Lectures related to Homotopy Theory: Cylinders and Path Objects

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.

Homotopical Algebra: Introduction

Introduces the course on homotopical algebra, exploring the power of analogy in pure mathematics.

Model Categories and Homotopy Theory: Functorial Connections

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

Homotopy Theory of Chain Complexes

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

Elementary Properties of Model Categories

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

Homotopy Category of a Model Category

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

Active Learning: Functors and Geometric Realization

Covers the computation of nerves and geometric realization in simplicial sets, along with functors into and out of the category of simplicial sets.

The Whitehead Lemma: Homotopy Equivalence in Model Categories

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

Serre model structure on Top

Explores the Serre model structure on Top, focusing on right and left homotopy.

Simplicial and Cosimplicial Objects: Examples and Applications

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

Model Categories: Properties and Structures

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

Serre model structure: Left and right homotopy

Explores the Serre model structure, focusing on left and right homotopy equivalences.

Adjunctions and Limits: Exploring Functors and Co-limits

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

Topology: Homotopy and Cone Attachments

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

Homology of Riemann Surfaces

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Homotopy theory of chain complexes

Explores the homotopy theory of chain complexes, focusing on defining cylinder objects and left homotopy.

Homotopy Categories: Model Structures

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

Model Category: Definition and Elementary Properties

Covers the definition and properties of a model category, including fibrations, cofibrations, weak equivalences, and more.

Construction of the homotopy category

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

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.