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 category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C that makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms.
In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest, most general compact Hausdorff space "generated" by X, in the sense that any continuous map from X to a compact Hausdorff space factors through βX. If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective.
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e. it does not admit a faithful functor to the category of sets.
In mathematics, a sheaf is a tool for systematically tracking data attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set.
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.
In category theory, a branch of mathematics, a monad is an endofunctor, together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group, the finite abelian groups, and the additive group of the integers, the real numbers, and every finite dimensional vector space over the reals or a p-adic field.
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology.
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper.
In general topology and related areas of mathematics, the initial topology on a set , with respect to a family of functions on , is the coarsest topology on X that makes those functions continuous.
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as that remains unchanged during subsequent discussion, and is kept track of during all operations.
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, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories without adjoined products may still have an exponential law.
This is a glossary of properties and concepts in category theory in mathematics.
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 mathematics, homotopy theory is a systematic study of situations in which maps 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 category theory (specifically the study of higher categories).
