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Topics referred to by the same term
topological category may refer to: A concept in categorical topology; see topological functor A category enriched over the category of topological spaces;
Topological_category
Category whose objects are topological spaces and whose morphisms are continuous maps
mathematics, the category of topological spaces, often denoted T o p {\displaystyle \mathbf {Top} } , is the category whose objects are topological spaces and
Category of topological spaces
Category_of_topological_spaces
In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous
Category of topological vector spaces
Category_of_topological_vector_spaces
Categorical treatment of topological spaces
In category theory, a discipline in mathematics, a topological category is a category that is enriched over the category of compactly generated Hausdorff
Topological category (enriched category theory)
Topological_category_(enriched_category_theory)
Generalization of category theory
(sometimes simply called topological categories) are categories enriched over some convenient category of topological spaces, e.g. the category of compactly generated
Higher_category_theory
properties to the forgetful functor from the category of topological spaces. The domain of a topological functor admits construction similar to initial topology
Topological_functor
Mathematical discipline
the investigation of topological categories and their relationships to each other." Category of topological spaces Topological functor Preuss 1988, Introduction
Categorical_topology
Tool to classify manifolds within a homotopy type in dim > 4
surgery exact sequence depending on the category of manifolds we work with: smooth (DIFF), PL, or topological manifolds and whether we take Whitehead
Surgery_exact_sequence
Concept in math
many other categories in geometry and algebra. The category of topological spaces Top has topological spaces as objects and as morphisms the continuous
Homotopy_category
Mathematical term
called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint
Disjoint_union_(topology)
Algebraic structure formed from a collection of algebraic structures
{B} \end{bmatrix}}.} A topological vector space (TVS) X , {\displaystyle X,} such as a Banach space, is said to be a topological direct sum of two vector
Direct_sum
Mathematical space with a notion of closeness
Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental
Topological_space
Category equipped with a faithful functor to the category of sets
interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself.
Concrete_category
Topological space with a distinguished point
metric maps Category of sets – Category whose objects are sets and whose morphisms are functions Category of topological spaces – Category whose objects
Pointed_space
Type of monoidal category
A modular tensor category (or modular fusion category) is a type of monoidal category that plays a role in the areas of topological quantum field theory
Modular_tensor_category
Mathematical functor
topological half-exact functor F is a functor from a fixed topological category (for example CW complexes or pointed spaces) to an abelian category (most
Topological half-exact functor
Topological_half-exact_functor
Concept in mathematics
continuous. Every topological group is a topological monoid. H-space Steenrod, N.E. (1968). "Milgram's classifying space of a topological group". Topology
Topological_monoid
Type of order at absolute zero
elementary particles; (4) topological entanglement entropy that reveals the entanglement origin of topological order, etc. Topological order is important in
Topological_order
"Small" subset of a topological space
meagre set (also called a meager set or a set of first category) is a subset of a topological space that is a countable union of subsets whose closures
Meagre_set
Generalization of a category
Given a topological space X, one can define its singular set S(X), also known as the fundamental ∞-groupoid of X. S(X) is a quasi-category in which every
Quasi-category
Vector space with a notion of nearness
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures
Topological_vector_space
Type of quantum computer
processors, the first used a toric code with twist defects as a topological degeneracy (or topological defect) while the second used a different but related protocol
Topological_quantum_computer
Field theory involving topological effects in physics
mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While
Topological quantum field theory
Topological_quantum_field_theory
Branch of mathematics
invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows
Topology
Two sets with a small overlap
is used in some other sense, or in the sense of measure theory or topological category. Here are some alternative definitions of "almost disjoint" that
Almost_disjoint_sets
Mapping which preserves all topological properties of a given space
are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space
Homeomorphism
Category used in algebraic topology
convenient to use in proofs. There is also such a category for the CGWH analog of pointed topological spaces, defined by requiring maps to preserve base
Category of compactly generated weak Hausdorff spaces
Category_of_compactly_generated_weak_Hausdorff_spaces
Simplicial set constructed from the objects and morphisms of a small category
geometric realization of this simplicial set is a topological space, called the classifying space of the category C. These closely related objects can provide
Nerve_(category_theory)
Type of topological space
a discrete subspace of some given topological space ( Y , τ ) {\displaystyle (Y,\tau )} refers to a topological subspace of ( Y , τ ) {\displaystyle
Discrete_space
Category in mathematics
Spherical fusion categories give rise to a family of three-dimensional topological state sum models (a particular formulation of a topological quantum field
Spherical_category
On topological spaces where the intersection of countably many dense open sets is dense
each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many
Baire_category_theorem
Category theory concept
schemes, smooth manifolds, or topological spaces. These categories encode objects relative to a fixed object, such as the category of schemes over S {\displaystyle
Overcategory
Concept in category theory
example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another
Fibred_category
Algebraic theory
algebraic theory of topological quantum information is a collection of algebraic techniques developed and applied to topological aspects of condensed
Algebraic theory of topological quantum information
Algebraic_theory_of_topological_quantum_information
Collection of objects and morphisms
include Set, the category of sets and set functions; Ring, the category of rings and ring homomorphisms; and Top, the category of topological spaces and continuous
Category_(mathematics)
Analysis of datasets using techniques from topology
In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information
Topological_data_analysis
Group that is a topological space with continuous group operations
In mathematics, topological groups are groups and topological spaces at the same time, where the group operations are required to be continuous. This
Topological_group
Mathematical construction used in homotopy theory
realization in the category CGHaus of compactly-generated Hausdorff topological spaces. Intuitively, the realization of X is the topological space (in fact
Simplicial_set
Concept in topology
nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details. A topological space X {\displaystyle
Baire_space
Mapping between categories
obtain a functor from the category of pointed topological spaces to the category of groups. In the category of topological spaces (without distinguished
Functor
topological topos, or more recent incarnations such as condensed sets or pyknotic sets. These attempt to embed the category of (certain) topological spaces
Generalized_space
Category whose objects are sets and whose morphisms are functions
concrete categories, such as the category of groups or the category of topological spaces. Category of topological spaces Set theory Small set (category theory)
Category_of_sets
Mathematical concept in category theory
object in Top, the category of topological spaces (with the monoidal structure induced by the product topology), is a topological monoid. A monoid object
Monoid_(category_theory)
Concept in functional analysis
between Fréchet spaces to be a topological homomorphism. A TVS embedding or a topological monomorphism is an injective topological homomorphism. Equivalently
Topological_homomorphism
of a topological category (in the musical case: time, position and pitch). Since the set of gestures of given skeleton and topological category defines
Musical_gesture
in quantum mechanics. Unitary modular tensor categories are relevant to the algebraic theory of topological quantum information since they conjecturally
Unitary modular tensor category
Unitary_modular_tensor_category
Mathematical category with weak equivalences, fibrations and cofibrations
relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by Daniel
Model_category
Higher categorical generalization of a topos
∞-topos is the ∞-category of sheaves of spaces on some topological space. But the notion is more flexible; for example, the ∞-category of étale sheaves
∞-topos
mathematics, a completely metrizable space (metrically topologically complete space) is a topological space (X, T) for which there exists at least one metric
Completely_metrizable_space
is both a monomorphism and epimorphism) is an isomorphism. The category of topological spaces is not balanced (since continuous bijections are not necessarily
Balanced_category
same proofs in the differentiable and PL categories. The proofs are much harder in the topological category, requiring the theory of Robion Kirby and
Whitehead_torsion
Multiple equivalent ways to define a topological space
arise from a topological structure or not. Such questions are greatly clarified by the topological axioms based on convergence. A topological space is a
Axiomatic foundations of topological spaces
Axiomatic_foundations_of_topological_spaces
Mathematical game on a topological space
In mathematics, a topological game is an infinite game of perfect information played between two players on a topological space. Players choose objects
Topological_game
Area of mathematics using condensed sets
replaces a topological space by a certain sheaf of sets, in order to solve some technical problems of doing homological algebra on topological groups. Essentially
Condensed_mathematics
Tool to track locally defined data attached to the open sets of a topological space
sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated
Sheaf_(mathematics)
Special dagger category that is compact
of compact topological groups from their category of finite-dimensional continuous unitary representations (that is, Tannakian categories). They also
Dagger_compact_category
General theory of mathematical structures
introduced categories for understanding and formalizing the processes (functors) that relate topological structures to algebraic structures (topological invariants)
Category_theory
Algebraic surface
maximize the systole length among compact Riemann surfaces in its topological category (that is, surfaces having the same genus) despite maximizing the
Bring's_curve
Type of topological space
In topology, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important
Topological_manifold
French mathematician (1928–2014)
Dieudonné and Laurent Schwartz. His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational
Alexander_Grothendieck
Concept in algebraic topology
A {\displaystyle X/A} . There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space X {\displaystyle
Topological_pair
(reasonable) topological space X is a Kan complex and the homotopy category of it is the fundamental groupoid of X. Let C be an ∞-category. If f , g :
Homotopy category of an ∞-category
Homotopy_category_of_an_∞-category
Branch of mathematics
from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though
Algebraic_topology
topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category
Fundamental_groupoid
Generalization of the notion of convergence that is found in general topology
from any topological space. An example of convergence that is in general non-topological is almost everywhere convergence. Many topological properties
Convergence_space
Symmetric monoidal closed category equipped with a dualizing object
*-autonomous. On the other hand, the category of topological vector spaces contains an extremely wide full subcategory, the category Ste of stereotype spaces, which
*-autonomous_category
Topics referred to by the same term
name of the Australian Federation Party Top (category theory), a category of topological spaces or topological manifolds Top (algebra), in module theory
Top
Set of arguments where two or more functions have the same value
the equaliser definition. Coincidence theory, a topological approach to equaliser sets in topological spaces. Pullback, a special limit that can be constructed
Equaliser_(mathematics)
Algebraic structure
is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic
Interior_algebra
Category whose objects are metric spaces and whose morphisms are metric maps
functions Category of topological spaces – Category whose objects are topological spaces and whose morphisms are continuous maps Category of topological spaces
Category_of_metric_spaces
Topological vector space
homeomorphism, topological embedding, quotient map) then so is every fi : Xi → X. Direct limits in the categories of topological spaces, topological vector spaces
LF-space
Constructs a fiber bundle from a base space, fiber and a set of transition functions
while keeping all other data the same. Let X and F be topological spaces and let G be a topological group with a continuous left action on F. Given an open
Fiber bundle construction theorem
Fiber_bundle_construction_theorem
Generalized topological space
Chu spaces generalize the notion of topological space by dropping the requirements that the set of open sets be closed under union and finite intersection
Chu_space
Mathematical set with some added structure
linear and topological structures underlie the linear topological space (in other words, topological vector space) structure. A linear topological space is
Space_(mathematics)
Certain generalizations of groups
A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous
Group_object
Concept in functional analysis
{\displaystyle X} . In the category of topological vector spaces, that algebraic decomposition becomes less useful. The definition of a topological vector space requires
Complemented_subspace
Structure preserving map derived canonically from another map
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
Induced_homomorphism
Inclusion of one mathematical structure in another, preserving properties of interest
Y {\displaystyle f:X\to Y} between topological spaces X {\displaystyle X} and Y {\displaystyle Y} is a topological embedding if f {\displaystyle f} yields
Embedding
History of maths
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
the category of pointed topological spaces and the category of pointed simplicial sets are not stable model categories. Any stable model category is equivalent
Stable_model_category
Mathematical structures in category theory
"category of right modules over C {\displaystyle C} ". The category of presheaves on a topological space X {\displaystyle X} is a functor category: we
Functor_category
Generalized object in category theory
product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most
Product_(category_theory)
Type of category in category theory
important example arises from topological spaces. If X is a topological space, then the open sets in X form the objects of a category O(X) for which there is
Cartesian_closed_category
Mathematical category
category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category
Topos
Type of regular Hausdorff space
particular, every topological manifold is Tychonoff. Every totally ordered set with the order topology is Tychonoff. Every topological group is completely
Tychonoff_space
Algebraic structure associated with a topological space
important in the study of topological spaces. Under nice conditions in which distinct homology theories for a single topological space produce the same homology
Homology_(mathematics)
Mathematical structure
{\displaystyle {\mathcal {C}}} act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a
Grothendieck_topology
Category whose objects are manifolds and whose morphisms are differentiable maps
U : Manp → Top to the category of topological spaces which is faithful and assigns to each manifold the underlying topological space and to each p-times
Category_of_manifolds
Branch of algebraic topology
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas
Topological_K-theory
Topological manifold with a piecewise linear structure on it
than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. PL sits between DIFF (the category of smooth
Piecewise_linear_manifold
so-called category of Flows, which admits a nontrivial model category structure. He introduced a topological version (here a topological category means a
Directed_algebraic_topology
Type of topological group
topology, making it a discrete topological group. Since every map from a discrete space is continuous, the topological homomorphisms between discrete
Discrete_group
Simplicial object in the category of topological spaces
in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces
Simplicial_space
Mathematical property of a space
and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms
Topological_property
Operation in algebra and mathematics
forgetful functor from the category of compact Hausdorff spaces to sets is monadic. However the forgetful functor from all topological spaces to sets is not
Monad_(category_theory)
Type of topological group in mathematics
targets Topological group – Group that is a topological space with continuous group operations Topological module Topological ring Topological semigroup
Locally_compact_group
Right inverse of a morphism
Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected
Section_(category_theory)
In mathematics, specifically in category theory, an (n, m)-category is an n-category all of whose j-morphisms for j > m {\displaystyle j>m} are invertible
(n,_m)-category
Concept in algebraic topology
applied to all topological spaces, and so singular homology is expressible as a functor from the category of topological spaces to the category of graded abelian
Singular_homology
Certain topology in mathematics
to accept a proper class as a topological space, then we may similarly view the class of all ordinals as a topological space with the order topology.
Order_topology
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