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Simplicial object in the category of topological spaces
In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the
Simplicial_space
Type of mathematical set
In mathematics, a simplicial complex is a structured set of simplices (for example, points, line segments, triangles, and their n-dimensional counterparts)
Simplicial_complex
Mathematical construction used in homotopy theory
higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization
Simplicial_set
n-dimensional ball. A simplicial manifold is also a simplicial object in the category of manifolds. This is a special case of a simplicial space in which, for
Simplicial_manifold
Concept in algebraic topology
connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are n-simplices, the n-dimensional
Simplicial_homology
Representation of mathematical space
the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits such a homeomorphism
Triangulation_(topology)
combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as
Simplicial_sphere
simplex always span a simplex. Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated; this
Simplicial_map
Simplicial set constructed from the objects and morphisms of a small category
C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called
Nerve_(category_theory)
homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant
Simplicial_presheaf
Mathematical object
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking
Abstract_simplicial_complex
Method for computing topological features of a space at different spatial resolutions
be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested
Persistent_homology
Concepts in algebraic topology
can construct the homotopy colimit using a simplicial replacement of the diagram. This is a simplicial space, srep ( D ) ∙ {\displaystyle {\text{srep}}(D)_{\bullet
Homotopy_colimit_and_limit
Construction for categories
join of simplicial sets is an operation making the category of simplicial sets into a monoidal category. In particular, it takes two simplicial sets to
Join_(simplicial_sets)
Mathematical concept in topology
theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object
Simplicial_group
Abstraction useful in the construction and triangulation of topological spaces
Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the
Delta_set
In topology, a branch of mathematics, a collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses
Collapse_(topology)
Branch of mathematics
combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead
Algebraic_topology
In algebraic topology, a simplicial homotopy is an analog of a homotopy between topological spaces for simplicial sets. Precisely,pg 23 if f , g : X →
Simplicial_homotopy
Derived descent
equivalent: in an appropriate setting, given a map a from a simplicial space X to a space S, a ∗ : D + ( S ) → D + ( X ) {\displaystyle a^{*}:D^{+}(S)\to
Cohomological_descent
Branch of mathematics
defined as particular kinds of simplicial sets. Since simplicial sets are sort of abstract spaces (if not topological spaces), it is possible to develop
Homotopy_theory
Research field in deep learning
meshes, time series, scalar fields graphs, or general topological spaces like simplicial complexes and CW complexes. TDL addresses this by incorporating
Topological_deep_learning
Mathematical space with a notion of closeness
Similarly, every simplex and every simplicial complex inherits a natural topology from the ambient Euclidean space R N {\displaystyle \mathbb {R} ^{N}}
Topological_space
Algebraic structure associated with a topological space
3-simplex.) Simplicial homology can in turn be generalized to singular homology, which allows more general maps of simplices into the topological space. Replacing
Homology_(mathematics)
In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn
Fibration_of_simplicial_sets
Continuous mappings can be approximated by ones that are piecewise simple
mappings between spaces that are built up from simplices—that is, finite simplicial complexes. The general continuous mapping between such spaces can be represented
Simplicial approximation theorem
Simplicial_approximation_theorem
{\displaystyle \mathbb {R} } -trees) are a class of metric spaces generalising simplicial trees. They arise naturally in many mathematical contexts, in
Real_tree
Topics referred to by the same term
topological spaces Join (category theory), an operation combining two categories Join (simplicial sets), an operation combining two simplicial sets Join
Join
History of maths
1894 Henri Poincaré Fundamental group of a topological space. 1895 Henri Poincaré Simplicial homology. 1895 Henri Poincaré Fundamental work Analysis
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Concept in mathematics
geodesic rays. When X = T {\displaystyle X=T} is a simplicial regular tree the boundary is just the space of ends, which is a Cantor set. Fixing a point x
Hyperbolic_metric_space
example is Segal's so-called Γ-space, which may be thought of as a generalization of simplicial abelian group (or simplicial abelian monoid). More precisely
Gamma-object
Mathematical category with weak equivalences, fibrations and cofibrations
spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of R-modules
Model_category
Map between simplicial sets with lifting property
}([i],[n])} Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n {\displaystyle n}
Kan_fibration
Concept in algebraic topology
particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to the subspace Xn that
N-skeleton
Complex in algebraic topology
analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information
Čech_complex
Topological manifold whose homology coincides with that of a sphere
to simplicial complexes. In particular, the example originally given by Galewski and Stern is not triangulable. Eilenberg–MacLane space Moore space (algebraic
Homology_sphere
Generalization of the notion of convergence that is found in general topology
In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a convergence that satisfies certain
Convergence_space
Multi-dimensional generalization of triangle
optimization method with inequality constraints Simplicial complex Simplicial homology Simplicial set Space frame Spectrahedron Ternary plot Elte, E.L. (2006)
Simplex
n-simplex. The facets of an omnitruncated simplicial honeycomb are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations
Omnitruncated simplicial honeycomb
Omnitruncated_simplicial_honeycomb
Topological space with only one nontrivial homotopy group
realization of simplicial abelian groups. This gives an explicit presentation of simplicial abelian groups which represent Eilenberg–MacLane spaces. Another
Eilenberg–MacLane_space
Endofunctor on the category of simplicial sets
mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable
Extension_(simplicial_set)
Topological data
simplex tree is a type of trie used to efficiently represent any general simplicial complex. Through its nodes, this data structure notably represents all
Simplex_tree
Roughly, the number of k-dimensional holes on a topological surface
distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact
Betti_number
Mathematical group of the homotopy classes of loops in a topological space
universal covering space of a finite connected simplicial complex X {\displaystyle X} can also be described directly as a simplicial complex using edge-paths
Fundamental_group
Method for dividing a simplicial complex
is an operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations:
Barycentric_subdivision
Computational problem in algebraic topology
family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each
Simplicial complex recognition problem
Simplicial_complex_recognition_problem
A subdivision (also called refinement) of a simplicial complex is another simplicial complex in which, intuitively, one or more simplices of the original
Subdivision (simplicial complex)
Subdivision_(simplicial_complex)
Mathematics glossary
definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations
Glossary of algebraic topology
Glossary_of_algebraic_topology
Application of homotopy to algebraic varieties
the category of simplicial objects on S h v ( S m S ) N i s {\displaystyle Shv(Sm_{S})_{Nis}} . Such an object is also called a simplicial sheaf on S m S
A¹_homotopy_theory
All points in the topological closure not belonging to the interior
idempotence. In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary
Boundary_(topology)
category D. Set, the category of (small) sets. sSet, the category of simplicial sets. "weak" instead of "strict" is given the default status; e.g., "n-category"
Glossary_of_category_theory
Quotient of a weakly contractible space by a free action
example, the bar construction), that gave concrete descriptions of BG as a simplicial complex for an arbitrary discrete group. Such constructions make evident
Classifying_space
Topological space
attached to the tunnels between the rooms, which make this simplicial complex contractible. Dogbone space Dunce hat (topology) List of topologies Bing's house
House_with_two_rooms
Mathematical concept
it has been shown that for any finite abstract simplicial complex K, there is a finite topological space XK and a weak homotopy equivalence f : |K| → XK
Finite_topological_space
Generalization of category theory
k) categories for any k. Simplicially enriched categories, or simplicial categories, are categories enriched over simplicial sets. However, when we look
Higher_category_theory
Branch of geometry that studies combinatorial properties and constructive methods
graphs Voronoi diagrams and Delaunay triangulations A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points
Discrete_geometry
The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local
Link_(simplicial_complex)
Equivalence between the categories of chain complexes and simplicial abelian groups
n and zero in all other degrees, the corresponding simplicial group is the Eilenberg–MacLane space K ( A , n ) {\displaystyle K(A,n)} . The Dold–Kan correspondence
Dold–Kan_correspondence
Separable space Lindelöf space Sigma-compact space Connected space Simply connected space Path connected space T0 space T1 space Hausdorff space Completely
List of general topology topics
List_of_general_topology_topics
Branch of mathematics
differential graded algebras (over Q {\displaystyle \mathbb {Q} } ), simplicial commutative rings or E ∞ {\displaystyle E_{\infty }} -ring spectra from
Derived_algebraic_geometry
Type of topological space
dimensions in specific ways. The notion generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was
CW_complex
Computer animation technique
"Practical Experiences with Pose Space Deformation" (PDF), ACM SIGGRAPH Cohen Bengio, Julien; Goldenthal, Rony (2013). "Simplicial interpolation for animating
Pose_space_deformation
Category enriched over the category of simplicial sets
In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. Simplicially enriched categories are often
Simplicially enriched category
Simplicially_enriched_category
Technique in topological data analysis
bifiltration is a collection of filtered simplicial complexes, typically built upon a set of data points in a metric space, that captures shape and density information
Subdivision_bifiltration
Non-orientable surface with one edge
come from an abstract simplicial complex, because all three triangles share the same three vertices, while abstract simplicial complexes require each
Möbius_strip
Branch of mathematics
topological data analysis is to: Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. Analyse these topological
Topology
Subject area in mathematics
to a simplicial complex or cell complex in such a way that each additional simplex or cell deformation retracts into a subdivision of the old space. Part
Algebraic_K-theory
filtration is used to create a discrete, simplicial model on point cloud data embedded in an ambient metric space. The Vietoris–Rips filtration is a multiscale
Vietoris–Rips_filtration
Mathematical category
pro-simplicial set is pro-finite. Mathematics portal History of topos theory Homotopy hypothesis ∞-topos Quasitopos Geometric logic Generalized space Illusie
Topos
Generalized manifold
spaces of non-positive curvature is due to Gromov. In this context triangles of groups correspond to non-positively curved 2-dimensional simplicial complexes
Orbifold
Branch of topology
topological spaces in general. Every manifold has a natural topology, since it is locally Euclidean. Similarly, every simplex and every simplicial complex
General_topology
considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is
Graph_homology
Steinitz is related to the notion of an abstract simplicial complex and it differs from a simplicial complex by the property that its elements are not
Abstract_cell_complex
Topological invariant in mathematics
(When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic
Euler_characteristic
Tool to track locally defined data attached to the open sets of a topological space
1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering. 1938 Hassler Whitney gives a 'modern' definition
Sheaf_(mathematics)
This means that one can consider symmetric products of objects like simplicial sets as well. Moreover, if the category is cartesian closed, the distributive
Symmetric_product_(topology)
Probability distribution
normalizes generalized gamma variates, one obtains variates from the simplicial generalized beta distribution (SGB). On the other hand, SGB variates can
Dirichlet_distribution
Abstract simplicial complex describing a graph's cliques
graph. The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed under the operation of
Clique_complex
Gives a homomorphism from homotopy groups to homology groups
^{n}} -group of an n-cube of spaces. The Hurewicz theorem for topological spaces can also be stated for n-connected simplicial sets satisfying the Kan condition
Hurewicz_theorem
Math concept
set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas
Polyhedral_complex
Algebraic topology uses abstract algebra to study topological spaces
Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set Simplicial
List of algebraic topology topics
List_of_algebraic_topology_topics
Category of non-empty finite ordinals and order-preserving maps
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving
Simplex_category
All even-degree subgraphs of a graph
even number of spanning forests. An undirected graph may be viewed as a simplicial complex with its vertices as zero-dimensional simplices and the edges
Cycle_space
contain a given point. The simplicial depth of a point p {\displaystyle p} in d {\displaystyle d} -dimensional Euclidean space, with respect to a set of
Simplicial_depth
Topological space formed from distances
a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance
Vietoris–Rips_complex
Mathematical ring
ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within
Stanley–Reisner_ring
Abstract homotopical model for topological spaces
homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model
∞-groupoid
Convex polyhedron projected from hypercube
zonohedron corresponds in this way to a simplicial arrangement, one in which each face is a triangle. Simplicial arrangements of great circles correspond
Zonohedron
Any collection of sets, or subsets of a set
abstract simplicial complex with an additional property called the augmentation property. Every filter is a family of sets. A convexity space is a set
Family_of_sets
Discrete (i.e., incremental) version of infinitesimal calculus
wedge product is defined on every cube seen as a vector space of the same dimension. For simplicial complexes, the wedge product is implemented as the cup
Discrete_calculus
Higher categorical generalization of a topos
Abstract homotopical model for topological spaces Simplicial set Kan complex – Concept in the theory of simplicial sets. Lurie 2009, Definition 6.1.0.4. Lurie
∞-topos
Operation in topology
A} and B {\displaystyle B} are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows: The vertex set
Join_(topology)
Analogs of homology groups for algebraic varieties
topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular
Chow_group
Theorems generalizing the Brouwer fixed-point theorem
over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray
Fixed-point theorems in infinite-dimensional spaces
Fixed-point_theorems_in_infinite-dimensional_spaces
They are used analogously to simplicial complexes and CW complexes in the computation of the homology of topological spaces. Non-positively curved and CAT(0)
Cubical_complex
the other hand, the category of pointed topological spaces and the category of pointed simplicial sets are not stable model categories. Any stable model
Stable_model_category
Construction for n-dimensional noise functions
An implementation typically involves four steps: coordinate skewing, simplicial subdivision, gradient selection, and kernel summation. An input coordinate
Simplex_noise
commutative ring spectra. A theorem of Jardine says that such presheaves form a simplicial model category, where F →G is a weak equivalence if the induced map of
Sheaf_of_spectra
Branch of topology
homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open
Low-dimensional_topology
Generalization of a category
Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between
Quasi-category
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