Lecture

The Nerve and Geometric Realization

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In course

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous examples of model categories and their applications in ...

Description

This lecture explores the computation of the nerve of small categories, showing that the nerve construction commutes with products. The slides, presented by Kathryn Hess, delve into the geometric realization of nerves, discussing various aspects such as topological sets and adjoint relationships. The lecture provides insights into the relationship between nerves and geometric structures, offering a deep dive into the mathematical concepts involved.

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Related concepts (32)

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.

Fuzzy set

In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].

Universal set

In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.

Facial nerve

The facial nerve, also known as the seventh cranial nerve, cranial nerve VII, or simply CN VII, is a cranial nerve that emerges from the pons of the brainstem, controls the muscles of facial expression, and functions in the conveyance of taste sensations from the anterior two-thirds of the tongue. The nerve typically travels from the pons through the facial canal in the temporal bone and exits the skull at the stylomastoid foramen.

Vagus nerve

The vagus nerve, also known as the tenth cranial nerve, cranial nerve X, or simply CN X, is a cranial nerve that carries sensory fibers that create a pathway that interfaces with the parasympathetic control of the heart, lungs, and digestive tract. It comprises two nerves—the left and right vagus nerves—but they are typically referred to collectively as a single subsystem. The vagus is the longest nerve of the autonomic nervous system in the human body and comprises both sensory and motor fibers.

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