A CW complex (pronounced "C-W complex") is a type of topological space that is particularly useful in algebraic topology. The term "CW" stands for "cellular" and "weak," referring to the construction method used to create such complexes. A CW complex is constructed using "cells," which are basic building blocks, typically in the shape of disks of different dimensions.

A pseudocircle is a mathematical concept related to the field of geometry, specifically in the study of topology and combinatorial geometry. The term can refer to a set of curves or shapes that exhibit certain properties similar to a circle but may not conform to the strict definition of a circle. In some contexts, a pseudocircle can also refer to a simple closed curve that is homeomorphic to a circle but may not have the same geometric properties as a traditional circle.

The Gysin homomorphism is a concept from algebraic topology and algebraic geometry, particularly in the study of cohomology theories, intersection theory, and the topology of manifolds. It is most commonly associated with the theory of fiber bundles and the intersection products in cohomology.

The Hopf construction is a mathematical procedure used in topology to create new topological spaces from given ones, particularly in the context of fiber bundles and homotopy theory. The method was introduced by Heinz Hopf in the early 20th century. A common application of Hopf construction involves taking a topological space known as a sphere and forming what is called a "Hopf fibration.

The Riemann–Hurwitz formula is a fundamental result in algebraic geometry and complex geometry that relates the properties of a branched cover of Riemann surfaces (or algebraic curves) to the properties of its base surface and the branching behavior of the cover.

In topology, *Shelling* refers to a particular process used in the field of combinatorial topology and geometric topology, primarily focusing on the study of polyhedral complexes and their properties. The concept is related to the process of incrementally building a complex by adding faces in a specific order while maintaining certain combinatorial or topological properties, such as connectivity or homotopy type.

In mathematics, the term "solenoid" can refer to a few different concepts depending on the context, particularly in topology. The most common usage refers to a specific type of topological space, often related to concepts in algebraic topology. ### Topological Solenoid A **topological solenoid** can be thought of as a compact, connected, and locally connected topological space that can be constructed as an inverse limit of circles (S¹).

Twisted Poincaré duality is a concept in algebraic topology that extends classical Poincaré duality.

Category theory is a branch of mathematics that deals with abstract structures and relationships between them. A category consists of objects and morphisms (arrows) that represent relationships between those objects. The central concepts of category theory include: 1. **Objects:** These can be anything—sets, spaces, groups, or more abstract entities. 2. **Morphisms:** These are arrows that represent relationships or functions between objects.

In category theory, a "cosmos" is a concept that extends the idea of a category to a more general framework, allowing for the study of "categories of categories" and related structures. Specifically, a cosmos is a category that is enriched over some universe of sets or types, which allows for a more flexible approach to discussing categories and their properties.

The concept of a Giraud subcategory arises in the context of category theory, particularly in the study of suitable subcategories of a given category. Giraud subcategories are named after the mathematician Jean Giraud, and they are important in the study of sheaf theory and topos theory. A Giraud subcategory is typically defined as a full subcategory of a topos (or a category with certain desirable properties) that retains the essential features of "nice" categories.

Grothendieck's Galois theory is an advanced branch of algebraic geometry and algebraic number theory that generalizes classical Galois theory. Introduced by Alexander Grothendieck in the 1960s, it focuses on the relationship between fields, algebraic varieties, and their coverings, especially in the context of schemes.

The Grothendieck construction is a method in category theory and algebraic topology that allows for the construction of a new category from a functor. Specifically, it is used to "glue together" objects from a family of categories indexed by another category through a functor.

An **indexed category** is a generalization of the concept of categories in category theory, which allows for a more structured way to organize objects and morphisms. In traditional category theory, a category consists of a collection of objects and morphisms (arrows) between them. An indexed category extends this by organizing a category according to some indexing set or category, which provides a way to manage multiple copies of a particular structure.

In category theory, an **isomorphism-closed subcategory** is a subcategory of a given category that is closed under isomorphisms. This means that if an object is in the subcategory, then all objects isomorphic to it are also included in the subcategory. To elaborate further, let \( \mathcal{C} \) be a category and let \( \mathcal{D} \) be a subcategory of \( \mathcal{C} \).

A **Krull–Schmidt category** is a concept in category theory, particularly in the study of additive categories and their decomposition properties. It is named after mathematicians Wolfgang Krull and Walter Schmidt. In a Krull–Schmidt category, every object can be decomposed into indecomposable objects in a manner that is unique up to isomorphism and ordering.

Pointless topology, also known as "point-free topology," is a branch of topology that focuses on the study of topological structures without reference to points. Instead of using points as the fundamental building blocks, it emphasizes the relationships and structures formed by open sets, closed sets, or more general constructs such as locales or spaces. In typical point-set topology, a topological space is defined as a set of points along with a collection of open sets that satisfy certain axioms.

Simplicial localization is a concept from algebraic topology and category theory that is concerned with the process of localizing simplicial sets or simplicial categories. The process is usually aimed at constructing a new simplicial set that reflects the homotopical or categorical properties of the original set while allowing one to "invert" certain morphisms or objects. ### Background Concepts 1. **Simplicial Sets:** A simplicial set is a combinatorial structure that encodes topological information.