MATH-436: Homotopical algebra

Lectures in this course (54)

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

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

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

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

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

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.

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

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

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

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.

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

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

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