Concept

Abstract simplicial complex

Related lectures (32)

Defines and studies the Joyal Model structure, focusing on the properties of quasi-categories within this framework.

Introduces two fundamental examples of simplicial sets: the nerve of a small category and the singular simplicial set of a topological space.

Limits and colimits as adjoints: Chapter 1(c)

Explores the characterization of limits and colimits as adjoints in category theory.

Geometric realization: Simplicial sets

Covers the construction of a left adjoint to the singular set functor, comparing the homotopy theory of topological spaces with that of simplicial sets.

Simplices

Covers the concept of simplices in delta complexes and explains the standard n-simplex and the ordering of vertices.

Simplicial Homology: Examples

Explores examples of simplicial homology by equipping known topological spaces with delta complex structures.

Homology and the fundamental group

Explores simplicial and singular homology, reduced homology groups, and their connection to the fundamental group.

Cellular Approximation: Homotopy and CW Complexes

Explores the cellular approximation theorem for CW complexes and its implications on homotopy groups.

From Simplicial Categories to Quasicategories

Introduces the construction of quasi-categories from Kan enriched categories through defining simplicially enriched categories and constructing the simplicial nerve functor.

Homology and Homotopy

Explores the comparison of long exact sequences for vibrations and the relationship between homotopy and homology groups.

From (Model) Categories to Quasicategories

Explores the relationship between categories and quasi-categories.

Review of Simplicial Sets

Covers limits, colimits, standard simplices, mapping spaces, boundaries, horns, and spines.