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Quasi-categories
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 re ... [...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] |
Simplicial Set
In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization. This realization consists of geometric simplices, glued together according to the rules of the simplicial set. Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space for the purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is equivalent to the familiar homotopy category of topological spaces. Formally, a simplicial set may be defined as a contravariant functor from the simplex category to the category of sets. Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Simplic ... [...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] |
Homotopy Hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-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 ∞-groupoid. Formulations A precise formulation of the hypothesis very strongly depends on the definition of an ∞-groupoid. One definition is that, mimicking the ordinary category case, an ∞-groupoid is an ∞-category in which each morphism is invertible or equivalently its homotopy category of an ∞-category, homotopy cat ... [...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] |
∞-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] |
∞-category Of Kan Complexes
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. For various kinds of fibrations for simplicial sets, see Fibration of simplicial sets. 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 \mathbb^ 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 ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-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] |
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 ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-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: ∞-categories and ∞-topoi (which are a special case of the former). The first five of the book's seven chapters comprise a rigorous development of general ∞-category theory in the language of quasicategories, a special class of simplicial set which acts as a model for ∞-categories. The path of this development largely parallels classical category theory, with the notable exception of the ∞-categorical Grothendieck construction; this correspondence, which Lurie refers to ... [...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] |
