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Lecture

Active Learning: Functors and Geometric Realization

Description

This lecture discusses the computation of nerves for various categories and provides a geometric realization of an interesting simplicial set. The instructor presents examples of functors into and out of the category of simplicial sets, illustrating the relationships between different structures. The lecture includes detailed examples that demonstrate how to define products in simplicial sets and explores the concept of geometric realization in this context. The instructor emphasizes the importance of understanding these functors and their applications in category theory. Additionally, the lecture addresses questions related to the construction of simplicial sets associated with groups, providing insights into the connections between algebraic and geometric perspectives. The discussion is enriched with visual aids and examples that clarify the concepts being presented, making it accessible for students and researchers interested in category theory and its applications.

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Topology: Algebraic topology

Related concepts (32)

Simplicial set

In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.

Higher category theory

In mathematics, higher category theory is the part of at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental . An ordinary has and morphisms, which are called 1-morphisms in the context of higher category theory.

Topology

In mathematics, topology (from the Greek words τόπος, and λόγος) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.

Abstract simplicial complex

In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).

Homotopy theory

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and (specifically the study of ). In homotopy theory and algebraic topology, the word "space" denotes a topological space.

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