Fundamental groupoidIn 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 , the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids. 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.
Loop spaceIn topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental group π1(X).
Homotopy theoryIn 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 studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and (specifically the study of ). In homotopy theory and algebraic topology, the word "space" denotes a topological space.
GroupoidIn mathematics, especially in and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Group with a partial function replacing the binary operation; in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group.
Homotopy type theoryIn mathematical logic and computer science, homotopy type theory (HoTT hɒt) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory applies. This includes, among other lines of work, the construction of homotopical and models for such type theories; the use of type theory as a logic (or internal language) for abstract homotopy theory and ; the development of mathematics within a type-theoretic foundation (including both previously existing mathematics and new mathematics that homotopical types make possible); and the formalization of each of these in computer proof assistants.
FibrationThe notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. A mapping satisfies the homotopy lifting property for a space if: for every homotopy and for every mapping (also called lift) lifting (i.e. ) there exists a (not necessarily unique) homotopy lifting (i.e.
Finite topological spaceIn mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". Let be a finite set.
HomotopyIn topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (həˈmɒtəpiː, ; ˈhoʊmoʊˌtoʊpiː, ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces.
Sierpiński spaceIn mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology.
General topologyIn mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points.
