Homotopy theory

Homotopy theory is a branch of algebraic topology that studies the properties of topological spaces through the concept of homotopy, which is a mathematical equivalence relation on continuous functions. The main focus of homotopy theory is to understand the ways in which spaces can be transformed into each other through continuous deformation.

In homotopy theory, a branch of topology, theorems often deal with properties of spaces and maps (functions between spaces) that remain invariant under continuous deformations, such as stretching and bending, but not tearing or gluing.

A **2-group** is a concept in group theory, a branch of mathematics. In particular, a 2-group is a group in which every element has an order that is a power of 2.

Adams filtration is a concept in homotopy theory, particularly in the study of stable homotopy groups of spheres and related areas. It is named after the mathematician Frank Adams, who developed this theory in the mid-20th century. Adams filtration is associated with the idea of understanding the stable homotopy category through a hierarchical structure that helps in studying and organizing the stable homotopy groups of spheres.

Adams spectral sequences are a sophisticated tool used in algebraic topology and homotopy theory, particularly in the study of stable homotopy groups of spheres and related objects. They are named after Frank Adams, who developed the theory in the 1960s. Here's an overview of the key concepts associated with Adams spectral sequences: 1. **Spectral Sequences**: These are mathematical constructs used to compute homology or cohomology groups in a systematic way.

A¹ homotopy theory is a branch of algebraic topology that is concerned with the study of homotopy theories in the context of algebraic varieties over a field, particularly a field with a non-Archimedean valuation or more generally over a base scheme. It is primarily developed in the framework of stable and unstable homotopy types, where the concepts of homotopy can be adapted to the settings of algebraic geometry.

Bousfield localization is a technique in homotopy theory, a branch of algebraic topology, that focuses on constructing new model categories (or topological spaces) from existing ones by inverting certain morphisms (maps). The concept was introduced by Daniel Bousfield in the context of stable homotopy theory, but it has since found applications in various areas of mathematics.

The classifying space for the unitary group \( U(n) \), denoted as \( BU(n) \), is an important object in algebraic topology and represents the space of principal \( U(n) \)-bundles.

In topology, a cofibration is a specific type of map between topological spaces that satisfies certain conditions. Cofibrations play a crucial role in homotopy theory and the study of fibration and cofibration sequences. They are often defined in terms of the homotopy extension property. ### Definition: A map \( i : A \to X \) is called a **cofibration** if it satisfies the homotopy extension property with respect to any space \( Y \).

Coherency in homotopy theory refers to the study of higher categorical structures and their relationships, particularly in the context of homotopy types, homotopy types as types, and the coherence conditions that arise in higher-dimensional category theory.

In algebraic topology, cohomotopy is a concept closely related to the more familiar notions of homotopy and cohomology. While homotopy typically deals with the idea of deformation of spaces and maps between them, cohomotopy focuses on a related set of questions but from the perspective of cohomology theories and spaces of maps.

In topology, a **compactly generated space** is a type of topological space that can be characterized by its relationship with compact subsets. Specifically, a topological space \( X \) is said to be compactly generated if it is Hausdorff and a topology on \( X \) can be described in terms of its compact subsets.

In the field of topology, a **contractible space** is a type of topological space that is homotopically equivalent to a single point. This means that there exists a continuous deformation (a homotopy) that can transform the entire space into a point while keeping the structure of the space intact.

The Cotangent complex is a fundamental construction in algebraic geometry and homotopy theory, especially within the context of derived algebraic geometry. It can be seen as a tool to study the deformation theory of schemes and their morphisms.

Cotriple homology is a concept that arises in the context of homological algebra and category theory. It is associated with the study of coalgebras and cohomological methods, akin to how traditional homology theories apply to algebraic structures like groups, rings, and spaces.

Desuspension is not a widely recognized term in academic or technical literature, and its meaning can depend on the context in which it is used. However, in a general sense, "desuspension" can refer to the process of removing particles or substances that are suspended in a liquid or gas phase.

An Eilenberg-MacLane space is a fundamental concept in algebraic topology, named after mathematicians Samuel Eilenberg and Saunders Mac Lane. It is used to study topological properties related to cohomology theories and homotopy theory.

Equivariant stable homotopy theory is a branch of algebraic topology that studies the stable homotopy categories of topological spaces or spectra with a group action, particularly focusing on the actions of a compact Lie group or discrete group. The theory extends classical stable homotopy theory, which examines stable phenomena in topology, into the context where symmetry plays an important role.

The term "exterior space" can refer to different concepts depending on the context in which it is used. Here are a few interpretations: 1. **Architecture and Urban Planning**: In this context, exterior space often refers to outdoor areas surrounding buildings or structures. This can include gardens, parks, plazas, patios, and other outdoor environments that are designed for public or private use. It emphasizes the design and arrangement of these spaces to enhance usability, aesthetic appeal, and connectivity with the built environment.

A *fibrant object* is a concept from homotopy theory and category theory, particularly in the context of model categories. A model category is a category equipped with both a notion of weak equivalences and a well-behaved notion of fibrations and cofibrations. Fibrant objects in this setting are those that satisfy certain conditions which make them "nice" from the point of view of homotopy.

"Frank Adams" could refer to different people, places, or concepts, depending on the context. Here are a few possibilities: 1. **Historical Figures**: There may be individuals named Frank Adams who have made contributions in various fields, such as politics, arts, or science. 2. **Fictional Characters**: Frank Adams could be a character in literature, film, or television.

The Generalized Whitehead product is a concept in algebraic topology, specifically within the context of homotopy theory. It generalizes the classical Whitehead product, which arises in the study of higher homotopy groups and the structure of loop spaces. ### Background In algebraic topology, the Whitehead product is a way of constructing a new homotopy class of maps from two existing homotopy classes.

The Generalized Poincaré Conjecture extends the classical Poincaré Conjecture, which is a statement about the topology of 3-dimensional manifolds. The original Poincaré Conjecture, proposed by Henri Poincaré in 1904, asserts that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

The Halperin conjecture is a statement in the field of topology, specifically relating to the study of CW complexes and their homotopy groups. Formulated by the mathematician and topologist Daniel Halperin in the 1970s, the conjecture predicts certain properties regarding the homotopy type of a space based on the behavior of its fundamental group and higher homotopy groups.

Homotopical connectivity is a concept from algebraic topology, a branch of mathematics that studies topological spaces through the lens of homotopy theory. It provides a way to classify topological spaces based on their "connectedness" in a homotopical sense. In more detail, homotopical connectivity can be understood through the following concepts: 1. **Connectedness**: A topological space is called connected if it cannot be divided into two disjoint open sets.

Homotopy is a concept in topology, a branch of mathematics that studies the properties and structures of spaces that are preserved under continuous transformations. More specifically, homotopy provides a way to classify continuous functions between topological spaces based on their ability to be deformed into one another.

The Homotopy Analysis Method (HAM) is a powerful and versatile mathematical technique used to solve nonlinear differential equations. Developed by Liao in the late 1990s, HAM is founded on the principles of homotopy from topology and provides a systematic approach to find approximate analytical solutions. ### Core Concepts of HAM: 1. **Homotopy**: In topology, homotopy refers to a continuous transformation of one function into another.

The homotopy category is a fundamental concept in algebraic topology and homotopy theory that captures the idea of "homotopy equivalence" between topological spaces (or more generally, between objects in a category) in a categorical framework. To understand the homotopy category, we begin with the following components: 1. **Topological Spaces and Continuous Maps**: In topology, we often deal with spaces that can be continuously deformed into each other.

In categorical topology, the concepts of homotopy colimits and homotopy limits extend the classical constructions of colimits and limits to a homotopical setting, allowing us to analyze and compare spaces in a way that respects their topological properties.

In algebraic topology, a homotopy group is an important algebraic invariant that captures the topological structure of a space. The most common homotopy groups are the fundamental group and higher homotopy groups. 1. **Fundamental Group (\(\pi_1\))**: The fundamental group is the first homotopy group and provides a measure of the "loop structure" of a space.

Homotopy groups of spheres are a fundamental topic in algebraic topology that encapsulate information about the topology of higher-dimensional spheres. More formally, the \(n\)-th homotopy group of the \(n\)-dimensional sphere \(S^n\), denoted \(\pi_n(S^n)\), is defined as the set of homotopy classes of based continuous maps from the \(n\)-dimensional sphere \(S^n\) to itself.

The Homotopy Hypothesis, often discussed in the context of higher category theory and homotopy theory, is a conjecture in mathematics concerning the relationship between homotopy types and higher categorical structures. It essentially posits that certain categories, specifically (\(\infty\)-categories), can be equivalently described in terms of homotopy types.

A **homotopy sphere** is a mathematical concept in the field of topology, specifically in geometric topology. It refers to a manifold that is homotopically equivalent to a sphere. This means that, while a homotopy sphere may not be geometrically the same as a standard sphere (such as the 2-sphere \( S^2 \) in three-dimensional space), it shares the same topological properties related to how paths can be continuously deformed within it.

The Hopf invariant is a topological invariant that arises in the study of mappings between spheres, particularly in the context of homotopy theory and homotopy groups of spheres. Named after Heinz Hopf, the invariant provides a way to classify certain types of mappings and can be used to distinguish between different homotopy classes of maps.

Hypercovering typically refers to a concept in topology and algebraic geometry that involves certain types of coverings related to sheaves, schemes, or topological spaces. In general, a hypercover is a tool used to construct derived functors or to study the properties of spaces in a more refined manner. In particular, in the context of sheaf theory, a hypercover is a type of covering that allows for 'higher' covering conditions.

The term "Infinite Loop Space Machine" is not a standard term in computer science or technology, but it seems to evoke concepts from various areas of computing, particularly in programming, hardware design, or theoretical computer science. 1. **Infinite Loop**: In programming, an infinite loop is a sequence of instructions that, when executed, repeats indefinitely. This can happen due to a loop condition that always evaluates to true.

The **iterated monodromy group** is a concept from the field of dynamical systems and algebraic geometry, particularly in the context of studying polynomial maps and their dynamics. It serves as a tool to understand the action of a polynomial or rational function on its fibers, especially in relation to their dynamical behavior.

A J-homomorphism is a concept in topology, specifically within the field of homotopy theory, that relates to stable homotopy groups and the homotopy type of spheres. It arises in the context of studying the relationships between various homotopy groups of spheres and stable homotopy theory. The J-homomorphism is an important tool in algebraic topology, particularly in the study of the stable homotopy category.

A **Kan fibration** is a concept from category theory, particularly in the context of simplicial sets and homotopy theory. It generalizes the notion of a fibration in topological spaces to simplicial sets, allowing one to work with homotopical algebra. To understand Kan fibrations, we must first familiarize ourselves with simplicial sets.

In topology, the localization of a topological space is a method of constructing a new topological space from an existing one by focusing on a particular subset of the original space. The concept of localization can be understood in several contexts, such as the localization of rings or sheaves, but here I will outline the localization of a topological space itself, particularly in algebraic topology. ### 1.

In mathematics, particularly in algebraic topology, the term "loop space" refers to a certain kind of space that captures the idea of loops in a given topological space. Specifically, the loop space of a pointed topological space \( (X, x_0) \) is the space of all loops based at the point \( x_0 \).

A **model category** is a concept from category theory, which is a branch of mathematics that deals with abstract structures and relationships between them. Specifically, a model category provides a framework for doing homotopy theory in a categorical setting. It allows mathematicians to work with "homotopical" concepts such as homotopy equivalences, fibrations, and cofibrations in a systematic way.

In category theory, an \( N \)-group is a concept that extends the notion of groups to a more general framework, particularly in the context of higher-dimensional algebra. The term "N-group" can refer to different concepts depending on the specific area of study, but it is commonly associated with the study of higher categories and homotopy theory.

The Nilpotence Theorem, often referred to in the context of algebra, pertains primarily to the properties of nilpotent elements or nilpotent operators in various algebraic structures, such as rings and linear operators. In a general sense, an element \( a \) of a ring \( R \) is said to be **nilpotent** if there exists a positive integer \( n \) such that \( a^n = 0 \).

The Novikov conjecture is a significant hypothesis in the field of topology and geometry, particularly concerning the relationships between the algebraic topology of manifolds and their geometric structure. It was proposed by the Russian mathematician Sergei Novikov in the 1970s. At its core, the Novikov conjecture deals with the higher dimensional homotopy theory, specifically the relationship between the homotopy type of a manifold and the groups of self-homotopy equivalences of the manifold.

In topology, a **path** is a concept that describes a continuous function from the closed interval \([0, 1]\) into a topological space \(X\). More formally, a path can be defined as follows: A function \(f: [0, 1] \to X\) is called a path in \(X\) if it satisfies the following conditions: 1. **Continuity**: The function \(f\) is continuous.

A "phantom map" typically refers to a theoretical or conceptual representation in various contexts, including geography, fantasy mapping, or even in virtual reality and gaming. However, the term can also have specific meanings in different fields: 1. **Theoretical Geography or Cartography**: A phantom map might refer to a map that represents an area that doesn't exist in reality, such as a fictional world in a novel or game. It often serves as a tool for storytelling and world-building.

In category theory and related fields in mathematics, a **pointed space** is a type of topological space that has a distinguished point. More formally, a pointed space is a pair \((X, x_0)\), where \(X\) is a topological space and \(x_0 \in X\) is a specified point called the **base point** or **point of interest**.

A Postnikov system is a concept in algebraic topology, specifically in the study of homotopy theory. It is a type of construction used to analyze the homotopy type of a space by breaking it down into simpler pieces that reflect certain homotopical features. More formally, a Postnikov system consists of a tower of spaces and maps that encode the information of the homotopy groups of a space.

The Puppe sequence specifically refers to a numerical sequence mentioned in various mathematical discussions, although it might not be widely recognized or defined in mainstream mathematics.

A quasi-category is a concept from the field of category theory, specifically in homotopy theory. It is used to formalize the notion of "weak n-categories" where we want to study spaces that behave like categories, but where the laws of composition and associativity are only satisfied up to higher homotopies. Quasi-categories are defined in a more relaxed way compared to ordinary categories.

In category theory, a **Quillen adjunction** is a specific type of adjunction between two categories that arises within the context of homotopy theory, particularly when dealing with model categories.

Rational homotopy theory is a branch of algebraic topology that studies spaces using rational coefficients. It focuses on understanding the homotopy type of topological spaces by considering their behavior when coefficients are taken in the field of rational numbers \(\mathbb{Q}\).

Ravenel's conjectures are a series of conjectures in the field of algebraic topology, specifically concerning stable homotopy theory. Proposed by Douglas Ravenel in the 1980s, these conjectures are primarily about the relationships between stable homotopy groups of spheres and the structure of the stable homotopy category, particularly in relation to the stable homotopy type of certain spaces.

The Seifert–Van Kampen theorem is a fundamental result in algebraic topology that provides a method for computing the fundamental group of a topological space that can be decomposed into simpler pieces. Specifically, it relates the fundamental group of a space to the fundamental groups of its subspaces when certain conditions are satisfied.

Shape theory is a branch of mathematics that studies the properties and classifications of shapes in a more abstract sense. It primarily deals with the concept of "shape" in topological spaces and focuses on understanding how shapes can be analyzed and compared based on their intrinsic properties, rather than their exact geometrical measurements. One of the key aspects of shape theory is the idea that two shapes can be considered equivalent if they can be continuously transformed into one another without cutting or gluing.

Simple homotopy equivalence is a concept in algebraic topology that provides a way to compare topological spaces in terms of their deformation properties. More specifically, it focuses on the notion of homotopy equivalence under certain simplifications. Two spaces \( X \) and \( Y \) are said to be *simple homotopy equivalent* if there exists a sequence of simple homotopy equivalences between them.

Simple homotopy theory is a branch of algebraic topology that provides a way to study the properties of topological spaces through the lens of homotopy equivalence. It is particularly concerned with the study of CW complexes and involves a concept known as simple homotopy equivalence. ### Key Concepts 1. **Homotopy**: In general, homotopy is a relation between continuous functions, where two functions are considered equivalent if one can be transformed into the other through continuous deformation.

Simplicial homotopy is a branch of algebraic topology that studies topological spaces using simplicial complexes. It combines concepts from both homotopy theory and simplicial geometry. Here's a breakdown of what it involves and its significance: ### Key Concepts 1. **Simplicial Complexes**: A simplicial complex is a combinatorial structure made up of vertices, edges, triangles, and higher-dimensional simplices. It serves as a combinatorial model for topological spaces.

A **simplicial presheaf** is a specific type of presheaf that arises in the context of simplicial sets and homotopy theory. It is a functor from the category of simplicial sets (or a related category) to another category (usually the category of sets, or perhaps some other category of interest such as topological spaces, abelian groups, etc.).

In mathematics, particularly in the field of algebraic topology, a **simplicial space** is a topological space that is equipped with a simplicial structure. More specifically, a simplicial space is a contravariant functor from the simplex category, which comprises simplices of various dimensions and their face and degeneracy maps, to the category of topological spaces.

In mathematics, particularly in category theory and algebraic topology, the smash product is a specific operation that combines two pointed spaces or pointed sets. The smash product is denoted by \( X \wedge Y \), where \( X \) and \( Y \) are pointed spaces, meaning that each has a distinguished 'base point.

Sobolev mapping refers to the concept of mappings (or functions) between two spaces that belong to Sobolev spaces, which are a class of function spaces that consider both the functions and their weak derivatives.

Spanier–Whitehead duality is a concept in algebraic topology, named after the mathematicians Edwin Spanier and Frank W. Whitehead. It provides a duality between certain types of topological spaces regarding their homotopy and homology theories. More specifically, it relates the category of pointed spaces to the category of pointed spectra, allowing one to translate problems in unstable homotopy theory into stable homotopy theory, and vice versa.

In topology, the term "spectrum" often refers to the spectrum of a topological space or a mathematical structure associated with it. Two commonly encountered contexts in which the term "spectrum" is used include algebraic topology and categorical topology. Here are some explanations of both contexts: 1. **Spectrum in Algebraic Topology**: In algebraic topology, the term "spectrum" can refer to a sequence of spaces or a generalized space arising in stable homotopy theory.

Stable homotopy theory is a field in algebraic topology that studies the properties of spaces and spectra that remain invariant under suspensions (or shifts). It arises from the observation that the homotopy groups of spheres, which are foundational objects in topology, exhibit a highly structured and rich behavior when examined in a stable context.

The **stable module category** is a concept from modern algebra related to the representation theory of finite-dimensional algebras and the study of stable homotopy theory. It serves as a framework that can simplify certain computations and analyses in algebra. ### Key Concepts 1. **Modules**: In this context, consider a finite-dimensional algebra \( A \) over a field (or a more general ring). A module over this algebra is a mathematical structure that generalizes the notion of vector spaces.

The Sullivan conjecture, proposed by mathematician Dennis Sullivan in the 1970s, pertains to the areas of topology and dynamical systems. Specifically, it deals with the interaction between topology and algebraic geometry concerning the existence of certain types of invariants. The conjecture states that any two homotopy equivalent aspherical spaces have homeomorphic fundamental groups.

In topology, the term "suspension" refers to a specific construction that produces a new topological space from an existing one. Given a topological space \(X\), the suspension of \(X\), denoted as \(\text{Susp}(X)\), is formed in the following way: 1. **Start with X**: Take a topological space \(X\).

The Toda bracket is a mathematical construction from algebraic topology, specifically in the context of homotopy theory. It arises in the study of homotopy groups of spheres and the stable homotopy category. The Toda bracket provides a way to construct new homotopy classes from existing ones and is particularly useful in establishing relations between them.

In the context of category theory and algebraic topology, a topological half-exact functor is a type of functor that reflects certain properties related to homotopy and convergence, particularly in the context of topological spaces, simplicial sets, or other similar structures. While the term "topological half-exact functor" is not widely standardized or commonly used in the literature, it's likely referring to concepts related to exactness in categorical contexts.

Topological rigidity is a concept in topology and differential geometry that refers to the behavior of certain spaces or structures under continuous deformations. A space is considered topologically rigid if it cannot be continuously deformed into another space without fundamentally altering its intrinsic topological properties. More formally, a topological space \(X\) is said to be rigid if any homeomorphism (a continuous function with a continuous inverse) from \(X\) onto itself must be the identity map.

In the context of mathematics, particularly in topology and algebraic geometry, the term "universal bundle" can refer to different concepts depending on the specific field of study. However, it commonly pertains to a type of fiber bundle that serves as a sort of "universal" example for a given class of objects. 1. **Universal Bundle in Algebraic Geometry**: In algebraic geometry, a universal bundle often refers to a family of algebraic varieties parameterized by a base space.

In homotopy theory, the concept of *weak equivalence* is central to the study of topological spaces and their properties under continuous deformations. Two spaces (or more generally, two objects in a suitable category) are said to be weakly equivalent if they have the same homotopy type, meaning there exists a continuous mapping between them that induces isomorphisms on all homotopy groups.

In topology, a space is said to be *weakly contractible* if it satisfies a certain condition regarding homotopy and homotopy groups.

In topology, a wedge sum is a specific way of combining two or more topological spaces into a single space. The construction involves taking a collection of spaces and identifying a single point from each space. The basic idea is as follows: 1. **Choose Spaces**: Consider two or more topological spaces, say \(X_1, X_2, \ldots, X_n\).

The Whitehead product is a concept from algebraic topology, specifically in the context of algebraic K-theory and homotopy theory. It is named after the mathematician G. W. Whitehead and plays a significant role in the study of higher homotopy groups and the structure of loop spaces. In general, the Whitehead product is a binary operation that can be defined on the homotopy groups of a space.

Étale homotopy type is a concept used in algebraic topology and algebraic geometry, specifically in the context of the study of schemes and the homotopical properties of algebraic varieties over a field. It is a way to describe the "shape" of a scheme using notions from homotopy theory.

An ∞-groupoid is a fundamental structure in higher category theory and homotopy theory that generalizes the notion of a groupoid to higher dimensions. In this context, we can think of a groupoid as a category where every morphism is invertible. An ∞-groupoid extends this idea by allowing not only objects and morphisms (which we typically think of in standard category theory), but also higher-dimensional morphisms, representing "homotopies" between morphisms.