Homotopical Algebra: Theory and Applications

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Related lectures (53)

Homotopical Algebra Log in to Mediaspace to watch this video

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

Active Learning Session Log in to Mediaspace to watch this video

Explores natural transformations in group theory and category theory, emphasizing functor composition and morphism composition.

Category Theory: G-sets and Left Adjoint Functor Log in to Mediaspace to watch this video

Explores the construction of G-sets and left adjoint functors in category theory.

Group Theory: Adjoint Functors and G-sets Log in to Mediaspace to watch this video

Explores adjunction between functors, composition of applications, G-equivariance, and natural transformations in G-sets.

Group Actions: Equivariance and Functors Log in to Mediaspace to watch this video

Explores equivariance in group actions and functors, demonstrating its importance in group theory.

Limits and Colimits: Understanding Categories Log in to Mediaspace to watch this video

Explores limits and colimits in category theory, discussing their definitions, properties, and applications, including the non-existence of limits in certain categories and the relationships between limits and colimits under functors.

Active Learning: Group Theory Log in to Mediaspace to watch this video

Explores adjoint functors, points fixes, orbits, and non-trivial actions in group theory.

Equivalences of Categories Log in to Mediaspace to watch this video

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

Group Actions and Functors Log in to Mediaspace to watch this video

Covers group quotient, group actions, and functors connecting sets and G-sets.

Introduction to Category Theory: Functors Log in to Mediaspace to watch this video

Covers the concept of functors in category theory, including composition, identity functors, and forgotten functors.

Natural Transformations in Algebra Log in to Mediaspace to watch this video

Explores natural transformations in algebra, defining functors and isomorphisms.

Free Abelian Groups: Group Theory Log in to Mediaspace to watch this video

Explores the concept of free abelian groups as an important left adjoint functor.

Natural Transformations: Functors and Categories Log in to Mediaspace to watch this video

Explores functors, natural transformations, and the theory of groups, emphasizing the importance of comparisons and structure preservation.

Lie Algebra Homomorphisms Log in to Mediaspace to watch this video

Explores the association of Lie algebras to algebraic groups and homomorphisms.

Abelian Groups: Free Abelian Groups Log in to Mediaspace to watch this video

Explores free abelian groups, homomorphisms, and exact sequences in group theory.

Sheafification: Natural Procedure and Adjoints Log in to Mediaspace to watch this video

Covers sheafification, adjoint functors, and challenges in ensuring sheaf properties.

Morphism of Groups Log in to Mediaspace to watch this video

Covers the concept of morphism of groups, actions on sets, and automorphisms.

The Lemma of the Five: Abelian Groups Log in to Mediaspace to watch this video

Presents the Lemma of the Five, a fundamental result in group theory.

Coxeter Groups: Generators and Relations Log in to Mediaspace to watch this video

Explores Coxeter groups, emphasizing generators, relations, and unique presentations in group theory.

Quillen Equivalences Log in to Mediaspace to watch this video

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

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