Category A may refer to any of the following Category A Listed building Scotland Category A Prison UK Category A Bioterrorism agent Category A services Canada The most serious category of disease recognized by the Centers for Disease Control and Prevention USA disambig ... more details
common categorization of hurricane intensities Lexical category Categories of New Testament manuscripts See also intitle Category Categorical disambiguation Categorization Category 1 disambiguation Category 2 disambiguation Category 3 disambiguation Category 4 disambiguation Category 5 disambiguation Category 6 disambiguation Category 7 The End of the World Category A disambiguation Category B disambiguation Category C disambiguation disambig Category Greek loanwords ceb Kategoriya cs Kategorie ... ujednoznacznienie pt Categoria ro Categorie dezambiguizare ru simple Category ... more details
In mathematics, simplicial category may refer to Simplex category , the category of finite ordinals and order preserving functions Simplicially enriched category , a category enriched over the category of simplicial sets. A simplicial object in the category of categories. References http ncatlab.org nlab show simplicial category simplicial category in ncatlab mathdab ... more details
Category 5 may refer to Category 5 album Category 5 album , an album from rock band, FireHouse Category 5 cable , used for carrying data Category 5 computer virus , as classified by Symantec Corporation Category 5 Records , a record label Category 5 Tropical Cyclone, on any of the Tropical cyclone scales Any of several hurricanes listed at List of Category 5 Atlantic hurricanes or List of Category 5 Pacific hurricanes Category 5 Pandemic, on the Pandemic Severity Index disambig ... more details
Orphan date January 2012 In mathematics, a , 1 category is a special case of an n , r category, consisting of an category in which all n morphisms for n 1 are equivalences. There are several models of , 1 categories, including Infinity category Segal category Simplicially enriched category Topological category Segal space Complete Segal space References http ncatlab.org nlab show 28infinity 2C1 29 category , 1 category at ncatlab CategoryCategory theory ... more details
Category B may refer to Category B Listed building Scotland Category B Prison UK Category B Bioterrorism agent Category B services Canadian television An intermediate category of disease recognized by the Centers for Disease Control and Prevention USA disambig ... more details
Category 1 can refer to Category 1 Tropical Cyclone on the Saffir Simpson Hurricane Scale Category 1 Pandemic on the Pandemic Severity Index Category 1 cable , an electrical standard for communications wiring disambig ... more details
Category 2 can refer to Category 2 Tropical Cyclone on the Saffir Simpson Hurricane Scale . Category 2 Pandemic on the Pandemic Severity Index Category 2 cable disambig ... more details
In mathematics, a topological category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory harv Lurie 2009 . See also Infinity category Simplicial category References Citation last1 Lurie first1 Jacob title Higher topos theory arxiv math.CT 0608040 publisher Princeton University Press series Annals of Mathematics Studies isbn 978 0 691 14049 0 978 0 691 14049 0 mr 2522659 year 2009 volume 170 CategoryCategory theory ... more details
Expert subject Mathematics date February 2009 DISPLAYTITLE n category In mathematics , n categories are a high order generalization of the notion of category theory category . The category of small n categories n Cat is defined by induction on n by the category 0 Cat is the category Set of category of sets sets and functions , the category n 1 Cat is the category of categories enriched category enriched over the category n Cat . An n category is then an object of n Cat . The monoidal category monoidal structure of Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite Product category theory products can be given a monoidal structure. The recursive construction of n Cat works fine because if a category C has finite products, the category of C enriched categories has finite products too. In particular, the category 1 Cat is the category Cat of category of small categories small categories and functor s. n categories have given rise to higher category theory , where several types of n categories are studied. The necessity of weakening the definition of an n category for homotopy homotopic purposes has led to the definition of Weak n category weak n categories . For distinction, the n categories as defined above are called strict. See also 2 category n category number Weak n category infinity category References cite book author Tom Leinster year 2004 title Higher Operads, Higher Categories publisher Cambridge University Press url http www.maths.gla.ac.uk tl book.html cite book author Eugenia Cheng, Aaron Lauda year 2004 title Higher Dimensional Categories an illustrated guide book url http www.cheng.staff.shef.ac.uk guidebook guidebook new.pdf Category Higher category theory categorytheory stub zh N ... more details
Category C may refer to any of the following Category C Listed building Scotland Category C Prison UK Category C Bioterrorism agent Pregnancy Category C Category C services Canadian television A Hooliganism hooligan Germany The least serious category of disease recognized by the Centers for Disease Control and Prevention USA disambig ... more details
Category 6 may refer to Category 6 Day of Destruction , a 2004 made for TV movie. Category 6 cable , a type of cable used for computer networking. A proposed hurricane level above Category 5, on the Saffir Simpson Hurricane Scale . disambig ... more details
Category 3 can refer to Category 3 cable , a specification for data cabling British firework classification Category 3 tropical cyclone on the Saffir Simpson Hurricane Scale . Category 3 Pandemic on the Pandemic Severity Index Category III, a rating in the Hong Kong motion picture rating system Category III, a capability level of aircraft instrument landing system ILS categories instrument landing system s disambig ... more details
In mathematics , especially category theory , a discrete category is a category whose only morphism s are the identity morphism s. It is the simplest kind of category. Specifically a category C is discrete if hom sub C sub X , X id sub X sub for all objects X hom sub C sub X , Y for all objects X Y Since by axioms, there is always the identity morphism between the same object, the above is equivalent to saying hom sub C sub X , Y is 1 when X Y and 0 when X is not equal to Y . Clearly, any class set theory class of objects defines a discrete category when augmented with identity maps. Any subcategory of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full category full . The limit category theory limit of any functor from a discrete category into another category is called a product category theory product , while the colimit is called a coproduct . References Robert Goldblatt 1984 . Topoi, the Categorial Analysis of Logic Studies in logic and the foundations of mathematics, 98 . North Holland. Reprinted 2006 by Dover Publications, and available http historical.library.cornell.edu cgi bin cul.math docviewer?did Gold010&id 3 online at http www.mcs.vuw.ac.nz rob Robert Goldblatt s homepage . CategoryCategory theory es Categor a discreta nl Discrete categorie ... more details
In mathematics, the simplex category or simplicial category or ordinal category is the category theory category of finite ordinals and order preserving maps. It is used to define simplicial set simplicial and cosimplicial objects. Formal definition The simplex category is usually denoted by math Delta math and is sometimes denoted by Ord . There are several equivalent descriptions of this category. math Delta math can be described as the category of finite ordinals as objects, thought of as totally ordered sets, and order preserving functions as morphisms . 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 math Delta math , that is a contravariant functor from math Delta math to another category. For instance, simplicial set s are contravariant with codomain category the category of sets. A cosimplicial object is defined similarly as a covariant functor originating from math Delta math . Note that in topology a simplicial object defined in this way would be called an augmented simplicial object because of the presence of an augmentation map. This map can be dropped to yield a traditionally defined simplicial object. An algebraic definition identifies math Delta math as the freely generated monoidal category on a single monoid object monoidal generator. This description is useful for understanding how any comonoid object in a monoidal category gives rise to a simplicial object since it can then be viewed as the image of a functor from math Delta text op math to the monoidal category containing the comonoid. Similarly ... monads can be viewed as monoid objects in functor category endofunctors categories . References ... category Simplex category in ncatlab Category Algebraic topology Category Homotopy theory Category Simplicial sets CategoryCategory theory zh ... more details
In category theory , a branch of mathematics , a closed category is a special kind of category mathematics category . In any category more precisely, in any locally small category , the morphisms between any two given objects x and y comprise a set mathematics set , the external hom x , y . In a closed category, these morphisms can be seen as comprising an object of the category itself, the internal hom x , y . Every closed category has a forgetful functor to the category of sets , which in particular takes the internal hom to the external hom. Definition A closed category can be defined as a category mathematics category V with a so called internal Hom functor math left right V op times V to V math , left Yoneda arrows natural in math B math and math C math and dinatural in math A math math L left B C right to left left A B right left A C right right math and a fixed object I of V such that there is a natural isomorphism math i A A cong left I A right math and a dinatural transformation math j A I to left A A right . , math Examples Cartesian closed category Cartesian closed categories are closed categories. In particular, any topos is closed. The canonical example is the category of sets . Compact closed category Compact closed categories are closed categories. The canonical example is the category mathematics category FdVect with finite dimensional vector spaces as objects and linear maps as morphisms. More generally, any monoidal closed category is a closed category. In this case, the object math I math is the monoidal unit. References Eilenberg, S. & Kelly, G.M. Closed categories Proceedings of the Conference on Categorical Algebra. La Jolla, 1965 Springer. 1966. pp. 421&ndash 562 Portal Category theory Category Closed categories categorytheory stub ... more details
Category 4 may refer to Category 4 cable , a cable that consists of four unshielded twisted pair wires with a data rate of 16 Mbit s and performance of up to 20 MHz Category 4 fireworks , British fireworks that are for sale only to professionals Category 4 hurricane , a hurricane in the second most intense category on the Saffir Simpson Hurricane Scale Category 4 influenza pandemic , an American influenza pandemic with a case fatality ratio between 1 and 2 Category four stadium , a football stadium of the highest quality as ranked by the UEFA disambig ... more details
Orphan date September 2011 In mathematics, more specifically in category theory especially in internal category theory an internal category ref Mac Lane, Moerdijk Sheaves in Geometry and Logic, Springer ref in a category math C math with pullback s consists of the following data two math C math objects math C 0,C 1 math named object of objects and object of morphisms respectively and four math C math arrows math d 0,d 1 C 1 rightarrow C 0, e C 0 rightarrow C 1,m C 1 times C 0 C 1 rightarrow C 1 math subject to coherence conditions expressing the axioms of category theory. Reflist CategoryCategory theory ... more details
A graded category is a mathematics mathematical concept. If math mathcal A math is a category theory category , then a math mathcal A math graded category is a category math mathcal C math together with a functor math F mathcal C rightarrow mathcal A math . Monoid s and group mathematics group s can be thought of as categories with a single element mathematics element . A monoid graded or group graded category is therefore one in which to each morphism is attached an element of a given monoid resp. group , its grade. This must be compatible with function composition composition , in the sense that compositions have the product grade. See also Graded algebra Slice category Unreferenced date February 2008 CategoryCategory theory cattheory stub ... more details
In mathematics , an autonomous category is a monoidal category where dual object s exist. ref Some authors use this term for a symmetric monoidal category symmetric closed monoidal category monoidal closed category , or for a biclosed monoidal category when symmetry is not assumed. ref Definition A left resp. right autonomous category is a monoidal category where every object has a left resp. right Dual object dual . An autonomous category is a monoidal category where every object has both a left and a right Dual object dual . ref Berman, pp 34 ref Rigid category is a synonym for autonomous category. In a symmetric monoidal category , the existence of left duals is equivalent to the existence of right duals, categories of this kind are called compact closed category compact closed categories . The concepts of autonomous category and autonomous category are directly related, specifically, every autonomous category is autonomous. A autonomous category may be described as a linearly distributive category with left and right negations such categories have two monoidal products linked with a sort of distributive law. In the case where the two monoidal products coincide and the distributivities are taken from the associativity isomorphism of the single monoidal structure, one obtains autonomous categories. Notes and references reflist Sources cite book last Yetter first David N. title Functorial Knot Theory year 2001 publisher World Scientific isbn 9810244436 cite book last Berman first Stephen coauthors Yuly Billi title Vertex Operator Algebras in Mathematics and Physics year 2003 publisher American Mathematical Society isbn 0821828568 categorytheory stub Category Monoidal categories ... more details
In cognitive psychology , a basic category is a category at a particular level of the category inclusion hierarchy i.e., a particular level of generality that is preferred by humans in learning and memory tasks. The term is associated with the work of psychologist Eleanor Rosch , who demonstrated basic category preferences in a number of classic experiments. Category Cognition cognitive psych stub bg references ... more details
In mathematics , a complete category is a category mathematics category in which all small limit category theory limit s exist. That is, a category C is complete if every diagram category theory diagram F J C where J is small category small has a limit in C . Duality category theory Dually , a cocomplete category is one in which all small colimit s exist. A bicomplete category is a category which is both ... to be practically relevant. Any category with this property is necessarily a Preordered class thin category for any two objects there can be at most one morphism from one object to the other. A weaker form of completeness is that of finite completeness. A category is finitely complete if all finite limits exists i.e. limits of diagrams indexed by a finite category J . Dually, a category is finitely ... that a category is complete if and only if it has Equaliser mathematics equalizers of all pairs of morphisms and all small product category theory product s. Since equalizers may be constructed from pullback category theory pullback s and binary products consider the pullback of f , g along the diagonal , a category is complete if and only if it has pullbacks and products. Dually, a category ... category theory pushout s and coproducts. Finite completeness can be characterized in several ways. For a category C , the following are all equivalent C is finitely complete, C has equalizers and all finite products, C has equalizers, binary products, and a terminal object , C has pullback category theory pullback s and a terminal object. The dual statements are also equivalent. A small category ..., Horst Herrlich, and George E. Strecker, theorem 12.7, page 213 ref A small complete category is necessarily thin. A posetal category vacuously has all equalizers and coequalizers, whence it is finitely ... restriction a posetal category with all products is automatically cocomplete, and dually ... Set , the category of sets Top , the category of topological spaces Grp , the category of groups ... more details
In mathematics , the category of manifolds , often denoted Man sup p sup , is the categorycategory theory category whose object category theory object s are manifold s of smooth function smoothness class C sup p sup and whose morphism s are p times continuously differentiable maps. This is a category because the function composition composition of two C sup p sup maps is again continuous. One is often interested only in C sup p sup manifolds modelled on spaces in a fixed category A , and the category of such manifolds is denoted Man sup p sup A . Similarly, the category of C sup p sup manifolds modelled on a fixed space E is denoted Man sup p sup E . One may also speak of the category of differentiable manifold smooth manifolds , Man sup sup , or the category of analytic manifolds, Man sup sup . Man sup p sup is a concrete category Like many categories, the category Man sup p sup is a concrete category , meaning its objects are Set mathematics sets with additional structure i.e. a topology and an equivalence class of atlases of charts defining a C sup p sup differentiable structure and its morphisms are function mathematics function s preserving this structure. There is a natural forgetful functor U Man sup p sup &rarr Top to the category of topological spaces which assigns to each manifold the underlying topological space the underlying set and to each p times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor U &prime Man sup p sup &rarr Set to the category of sets which assigns to each manifold the underlying set and to each p times continuously differentiable function the underlying function. References cite book last Lang first Serge title Differential manifolds publisher Addison Wesley Publishing Co., Inc. location Reading, Mass.&ndash London&ndash Don Mills, Ont. year 1972 CategoryCategory theoretic categories Manifolds Category Manifolds cattheory stub ... more details
Unreferenced date December 2009 In category theory , a branch of mathematics , the opposite category or dual category C sup op sup of a given category C is formed by reversing the morphism s, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, math C op op C math . Examples An example comes from reversing the direction of inequalities in a partial order . So if X is a Set mathematics set and a partial order relation, we can define a new partial order relation sub new sub by x sub new sub y if and only if y x . For example, there are opposite pairs child parent, or descendant ancestor. The category of Boolean algebra structure Boolean algebra s and Boolean homomorphism s is Equivalence of categories equivalent to the opposite of the category of Stone space s and continuous functions. The category of affine scheme s is equivalence category theory equivalent to the opposite of the category of commutative ring s. The Pontryagin duality restricts to an equivalence between the category of Compact space Definition compact Hausdorff space Hausdorff abelian group abelian topological group s and the opposite of the category of discrete abelian groups. By the Gelfand Neumark theorem, the category of localizable Sigma algebra measurable spaces with measurable function measurable maps is equivalent to the category of commutative Von Neumann algebra ... math C times D op cong C op times D op math see product category math Funct C,D op cong Funct C op ,D op math ref H. Herrlich, G. E. Strecker, Category Theory , 3rd Edition, Heldermann Verlag ... see functor category math F downarrow G op cong G op downarrow F op math see comma category References Reflist See also Dual object Dual category theory Duality mathematics Contravariant functor Opposite functor Citations DEFAULTSORT Opposite CategoryCategoryCategory theory cattheory stub nl Tegenovergestelde ... more details
In mathematics, a Segal category is a model of an infinity category introduced by harvtxt Hirschowitz Simpson 1998 , based on work of Graeme Segal in 1974. References cite arxiv first Andr last Hirschowitz first2 Carlos last2 Simpson title Descente pour les n champs Descent for n stacks arxiv math 9807049 class year 1998 citation first A. last Joyal title The theory of quasi categories and its applications, lectures at CRM Barcelona year 2008 url http www.crm.cat HigherCategories hc2.pdf pages 164 169 External links http ncatlab.org nlab show Segal category Segal category in ncatlab CategoryCategory theory ... more details
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