|
Accessible ∞-category
In mathematics, especially category theory, an accessible quasi-category is a quasi-category in which each object is an ind-object on some small quasi-category. In particular, an accessible quasi-category is typically large (not small). The notion is a generalization of an earlier 1-category version of it, an accessible category introduced by Adámek and Rosický. Definition 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 (ma ... is called accessible or more precisely \kappa-accessible if it is equivalent to the ∞-category of \kappa-ind objects on some small ∞-category for some regular cardinal \kappa. Facts A small ∞-category is accessible if and only if it is idempotent-complete. References * * Charles Rezk, Generalizing accessible ∞-categories, 202dr ... [...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] |
Quasi-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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accessible Category
The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates in the work of Grothendieck completed by 1969, and Gabriel and Ulmer (1971). It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic. A standard text book by Adámek and Rosický appeared in 1994. Accessible categories also have applications in homotopy theory.J. Rosick�"On combinatorial model categories" ''arXiv'', 16 August 2007. Retrieved on 19 January 2008.Rosický, J. "Injectivity and accessible categories." ''Cubo Matem. Educ'' 4 (2002): 201-211. Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript ''Les dérivateurs''. Some properties of accessible categories depe ... [...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] |
Idempotent-complete
In mathematics the Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category. Taking the Karoubi envelope of a preadditive category gives a pseudo-abelian category, hence for additive categories, the construction is sometimes called the pseudo-abelian completion. It is named for the French mathematician Max Karoubi. Given a category C, an idempotent of C is an endomorphism :e: A \rightarrow A with :e\circ e = e. An idempotent ''e'': ''A'' → ''A'' is said to split if there is an object ''B'' and morphisms ''f'': ''A'' → ''B'', ''g'' : ''B'' → ''A'' such that ''e'' = ''g'' ''f'' and 1''B'' = ''f'' ''g''. The Karoubi envelope of C, sometimes written Split(C), is the category whose objects are pairs of the form (''A'', ''e'') where ''A'' is an object of C and e : A \rightarrow A is an idempotent of C, and whose morphisms are the triples : (e, f, e^): (A, e) \rightarrow ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
