Lectures related to Homotopical Algebra: The Homotopy Category of a Model Category

Explores natural transformations in algebra, defining functors and isomorphisms.

Explains the construction of the homotopy category of a model category using cofibrant and fibrant replacement.

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

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

Covers the elementary properties of model categories, emphasizing the duality between fibrations and cofibrations.

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

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

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

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

Introduces the homotopy category of a model category with inverted weak equivalences and unique homotopy equivalences.

Explores the Whitehead Lemma, showing when a morphism is a weak equivalence.

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

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

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

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

Covers cylinders, path objects, and homotopy in model categories.

Introduces the course on homotopical algebra, exploring the power of analogy in pure mathematics.

Covers the transfer of model structures through adjunctions in the context of model categories.

Covers the properties and structures of model categories, focusing on factorizations, model structures, and homotopy of continuous maps.

Explores the left homotopy relation as an equivalence relation in model categories.