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β-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, β-category, Boardman complex, quategory) is a generalization of the notion of a Category (mathematics), category. The study of such generalizations is known as higher category theory. Overview Quasi-categories were introduced by . AndrΓ© 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 tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limits And Colimits In An β-category
In mathematics, especially category theory, limits and colimits in an β-category generalize limit (category theory), limits and colimit (category theory), colimits in a category. Like the counterparts in ordinary category theory, they play fundamental roles in constructions (e.g., Kan extensions) as well as characterizations (e.g., sheaf conditions) in higher category theory. Definition Let I be a simplicial set and C an β-category (a weak Kan complex). Fix a Grothendieck universe. Then, roughly, a limit of a functor f : I \to C amounts to the following isomorphism: :\operatorname(a_, f) \overset\to \operatorname(a, \varprojlim f) functorially in a, where a_ : I \to C denotes the constant functor with value a. A typical case is when I = \Delta is the simplex category or rather its opposite; in the latter case, the functor f is commonly called a simplicial diagram. Facts The ordinary category of sets has small limits and colimits. Similarly, *The β-category of β-categori ... [...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] |
Homotopy Hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the β-groupoids are space (mathematics), spaces. One version of the hypothesis was claimed to be proved in the 1991 paper by Mikhail Kapranov, Kapranov and Vladimir Voevodsky, Voevodsky. Their proof turned out to be flawed and their result in the form interpreted by Carlos Simpson is now known as the Simpson conjecture. In higher category theory, one considers a space-valued presheaf instead of a presheaf (category theory), set-valued presheaf in ordinary category theory. In view of homotopy hypothesis, a space here can be taken to an β-groupoid. Formulations A precise formulation of the hypothesis very strongly depends on the definition of an β-groupoid. One definition is that, mimicking the ordinary category case, an β-groupoid is an β-category in which each morphism is invertible or equivalently its homotopy category of an β-category, homotopy cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Fibration
In mathematics, especially homotopy theory, a cartesian fibration is, roughly, a map so that every lift exists that is a final object among all lifts. For example, the forgetful functor :\textrm \to \textrm from the category of pairs (X, F) of schemes and quasi-coherent sheaves on them is a cartesian fibration (see ). In fact, the Grothendieck construction says all cartesian fibrations are of this type; i.e., they simply forget extra data. See also: fibred category, prestack. The dual of a cartesian fibration is called an op-fibration; in particular, not a cocartesian fibration. A right fibration between simplicial sets is an example of a cartesian fibration. Definition Given a functor \pi : C \to S, a morphism f : x \to y in C is called \pi-cartesian or simply cartesian if the natural map :(f_*, \pi) : \operatorname(z, x) \to \operatorname(z, y) \times_ \operatorname(\pi(z), \pi(x)) is bijective. Explicitly, thus, f : x \to y is cartesian if given *g: z \to y and *u : \pi(z) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
β-Yoneda Embedding
In mathematics, especially category theory, the 2-Yoneda lemma is a generalization of the Yoneda lemma to 2-categories. Precisely, given a contravariant pseudofunctor F on a category ''C'', it says: for each object x in ''C'', the natural functor (evaluation at the identity) :\underline(h_x, F) \to F(x) is an equivalence of categories, where \underline(-, -) denotes (roughly) the category of natural transformations between pseudofunctors on ''C'' and h_x = \operatorname(-, x). Under the Grothendieck construction, h_x corresponds to the comma category C \downarrow x. So, the lemma is also frequently stated as: :F(x) \simeq \underline(C \downarrow x, F), where F is identified with the fibered category associated to F. As an application of this lemma, the coherence theorem for bicategories holds. Sketch of proof First we define the functor in the opposite direction :\mu : F(x) \to \underline(h_x, F) as follows. Given an object \overline in F(x), define the natural transformat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kan Complex
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'' β₯ 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'' β€ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core Of An β-category
In mathematics, especially category theory, the core of a category ''C'' is the category whose objects are the objects of ''C'' and whose morphisms are the invertible morphisms in ''C''.Pierre Gabriel, Michel Zisman, Β§ 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967/ref> In other words, it is the largest groupoid subcategory. As a functor C \mapsto \operatorname(C), the core is a right adjoint to the inclusion of the category of (small) groupoids into the category of (small) categories. On the other hand, the left adjoint to the above inclusion is the fundamental groupoid functor. For β-categories, \operatorname is defined as a right adjoint to the inclusion β-Grpd \hookrightarrow β-Cat. The core of an β-category C is then the largest β-groupoid contained in C. The core of ''C'' is also often written as C^. The left adjoint to the above inclusion is given by a localization of an β-category. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kan (mathematics)
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'' β₯ 0, recall that the Simplicial set#The standard n-simplex and the simplex category, standard n-simplex, \Delta^n, is the representable simplicial set :\Delta^n(i) = \mathrm_ ([i], [n]) Applying the Simplicial set#Geometric realization, geometric realization functor to this simplicial set gives a space homeomorphic to the Simplex#The standard simplex, topological standard n-simplex: the convex subspace of \mathbb^ consisting of all points (t_0,\dots,t_n) such that the coordinates are non-negativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Core Of A Category
In mathematics, especially category theory, the core of a category ''C'' is the category whose objects are the objects of ''C'' and whose morphisms are the invertible morphisms in ''C''.Pierre Gabriel, Michel Zisman, Β§ 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967/ref> In other words, it is the largest groupoid subcategory. As a functor C \mapsto \operatorname(C), the core is a right adjoint to the inclusion of the category of (small) groupoids into the category of (small) categories. On the other hand, the left adjoint to the above inclusion is the fundamental groupoid functor. For β-categories, \operatorname is defined as a right adjoint to the inclusion β-Grpd \hookrightarrow β-Cat. The core of an β-category C is then the largest β-groupoid contained in C. The core of ''C'' is also often written as C^. The left adjoint to the above inclusion is given by a localization of an β-category. ... [...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'' β₯ 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'' β€ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nerve Of A Category
In category theory, a discipline within mathematics, the nerve ''N''(''C'') 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 the classifying space of the category ''C''. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory. Motivation The nerve of a category is often used to construct topological versions of moduli spaces. If ''X'' is an object of ''C'', its moduli space should somehow encode all objects isomorphic to ''X'' and keep track of the various isomorphisms between all of these objects in that category. This can become rather complicated, especially if the objects have many non-identity automorphisms. The nerve provides a combinatorial way of organizing this data. Since simplicial sets have a good homotopy theory, one can ask questions about the mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
