental Theorem in homotopical algebra.

Quillen pairs and Quillen equivalences: Derived functors

Explores Quillen pairs, equivalences, and derived functors in homotopical algebra.

Model Categories and Homotopy Theory: Functorial Connections

Covers the relationship between model categories and homotopy categories through functors preserving structural properties.

Homological Algebra: Basics and Applications

Covers the basics of Homological algebra, focusing on Ext modules and their significance in modern mathematics.

Homotopy Theory of Chain Complexes

Explores the homotopy theory of chain complexes, focusing on model categories, weak equivalences, and the retraction axiom.

Existence of Left Derived Functors: Part 2

Concludes the proof of the existence of left derived functors and discusses total left and right derived functors.

Categories and Functors

Covers categories, functors, and presheaf categories, exploring the relationships between objects and morphisms.

Derived Functor Approach

Covers the derived functor approach to Čech cohomology, emphasizing the relationship between derived functors and sheaf theory.

Bar Construction: Homology Groups and Classifying Space

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.

Introduction to Derived Functors: Left and Right Derived Functors

Introduces left and right derived functors in homotopical algebra, emphasizing their uniqueness and providing an illustrative example.