Lectures related to Homotopy theory

Focuses on lifting properties in homotopy theory of chain complexes.

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

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

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

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

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

Explores acyclic fibrations and cylinder objects in chain complexes.

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

Explores the homotopy theory of chain complexes over a field, focusing on closure properties and decomposition.

Explores chain homotopy, projective complexes, and homotopy equivalences in chain complexes.

Explores homotopy equivalence in chain complexes, emphasizing path object construction and left/right homotopy characterization.

Explores the model structure on chain complexes over a field.

Introduces the concept of a universal covering construction with examples like Hawaiian rings.

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Explores the cylinder construction on a chain complex and its equivalence to chain homotopy.

Explores the cellular approximation theorem for CW complexes and its implications on homotopy groups.

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

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

Covers homotopy type theory, including equivalences, contractible spaces, pushouts, and applications.