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Twisted Diagonal (simplicial Sets)
In higher category theory in mathematics, the twisted diagonal of a simplicial set (for ∞-categories also called the twisted arrow ∞-category) is a construction, which generalizes the twisted diagonal of a category to which it corresponds under the nerve construction. Since the twisted diagonal of a category is the category of elements of the Hom functor, the twisted diagonal of an ∞-category can be used to define the Hom functor of an ∞-category. Twisted diagonal with the join operation For a simplicial set A define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:Cisinski 2019, 5.6.1. : \mathbf(A)_ =\operatorname((\Delta^m)^\mathrm*\Delta^n,A), : \operatorname(A) =\delta^*(\mathbf(A)). The canonical morphisms (\Delta^m)^\mathrm\rightarrow(\Delta^m)^\mathrm*\Delta^n\leftarrow\Delta^n induce canonical morphisms \mathbf(A)\rightarrow A^\mathrm\boxtimes A and \operatorname(A)\rightarrow A^\mathrm\times A. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Category Theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic Invariant (mathematics), invariants of topological space, spaces, such as the Fundamental groupoid, fundamental . In higher category theory, the concept of higher categorical structures, such as (), allows for a more robust treatment of homotopy theory, enabling one to capture finer homotopical distinctions, such as differentiating two topological spaces that have the same fundamental group but differ in their higher homotopy groups. This approach is particularly valuable when dealing with spaces with intricate topological features, such as the Eilenberg-MacLane space. Strict higher categories An ordinary category (m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisimplicial Set
In higher category theory in mathematics, a bisimplicial set is a simplicial object in the category of simplicial sets, which themselves are simplicial objects in the category of sets. Many concepts from homotopical algebra, which studies simplicial sets, can be transported over to the study of bisimplicial sets, which for example includes Kan fibrations and Kan complexes. Definition Bisimplicial sets are simplicial objects in the category of simplicial sets \mathbf, hence functors \Delta^\mathrm\rightarrow\mathbf with the simplex category \Delta. The category of bisimplicial sets is denoted: : \mathbf :=\mathbf(\Delta^\mathrm,\mathbf) \cong\mathbf((\Delta\times\Delta)^\mathrm,\mathbf) Let \operatorname_1,\operatorname_2\colon \Delta\times\Delta\rightarrow\Delta be the canonical projections, then there are induced functors \operatorname_1^*,\operatorname_2^*\colon \mathbf\rightarrow\mathbf by precomposition. For simplicial sets X and Y, there is a bisimplicial set X\boxtime ... [...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] |
Kan Fibration
In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan. For various kinds of fibrations for simplicial sets, see Fibration of simplicial sets. Definitions Definition of the standard n-simplex For each ''n'' ≥ 0, recall that the standard n-simplex, \Delta^n, is the representable simplicial set :\Delta^n(i) = \mathrm_ ( Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard n-simplex: the convex subspace of \mathbb^ consisting of all points (t_0,\dots,t_n) such that the coordinates are non-negative and sum to 1. Definition of a horn For each ''k'' ≤ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left Fibration
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] |
Diamond Operation
In higher category theory in mathematics, the diamond operation of simplicial sets is an operation taking two simplicial sets to construct another simplicial set. It is closely related to the join of simplicial sets and used in an alternative construction of the twisted diagonal. Definition For simplicial set X and Y, their ''diamond'' X\diamond Y is the pushout of the diagram: : X\times Y\times\Delta^1\leftarrow X\times Y\times\partial\Delta^1\rightarrow X+Y. One has a canonical map X\diamond Y\rightarrow\Delta^0\diamond\Delta^0 \cong\Delta^1 for which the fiber of 0 is X and the fiber of 1 is Y. Right adjoints Let Y be a simplicial set. The functor Y\diamond -\colon \mathbf\rightarrow Y\backslash\mathbf, X\mapsto(Y\mapsto X\diamond Y) has a right adjoint Y\backslash\mathbf\rightarrow\mathbf, (t\colon Y\rightarrow W)\mapsto t\backslash\backslash W (alternatively denoted Y\backslash\backslash W) and the functor -\diamond Y\colon \mathbf\rightarrow Y\backslash\mathbf, X\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Join (simplicial Sets)
In higher category theory in mathematics, the 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 construct another simplicial set. It is closely related to the diamond operation and used in the construction of the twisted diagonal. Under the nerve construction, it corresponds to the join of categories and under the geometric realization, it corresponds to the join of topological spaces. Definition For natural numbers m,p,q\in\mathbb, one has the identity:Cisinski 2019, 3.4.12. : \operatorname( +q+1 =\prod_\operatorname( \times\operatorname( , which can be extended by colimits to a functor a functor -*-\colon \mathbf\times\mathbf\rightarrow \mathbf, which together with the empty simplicial set as unit element makes the category of simplicial sets \mathbf into a monoidal category. For simplicial set X and Y, their ''join'' X*Y is the simplicial set: : (X*Y)_n =\prod_X_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opposite Simplicial Set
In higher category theory in mathematics, the opposite simplicial set (or dual simplicial set) is an operation extending the opposite category (or dual category). It generalizes the concept of inverting arrows from 1-categories to ∞-categories. Similar to the opposite category defining an involution on the category of small categories, the opposite simplicial sets defines an involution on the category of simplicial sets. Both correspond to each other under the nerve construction. Definition On the simplex category \Delta, there is an automorphism \rho\colon \Delta\rightarrow\Delta, which for a map f\colon rightarrow /math> is given by \rho(f)(i):=n-f(m-i). It fulfills \rho^2=\operatorname and is the only automorphism on the simplex category \Delta. By precomposition, it defines a functor \rho^*\colon \mathbf\rightarrow\mathbf on the category of simplicial sets \mathbf =\mathbf(\Delta,\mathbf). For a simplicial set X, the simplicial set X^\mathrm =\rho^*(X)is its ''opposite simp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hom Functor Of An ∞-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. The study of such generalizations is known as higher category theory. Overview Quasi-categories were introduced by . André Joyal has much advanced the study of quasi-categories showing that most of the usual basic category theory and some of the advanced notions and theorems have their analogues for quasi-categories. An elaborate treatise of the theory of quasi-categories has been expounded by . Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined. All the morphisms that can serve as composition of two given morphisms are re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hom Functor
In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Formal definition Let ''C'' be a locally small category (i.e. a category (mathematics), category for which hom-classes are actually Set (mathematics), sets and not proper classes). For all objects ''A'' and ''B'' in ''C'' we define two functors to the category of sets as follows: : The functor Hom(–, ''B'') is also called the ''functor of points'' of the object ''B''. Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms. The pair of functors Hom(''A'', –) and Hom(–, ''B'') are related in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Elements
In category theory, a branch of mathematics, the category of elements of a presheaf is a category associated to that presheaf whose objects are the elements of sets in the presheaf. It and its generalization are also known as the Grothendieck construction (named after Alexander Grothendieck) especially in the theory of descent, in the theory of stacks, and in fibred category theory. The Grothendieck construction is an instance of straightening (or rather unstraightening). Significance In categorical logic, the construction is used to model the relationship between a type theory and a logic over that type theory, and allows for the translation of concepts from indexed category theory into fibred category theory, such as Lawvere's concept of hyperdoctrine. The category of elements of a simplicial set is fundamental in simplicial homotopy theory, a branch of algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study top ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
