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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] |
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] |
Simplex Category
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definition The simplex category is usually denoted by \Delta. There are several equivalent descriptions of this category. \Delta can be described as the category of ''non-empty finite ordinals'' as objects, thought of as totally ordered sets, and ''(non-strictly) order-preserving functions'' as morphisms. The objects are commonly denoted = \ (so that is the ordinal n+1 ). The category is generated by coface and codegeneracy maps, which amount to inserting or deleting elements of the orderings. (See simplicial set for relations of these maps.) A simplicial object is a presheaf on \Delta, that is a contravariant functor from \Delta to another category. For instance, simplicial sets are contravariant with the codomain category being the catego ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
André Joyal
André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to join the ''Special Year on Univalent Foundations of Mathematics''. Research He discovered Kripke–Joyal semantics, the theory of combinatorial species and with Myles Tierney a generalization of the Galois theory of Alexander Grothendieck in the setup of locales. Most of his research is in some way related to category theory, higher category theory and their applications. He did some work on quasi-categories, after their invention by Michael Boardman and Rainer Vogt, in particular conjecturing and proving the existence of a Quillen model structure on the category of simplicial sets whose weak equivalences generalize both equivalence of categories and Kan equivalence of spaces, which is now known as Joyal model structure. He co-a ... [...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] |
(∞, 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] |
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 theory using higher category theory. The word "stacks" in the title refers to what are nowadays usually called " ∞-groupoids", one possible definition of which Grothendieck sketches in his manuscript. (The stacks of algebraic geometry, which also go back to Grothendieck, are not the focus of this manuscript.) Among the concepts introduced in the work are derivators and test categories. Some parts of the manuscript were later developed in: * * Overview of manuscript I. The letter to Daniel Quillen Pursuing stacks started out as a letter from Grothendieck to Daniel Quillen. In this letter he discusses Quillen's progress on the foundations for homotopy theory and remarked on the lack of progress since then. He remarks how some of his fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
