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Presentable ∞-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 r ... [...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] |
Simplicially-enriched Category
In mathematics, a simplicially enriched category, is a category enriched over the category of simplicial sets. 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, all face and degeneracy maps are bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ... 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. The hom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joyal Fibration
Joyal is a surname. Notable people with the surname include: *André Joyal (born 1943), Canadian mathematician * André Joyal (economist), Canadian economist * Dave Joyal (born 1983), American rock drummer * Eddie Joyal (born 1940), Canadian ice hockey player * Glenn Joyal, Manitoba judge *Serge Joyal Serge Joyal (born February 1, 1945) is a Canadian politician who served in the House of Commons of Canada from 1974 to 1984 and subsequently in the Senate of Canada from 1997 to 2020. Career A lawyer by profession, Joyal served as vice-presid ... (born 1945), Canadian senator See also * Fusing {{surname, Joyal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Final Object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): is terminal if for every object in there exists exactly one morphism . Initial objects are also called coterminal or universal, and terminal objects are also called final. If an object is both initial and terminal, it is called a zero object or null object. A pointed category is one with a zero object. A strict initial object is one for which every morphism into is an isomorphism. Examples * The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in this category. * In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Straightening (category Theory)
"Straightenin" is a song by American hip hop trio Migos. It was released through Quality Control and Motown on May 14, 2021, as the second single from their fourth studio album, ''Culture III''. Straightenin was the Migos last single to be released before the death of their member Takeoff the next year in 2022. Background In March 2021, Quavo previewed the song's music video on social media. The track and the accompanying video were released on May 14, 2021 with an announcement of the album ''Culture III''. Hong Kong rapper and producer Big Spoon accused Migos of copying his 2020 song "Magic Show é”術表演", which he compared to "Straightenin" in an Instagram video, saying, "I composed and produced the song by myself, did all the rapping, and played all the instruments so there are no samples or loops taken from elsewhere. I tweaked their song to the same tempo and key for comparison." Composition The song features "triplet flows and ad libs" quickly rapped over "booming bass ... [...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] |
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] |
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] |
∞-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] |
∞-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] |
∞-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] |
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] |
