Mapping Cones and Fiber Sequences

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Related lectures (32)

The Topological Künneth Theorem

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

CW Approximation Theorem

Explores the CW Approximation Theorem, constructing CW complexes from spaces to ensure isomorphism on homology groups.

Homotopy Invariance

Explores homotopy invariance, emphasizing the preservation of properties under continuous functions and their relationship with topological spaces.

Euler Characteristic: Surfaces and Homotopy

Explores the Euler characteristic of surfaces and homotopy properties.

Cylinder Construction on Chain Complex

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

Unifying left and right homotopy: The homotopy relation

Explores the equivalence of left and right homotopy for morphisms between bifibrant objects.

Derived functors: Two technical lemmas

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

Universal Covering Construction

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

Induced Homomorphisms on Relative Homology Groups

Covers induced homomorphisms on relative homology groups and their properties.

Retracte: Fundamental Group and Cell Attachment

Covers the concept of a subspace being a retract of another space and fundamental groups, including examples like contracting the teeth of a necklace.

Quasi-Categories: Active Learning Session

Covers fibrant objects, lift of horns, and the adjunction between quasi-categories and Kan complexes, as well as the generalization of categories and Kan complexes.

Homotopy Category of a Model Category

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

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