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(∞,1)-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. 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 mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic ''K''-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results. Motivation Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of ''R''-modules for a commutative ring ''R''. Homotopy th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. 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 spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...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. Definitions Definition of the standard n-simplex For each ''n'' ≥ 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 ℝn+1 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'' ≤ ''n'', this has a subcomplex \Lambda^n_k, the ''k''-th horn inside \Delta^n, corresponding to the boundary of the ''n''-simplex, with the ''k''-th face r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Globular Set
In category theory, a branch of mathematics, a globular set is a higher-dimensional generalization of a directed graph. Precisely, it is a sequence of sets X_0, X_1, X_2, \dots equipped with pairs of functions s_n, t_n: X_n \to X_ such that * s_n \circ s_ = s_n \circ t_, * t_n \circ s_ = t_n \circ t_. (Equivalently, it is a presheaf on the category of “globes”.) The letters "''s''", "''t''" stand for "source" and "target" and one imagines X_n consists of directed edges at level ''n''. A variant of the notion was used by Grothendieck to introduce the notion of 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 of simplicial sets (with the standard model structure). I .... Extending Grothendieck's work, gave a definition of a weak ∞-category in terms of globular sets. References Further reading *Dimitri Ara. On t ... [...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 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 invariants of spaces, such as their fundamental weak ∞-groupoid. Strict higher categories An ordinary category has objects and morphisms, which are called 1-morphisms in the context of higher category theory. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to ''n''-morphisms between (''n'' − 1)-morphisms gives an ''n''-category. Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category ''n''-Cat of (small) ''n''-categories is actually a ... [...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 of simplicial sets (with the standard 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 spaces. 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 presheaves on the globular category \mathbb. This is defined as the category whose objects are finite ordinals /math> and morphisms are given by \begin \sigma_n: \to +1\ \tau_n: \to +1\end such that the globular relations hold \begin \sigma_\circ\sigma_n &= \tau_\circ\sigma_n \\ \sigma_\circ\tau_n &= \tau_\circ\tau_n \end These encod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Infinity Category
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the American-style barn, for instance, is a large barn with a door at each end and individual stalls inside or free-standing stables with top and bottom-opening doors. The term "stable" is also used to describe a group of animals kept by one owner, regardless of housing or location. The exterior design of a stable can vary widely, based on climate, building materials, historical period and cultural styles of architecture. A wide range of building materials can be used, including masonry (bricks or stone), wood and steel. Stables also range widely in size, from a small building housing one or two animals to facilities at agricultural shows or race tracks that can house hundreds of animals. History The stable is typically historically the se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segal Space
In mathematics, a Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered as a simplicial discrete space, satisfies the Segal conditions iff it is the nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system. A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ... of a category. The condition for Segal spaces is a homotopical version of this. Complete Segal spaces were introduced by as models for (∞, 1)-categories. References * External links * *{{nlab, id=complete+Segal+space, title=Complete Segal space Category theory Simplicial sets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicially Enriched Category
In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...s. Simplicially enriched categories are often also called, more ambiguously, simplicial categories; the latter term however also applies to simplicial objects in Cat (the category of small categories). Simplicially enriched categories can, however, be identified with simplicial objects in Cat whose object part is constant, or more precisely, whose all face and degeneracy maps are bijective on objects. Simplicially enriched categories can model (∞, 1)-categories, but the dictionary has to be carefully built. Namely many notions, limits for example, are different from the limits in the sense of enriched category theory. Refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segal Category
In mathematics, a Segal category is a model of an infinity category introduced by , based on work of Graeme Segal Graeme Bryce Segal FRS (born 21 December 1941) is an Australian mathematician, and professor at the University of Oxford. Biography Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receiv ... in 1974. References * * External links *{{nlab, id=Segal+category, title=Segal category Category theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
