(∞, N)-category

	(∞, N)-category In mathematics, especially category theory, an (∞, ''n'')-category is a generalization of an ∞-category, where each ''k''-morphism is invertible for k > n. Thus, an ∞-category is an (∞, 1)-category, while an ∞-groupoid In category theory, a branch of mathematics, an ∞-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 standa ... is an (∞, 0)-category. See also * Joyal's theta category References nlab(infinity,n)-category in nLab Further reading (∞,n)-categoryin Japanese Homotopy theory Higher category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Category Theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	∞-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 (mathematics), 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 tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	∞-groupoid In category theory, a branch of mathematics, an ∞-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 ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism. The homotopy hypothesis states that ∞-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 ∞-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]
	Joyal's Theta Category In mathematics, especially category theory, Joyal's theta category \Theta is an alternative to the simplex category \Delta. It was introduced by André Joyal to give a definition of an ∞-category using \Theta-sets = presheaves on \Theta instead of simplicial sets = presheaves on \Delta. Namely, in the definition of Boardman and Vogt (which is the standard definition today), an ∞-category is defined as a simplicial set satisfying the weak Kan condition. In a similar way, Joyal proposed to define an ∞-category as a \Theta-set satisfying the weak Kan condition. In practice, the category \Theta is often used to define (∞, n)-categories. See also test category ''Pursuing Stacks'' () is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the work is a generalized homotopy theo ... Notes References * * Further reading * http://pantodon.j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Homotopy Theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Applications to other fields of mathematics Besides algebraic topology, the theory has also been used in other areas of mathematics such as: * Algebraic geometry (e.g., A1 homotopy theory, A1 homotopy theory) * Category theory (specifically the study of higher category theory, higher categories) Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid Pathological (mathematics), pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being Category of compactly generated weak Hausdorff spaces, compactly generated weak Hausdorff or a CW complex. In the same vein as above, a "Map (mathematics), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]