Lectures related to Derived Functors in Homotopical Algebra

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

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

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

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

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

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

Explores natural transformations in algebra, defining functors and isomorphisms.

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

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

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

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

Explores the existence of left derived functors in homotopical algebra, focusing on isomorphism conditions and natural transformations.

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

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

Focuses on proving the construction of the homotopy category and its properties, including preservation of composition and uniqueness of functors.

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

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

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

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

Explores Quillen equivalences, emphasizing the preservation of cofibrations and acyclic cofibrations.