In geometry, a line is a fundamental concept that represents a straight one-dimensional figure that extends infinitely in both directions. It has no thickness, width, or curvature, and is typically defined by at least two points. Lines can be described using a variety of properties: 1. **Definition**: A line is determined by any two distinct points on it.

In geometry, the term "parallel" refers to two or more lines or planes that are the same distance apart at all points and do not meet or intersect, no matter how far they are extended. This property is fundamental in understanding the behavior of lines within Euclidean geometry. ### Key Properties of Parallel Lines: 1. **Equidistant**: Parallel lines maintain a constant distance from each other, meaning the distance between them remains consistent along their entire length.

In geometry, a "slab" typically refers to a three-dimensional shape that is essentially a thick, flat object bounded by two parallel surfaces. This can be visualized as a rectangular prism with very small height relative to its length and width, resembling a sheet or a plate. In a more formal mathematical context, particularly in the study of convex geometry, a slab can be defined by two parallel hyperplanes in higher-dimensional spaces.

A spherical shell is a three-dimensional hollow structure that is shaped like a sphere. It is typically defined as the space between two concentric spherical surfaces — an outer surface and an inner surface. The shell has a certain thickness, which is the difference between the radii of the outer and inner surfaces. Key characteristics of a spherical shell include: 1. **Outer Radius (R_outer)**: The radius of the outer surface of the shell.

Bredon cohomology is a type of cohomology theory that is particularly useful in the context of spaces with group actions. It was introduced by Glen Bredon in the 1960s and is designed to study topological spaces with an additional structure of a group action, often leading to insights in equivariant topology.

De Rham cohomology is a mathematical concept from the field of differential geometry and algebraic topology that studies the topology of smooth manifolds using differential forms. It provides a bridge between analysis and topology by utilizing the properties of differential forms and their relationships through the exterior derivative. ### Key Concepts 1. **Differentiable Manifolds**: A differentiable manifold is a topological space that is locally similar to Euclidean space and has a well-defined notion of differentiability.

Elliptic cohomology is a branch of algebraic topology that generalizes classical cohomology theories using the framework of elliptic curves and modular forms. It is an advanced topic that blends ideas from algebraic geometry, number theory, and homotopy theory. ### Key Features 1.

Lie algebra cohomology is a mathematical concept that arises in the study of Lie algebras, which are algebraic structures used extensively in mathematics and physics to describe symmetries and conservation laws. Cohomology, in this context, refers to a homological algebra framework that helps in analyzing the structure and properties of Lie algebras.

In the context of cohomology, a pullback is a construction that allows you to take a cohomology class on a target space and "pull it back" to a cohomology class on a domain space via a continuous map. This is particularly common in algebraic topology and differential geometry. ### Formal Definition Let \( f: X \to Y \) be a continuous map between two topological spaces \( X \) and \( Y \).

Witt vector cohomology is a tool in algebraic geometry and number theory that utilizes Witt vectors to study the cohomological properties of schemes in the context of p-adic cohomology theories. Witt vectors are a generalization of the notion of numbers in a ring, particularly for fields of characteristic \( p \), and they allow the construction of an effective cohomology theory that preserves useful algebraic properties. ### Key Concepts 1.

Étale cohomology is a cohomological theory in algebraic geometry that provides a means to study the properties of algebraic varieties over fields, particularly in the context of fields that are not algebraically closed. It was developed in the mid-20th century, notably by Alexander Grothendieck, and is part of the broader framework of schemes in modern algebraic geometry.

In mathematics, particularly in the field of abstract algebra and category theory, a **category of groups** is a concept that arises from the framework of category theory, which is a branch of mathematics that deals with objects and morphisms (arrows) between them. ### Basic Definitions 1. **Category**: A category consists of: - A collection of objects. - A collection of morphisms (arrows) between those objects, which can be thought of as structure-preserving functions.

In category theory, a preordered set (or preordered set) is a set equipped with a reflexive and transitive binary relation. More formally, a preordered set \( (P, \leq) \) consists of a set \( P \) and a relation \( \leq \) such that: 1. **Reflexivity**: For all \( x \in P \), \( x \leq x \).

Emily Riehl is a mathematician known for her contributions to category theory, homotopy theory, and algebraic topology. She is an associate professor at Johns Hopkins University and has published several research papers in her areas of expertise. Riehl has also been involved in mathematical education, producing resources aimed at improving the teaching and understanding of mathematics, particularly in higher education. She is recognized for her work in making advanced mathematical concepts more accessible.

In mathematics, particularly in the field of algebraic geometry and homological algebra, a **derived category** is a concept that allows one to work with complexes of objects (such as sheaves, abelian groups, or modules) in a way that takes into account their morphisms up to homotopy. Derived categories provide a framework for studying how complex objects relate to one another and for performing calculations in a more flexible manner than is possible in the traditional context of abelian categories.

In category theory, a Kleisli category is a construction that allows you to work with monads in a categorical setting. A monad, in this context, is a triple \((T, \eta, \mu)\), where \(T\) is a functor and \(\eta\) (the unit) and \(\mu\) (the multiplication) are specific natural transformations satisfying certain coherence conditions.

Michael Shulman is a mathematician known for his work in the fields of algebra, category theory, and type theory. He has made contributions to the study of homotopy theory, higher categories, and the connections between mathematics and computer science, particularly in the context of programming languages and formal systems. Shulman has also been involved in research that bridges the gap between abstract mathematical theory and practical computational applications.

Samuel Eilenberg (1913-1998) was a renowned Polish-American mathematician known for his significant contributions to the fields of algebra, topology, and category theory. He was particularly influential in the development of algebraic topology and cohomology theories. Eilenberg is perhaps best known for the concept of Eilenberg-Mac Lane spaces, which are important in algebraic topology and homotopy theory.

Valeria de Paiva is a Brazilian mathematician known for her work in the field of type theory, particularly in the context of computer science and programming languages. She has made significant contributions to the development of mathematical frameworks that inform type systems in software, which are critical for ensuring code correctness and safety. Additionally, Valeria de Paiva has been involved in research related to category theory and its applications in functional programming. She is also noted for her engagement in teaching and collaboration within the academic community.