Quasicategories: An Alternative Homotopy Theory

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

Homotopy theory of chain complexes

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

Homotopical Algebra: Introduction

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

Construction of the homotopy category

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

Simplicial and Cosimplicial Objects: Examples and Applications

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

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.

Lifting properties and model categories

Covers the study of lifting properties in categories, focusing on the left and right lifting properties.

Homotopy Theory: Cylinders and Path Objects

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

Model Categories and Homotopy Theory: Functorial Connections

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

Quillen Equivalences

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