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Subdivision (simplicial Set)
In higher category theory in mathematics, the subdivision of simplicial sets (subdivision functor or Sd functor) is an endofunctor on the category of simplicial sets. It refines the structure of simplicial sets in a purely combinatorical way without changing constructions like the geometric realization. Furthermore, the subdivision of simplicial sets plays an important role in the extension of simplicial sets right adjoint to it. Definition For a partially ordered set I, let s(I) be the set of non-empty finite totally ordered subsets, which itself is partially ordered by inclusion. Every partially ordered set can be considered as a category. Postcomposition with the nerve N\colon \mathbf\rightarrow\mathbf defines the subdivision functor \operatorname\colon \Delta\rightarrow\mathbf on the simplex category by: : \operatorname(\Delta^n) :=N(s( ). On the full category of simplicial sets, the subdivision functor \operatorname\colon \mathbf\rightarrow\mathbf, similar to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric Subdivision
In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension to simplicial complexes is a canonical method to refining them. Therefore, the barycentric subdivision is an important tool in algebraic topology. Motivation The barycentric subdivision is an operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations: This substitution allows one to assign combinatorial invariants such as the Euler characteristic to the spaces. One can ask whether there is an analogous way to replace the continuous functions defined on the topological spaces with functions that are linear on the simplices and homotopic to the original maps (see also simplicial approximation). In general, such an assignment requires a refinement of the given complex, meaning that one replaces larger simplices with a union of smaller simplices. A standard way to c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D (both from C to D), then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subdivision
Subdivision may refer to: Arts and entertainment * Subdivision (metre), in music * ''Subdivision'' (film), 2009 * "Subdivision", an episode of ''Prison Break'' (season 2) * ''Subdivisions'' (EP), by Sinch, 2005 * "Subdivisions" (song), by Rush, 1982 Science, technology and mathematics * Subdivision (rank), a taxonomic rank * Subdivision (botany), or subphylum, a taxonomic rank * Subdivision (graph theory), adding new vertices to some edges of a graph, whereby replacing the edges by paths * Subdivision (simplicial complex) * Subdivision (simplicial set) * Subdivision surface, in computer graphics Other uses * Subdivision, an administrative division, a portion of a country * Subdivision (India), an administrative division in India * Subdivision (land) Subdivisions are land that is divided into pieces that are easier to sell or otherwise Real estate development, develop, usually via a plat. The former single piece as a whole is then known as a subdivision. Subdivisions ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subdivision (simplicial Complex)
A subdivision (also called refinement) of a simplicial complex is another simplicial complex in which, intuitively, one or more simplices of the original complex have been partitioned into smaller simplices. The most commonly used subdivision is the barycentric subdivision, but the term is more general. The subdivision is defined in slightly different ways in different contexts. In geometric simplicial complexes Let ''K'' be a geometric simplicial complex (GSC). A subdivision of ''K'' is a GSC ''L'' such that: * , ''K'', = , ''L'', , that is, the union of simplices in ''K'' equals the union of simplices in ''L'' (they cover the same region in space). * each simplex of ''L'' is contained in some simplex of ''K''. As an example, let ''K'' be a GSC containing a single triangle (with all its faces and vertices). Let ''D'' be a point on the face AB. Let ''L'' be the complex containing the two triangles and (with all their faces and vertices). Then ''L'' is a subdivision of ''K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quillen Adjunction
In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen. Formal definition Given two closed model categories C and D, a Quillen adjunction is a pair :(''F'', ''G''): C \leftrightarrows D of adjoint functors with ''F'' left adjoint to ''G'' such that ''F'' preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that ''G'' preserves fibrations and trivial fibrations. In such an adjunction ''F'' is called the left Quillen functor and ''G'' is called the right Quillen functor. Properties It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kan–Quillen Model Structure
In higher category theory, the Kan–Quillen model structure is a special model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called ''fibrations'', ''cofibrations'' and ''weak equivalences'', which fulfill the properties of a model structure. Its fibrant objects are all Kan complexes and it furthermore models the homotopy theory of CW complexes up to weak homotopy equivalence, with the correspondence between simplicial sets, Kan complexes and CW complexes being given by the geometric realization and the singular functor ( Milnor's theorem). The Kan–Quillen model structure is named after Daniel Kan and Daniel Quillen. Definition The Kan–Quillen model structure is given by: * Fibrations are Kan fibrations.Joyal 2008, Theorem 6.1. on p. 293 * Cofibrations are monomorphisms.Cisinski 2019, Theorem 3.1.8. * Weak equivalences are ''weak homotopy equivalences'', hence morphisms between simplicial sets, whose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anodyne Extension
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 inclusions \Lambda^n_i \subset \Delta^n, 0 \le i < n. A right fibration is defined similarly with the condition . A is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is exactly a map that is both a left and right fibration. Examples A right fibration is a cartesian fibration such that each fiber is a . In particular, a[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2. Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Equivalence (homotopy Theory)
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. A model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated homotopy category of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into isomorphisms. It is a useful observation that the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations. Topological spaces Model categories were defined by Quillen as an axiomatization of homotopy theory that applies to topological spaces, but also to many other categories in algebra and geometry. The example that started the subject is the category of topological spaces with Serre fibrations as fibrations and weak homotopy equival ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ex∞ Functor
In higher category theory in mathematics, the extension of simplicial sets (extension functor or Ex functor) is an endofunctor on the category of simplicial sets. Due to many remarkable properties, the extension functor has plenty and strong applications in homotopical algebra. Among the most well-known is its application in the construction of Kan complexes from arbitrary simplicial sets, which often enables without loss of generality to take the former for proofs about the latter. It is furthermore very well compatible with the Kan–Quillen model structure and can for example be used to explicitly state its factorizations or to search for weak homotopy equivalences. Definition Using the subdivision of simplicial sets, the extension of simplicial sets is defined as:Guillou, Definition 6 : \operatorname\colon \mathbf\rightarrow\mathbf, \operatorname(Y)_n :=\operatorname(\operatorname(\Delta^n),Y). Due to the Yoneda lemma, one also has \operatorname(Y)_n \cong\operatorname( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize. Definition Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
