In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids.
[...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points, [,,,]
Let X be a topological space. Consider the equivalence relation on continuous paths in X in which two continuous paths are equivalent if they are homotopic with fixed endpoints. The fundamental groupoid Π(X), or Π1(X), assigns to each ordered pair of points (p, q) in X the collection of equivalence classes of continuous paths from p to q. More generally, the fundamental groupoid of X on a set S restricts the fundamental groupoid to the points which lie in both X and S. This allows for a generalisation of the Van Kampen theorem using two base points to compute the fundamental group of the circle. [1]
As suggested by its name, the fundamental groupoid of X naturally has the structure of a groupoid. In particular, it forms a category; the objects are taken to be the points of X and the collection of morphisms from p to q is the collection of equivalence classes given above. The fact that this satisfies the definition of a category amounts to the standard fact that the equivalence class of the concatenation of two paths only depends on the equivalence classes of the individual paths. [2] Likewise, the fact that this category is a groupoid, which asserts that every morphism is invertible, amounts to the standard fact that one can reverse the orientation of a path, and the equivalence class of the resulting concatenation contains the constant path. [3]
Note that the fundamental groupoid assigns, to the ordered pair (p, p), the fundamental group of X based at p.
Given a topological space X, the path-connected components of X are naturally encoded in its fundamental groupoid; the observation is that p and q are in the same path-connected component of X if and only if the collection of equivalence classes of continuous paths from p to q is nonempty. In categorical terms, the assertion is that the objects p and q are in the same groupoid component if and only if the set of morphisms from p to q is nonempty. [4]
Suppose that X is path-connected, and fix an element p of X. One can view the fundamental group π1(X, p) as a category; there is one object and the morphisms from it to itself are the elements of π1(X, p). The selection, for each q in M, of a continuous path from p to q, allows one to use concatenation to view any path in X as a loop based at p. This defines an equivalence of categories between π1(X, p) and the fundamental groupoid of X. More precisely, this exhibits π1(X, p) as a skeleton of the fundamental groupoid of X. [5]
The fundamental groupoid of a (path-connected) differentiable manifold X is actually a Lie groupoid, arising as the gauge groupoid of the universal cover of X. [6]
Given a topological space X, a local system is a functor from the fundamental groupoid of X to a category. [7] As an important special case, a bundle of (abelian) groups on X is a local system valued in the category of (abelian) groups. This is to say that a bundle of groups on X assigns a group Gp to each element p of X, and assigns a group homomorphism Gp → Gq to each continuous path from p to q. In order to be a functor, these group homomorphisms are required to be compatible with the topological structure, so that homotopic paths with fixed endpoints define the same homomorphism; furthermore the group homomorphisms must compose in accordance with the concatenation and inversion of paths. [8] One can define homology with coefficients in a bundle of abelian groups. [9]
When X satisfies certain conditions, a local system can be equivalently described as a locally constant sheaf.
The homotopy hypothesis, a well-known conjecture in homotopy theory formulated by Alexander Grothendieck, states that a suitable generalization of the fundamental groupoid, known as the fundamental ∞-groupoid, captures all information about a topological space up to weak homotopy equivalence.
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups. The fundamental group of a topological space is denoted by .
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
In mathematics, a category is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
In mathematics, the Seifert–Van Kampen theorem of algebraic topology, sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space in terms of the fundamental groups of two open, path-connected subspaces that cover . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.
In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain. The pushout consists of an object P along with two morphisms X → P and Y → P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are and .
In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different categories, as discussed below.
In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y.
In mathematics, particularly category theory, a 2-group is a groupoid with a way to multiply objects, making it resemble a group. They are part of a larger hierarchy of n-groups. They were introduced by Hoàng Xuân Sính in the late 1960s under the name gr-categories, and they are also known as categorical groups.
In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental n-categories of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in concurrency, network traffic control, general relativity, noncommutative geometry, rewriting theory, and biological systems.
In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as its reduced homology groups. The most common version of its proof consists of showing that the composition of the homotopy group functors with the infinite symmetric product defines a reduced homology theory. One of the main tools used in doing so are quasifibrations. The theorem has been generalised in various ways, for example by the Almgren isomorphism theorem.
In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.
In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets. It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism.
This is a glossary of properties and concepts in algebraic topology in mathematics.
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 learned as an independent discipline.
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