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β-categories
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. 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 two given morphisms are re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Construction
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. More generally, the category of elements plays a key role in the proof that every weighted colimit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Topos Theory
''Higher Topos Theory'' is a treatise on the theory of β-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of β-topoi, the book is widely considered foundational to higher category theory. Since 2018, Lurie has been transferring the contents of ''Higher Topos Theory'' (along with new material) to Kerodon, an "online resource for homotopy-coherent mathematics" inspired by the Stacks Project. Topics ''Higher Topos Theory'' covers two related topics: β-categories and β-topoi (which are a special case of the former). The first five of the book's seven chapters comprise a rigorous development of general β-category theory in the language of quasicategories, a special class of simplicial set which acts as a model for β-categories. The path of this development largely parallels classical category theory, with the notable exception of the β-categorical Grothendieck construction; this correspondence, which Lurie refers to ... [...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] |
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] |
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] |
β-groupoid
In category theory, a branch of mathematics, an β-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category (mathematics), category of simplicial sets (with the standard model category, model structure). It is an β-category generalization of a groupoid, a category in which every morphism is an isomorphism. The homotopy hypothesis states that β-groupoids are equivalent to spaces up to homotopy. Globular Groupoids Alexander Grothendieck suggested in ''Pursuing Stacks'' that there should be an extraordinarily simple model of β-groupoids using globular sets, originally called hemispherical complexes. These sets are constructed as Sheaf (mathematics)#Presheaves, presheaves on the globular category \mathbb. This is defined as the category whose objects are finite ordinals [n] and morphisms are given by \begin \sigma_n: [n] \to [n+1]\\ \tau_n: [n] \to [n+1] \end such that the globular relations hold \ ... [...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] |
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] |
Nerve Functor
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 simplicial set, 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 ... [...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] |