Lectures related to Left Homotopy as an Equivalence Relation: The Homotopy Relation in a Model Category

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

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

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

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

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

Introduces left homotopy between morphisms and its preservation under postcomposition in a model category.

Explores the basic properties of left homotopy in model categories, focusing on weak equivalences and morphism relationships.

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

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

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

Explores the construction of cylinder objects in chain complexes over a field, focusing on left homotopy and interval chain complexes.

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

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

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

Explores sets of left homotopy equivalence classes of morphisms in model categories.

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

Explores the Serre model structure, focusing on left and right homotopy equivalences.

Explores natural transformations in algebra, defining functors and isomorphisms.

Covers the Fundamental Theorem of homotopical algebra, Quillen pairs, and derived functors.

Covers the definition and properties of a model category, including fibrations, cofibrations, weak equivalences, and more.