Lectures related to Homotopy Theory of Chain Complexes

Homotopy theory of chain complexes

Explores the homotopy theory of chain complexes, focusing on retractions and model category structures.

Transfer of Model Structures

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

Acyclic Models: Cup Product and Cohomology

Covers the cup product on cohomology, acyclic models, and the universal coefficient theorem.

Adjunctions and Limits: Exploring Functors and Co-limits

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

Group Cohomology

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

Model Categories and Homotopy Theory: Functorial Connections

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

Limits and colimits in Top

Covers the concepts of limits and colimits in the category of Topological Spaces, emphasizing the relationship between colimit and limit constructions and adjunctions.

Model Categories: Properties and Structures

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

Simplicial and Cosimplicial Objects: Examples and Applications

Covers simplicial and cosimplicial objects in category theory with practical examples.

Algebraic Kunneth Theorem

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

Elementary Properties of Model Categories

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

Natural Transformations in Algebra

Explores natural transformations in algebra, defining functors and isomorphisms.

Homotopy Categories: Model Structures

Explores homotopy categories in model structures, emphasizing weak equivalences and the Whitehead Lemma.

Quillen Equivalences

Explores Quillen equivalences, emphasizing the preservation of cofibrations and acyclic cofibrations.

Derived functors: Identity and Homotopy Categories

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

Homotopy Theory: Cylinders and Path Objects

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

Homotopy Category and Derived Functors

Explores the homotopy category of chain complexes and the relation between quasi-isomorphisms and chain homotopy equivalences.

Limits and Colimits in Functor Categories

Explores limits and colimits in functor categories, focusing on equalizers, pullbacks, and their significance in category theory.

Homotopy Theory of Chain Complexes

Explores the model structure on chain complexes over a field.

Construction of the homotopy category

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