Lectures related to Introduction to Derived Functors: Left and Right Derived Functors

Explores natural transformations in algebra, defining functors and isomorphisms.

Derived functors: Identity and Homotopy Categories

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

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

Covers the theory of groups and homotopical algebra, emphasizing natural transformations, identities, and isomorphism of categories.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Existence of Left Derived Functors

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

Homotopical Algebra: (Co)Limits

Explores the concept of (co)limits in homotopical algebra, discussing functor relations, special cases, and the universal properties of colimits and limits.

Quillen pairs and Quillen equivalences: Derived functors

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

Existence of Left Derived Functors: Part 2

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

Model Categories and Homotopy Theory: Functorial Connections

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

Transfer of Model Structures

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

Homotopical Algebra: The Homotopy Category of a Model Category

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

Adjunctions and Limits: Exploring Functors and Co-limits

Covers adjunctions and limits, focusing on functors, co-limits, and their applications in category theory.

Homotopy Theory of Chain Complexes

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

Derived Functors in Homotopical Algebra

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

Derived functors: Two technical lemmas

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

Equivalences of Categories

Explores examples of natural transformations, equivalence of categories, and adjunction with specific instances involving Un.

Functors and Adjunctions

Explores functors between categories and the conditions for left and right adjoints.

Functors: Categories, Functors, and Natural Transformations

Introduces functors, including identity and forgetful functors, and explores duality functors.

Homotopical Algebra: Examples and Adjunctions

Explores homotopical algebra examples and adjunctions, focusing on left and right adjoints in group functors and coproducts.