Homotopical Algebra: Introduction

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Related lectures (46)

Homotopy Theory: Cylinders and Path Objects

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

Construction of the homotopy category

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

Homotopy Category of a Model Category

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

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.

The Whitehead Lemma: Homotopy Equivalence in Model Categories

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

Elementary Properties of Model Categories

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

Homotopy Theory in Care Complexes

Explores the construction of cylinder objects in chain complexes over a field, focusing on left homotopy and interval chain complexes.

Lifting Properties in Model Categories: An Overview

Provides an overview of lifting properties in model categories, focusing on their definitions and implications for morphisms and commutative diagrams.

Homotopy theory of chain complexes

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

The Topological Künneth Theorem

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

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.

Left Homotopy as an Equivalence Relation: The Homotopy Relation in a Model Category

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

Quasicategories: An Alternative Homotopy Theory

Introduces quasicategories as an alternative approach to defining homotopy maps and categories.

Relative Homotopy Groups

Covers relative homotopy groups, establishing long exact sequences and defining boundary homomorphisms.

Topology: Homotopy and Cone Attachments

Discusses homotopy and cone attachments in topology, emphasizing their significance in understanding connected components and fundamental groups.

Quillen pairs and Quillen equivalences: Derived functors

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

Introduction to Model Categories

Explores lifting properties and model categories in topological spaces.

Sets of Left Homotopy Classes: The Homotopy Relation in a Model Category

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

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.

Topology: Homotopy and Projective Spaces

Discusses homotopy, projective spaces, and the universal property of quotient spaces in topology.

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