Disjoint unionFrom Wikipedia, the free encyclopediaContents1 Direct limit 11.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Algebraic objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Direct limit over a direct system in a category . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Related constructions and generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Disjoint union 42.1 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Set theory denition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Category theory point of view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Functor 63.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.1 Covariance and contravariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 Opposite functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.3 Bifunctors and multifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Relation to other categorical concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 Computer implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Limit (category theory) 114.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.1.1 Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iii CONTENTS4.1.2 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.1.3 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.1 Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.2 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.1 Existence of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.2 Universal property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3.3 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3.4 As representations of functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.5 Interchange of limits and colimits of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4 Functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4.1 Preservation of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.2 Lifting of limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.3 Creation and reection of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.5 A note on terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 214.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Chapter 1Direct limitIn mathematics, a direct limit (also called inductive limit) is a colimit of a directed family of objects. We willrst give the denition for algebraic structures like groups and modules, and then the general denition, which can beused in any category.1.1 Formal denition1.1.1 Algebraic objectsSee also: Directed set and Filtered categoryIn this section objects are understood to be sets with a given algebraic structure such as groups, rings, modules (over axed ring), algebras (over a xed eld), etc. With this in mind, homomorphisms are understood in the correspondingsetting (group homomorphisms, etc.).Start with the denition of adirectsystem of objects and homomorphisms. Let I, be a directed set. Let{Ai: i I} be a family of objects indexed by I and fij: Ai Aj be a homomorphism for all i j with thefollowing properties:1. fiiis the identity of Ai, and2. fik= fjk fij for all i j k .Then the pair Ai, fij is called a direct system over I.The underlying set of the direct limit, A, of the direct system Ai, fij is dened as the disjoint union of the Ai'smodulo a certain equivalence relation :limAi=iAi/ .Here, if xi Ai and xj Aj , xi xj if there is some k I such that fik(xi)=fjk(xj). Heuristically, twoelements in the disjoint union are equivalent if and only if they eventually become equal in the direct system. Anequivalent formulation that highlights the duality to the inverse limit is that an element is equivalent to all its imagesunder the maps of the directed system, i.e.xi fik(xi) .One naturally obtains from this denition canonical morphisms i: Ai A sending each element to its equivalenceclass. The algebraic operations on Aare dened via these maps in the obvious manner.An important property is that taking direct limits in the category of modules is an exact functor.12 CHAPTER 1. DIRECT LIMIT1.1.2 Direct limit over a direct system in a categoryThe direct limit can be dened in an arbitrary category C by means of a universal property. Let Xi, fij be a directsystem of objects and morphisms in C (same denition as above). The direct limit of this system is an object Xin Ctogether with morphisms i:Xi X satisfying i=j fij . The pair X, i must be universal in the sensethat for any other such pair Y, i there exists a unique morphism u : X Ymaking the diagramcommute for all i, j. The direct limit is often denotedX= limXiwith the direct system Xi, fij being understood.Unlike for algebraic objects, the direct limit may not exist in an arbitrary category. If it does, however, it is unique ina strong sense: given another direct limit X there exists a unique isomorphism X X commuting with the canonicalmorphisms.We note that a direct system in a category C admits an alternative description in terms of functors. Any directed posetI, can be considered as a small category I where the morphisms consist of arrows i j if and only if i j .A direct system is then just a covariant functor I C . In this case a direct limit is a colimit.1.2 ExamplesA collection of subsets Mi of a set M can be partially ordered by inclusion. If the collection is directed, itsdirect limit is the unionMi .Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system isisomorphic to Xm and the canonical morphism m: Xm X is an isomorphism.Let p be a prime number. Consider the direct system composed of the groups Z/pnZ and the homomorphismsZ/pnZ Z/pn+1Z induced by multiplication by p. The direct limit of this system consists of all the roots ofunity of order some power of p, and is called the Prfer group Z(p).Let F be a C-valued sheaf on a topological space X. Fix a point x in X. The open neighborhoods of x form adirected poset ordered by inclusion (U V if and only if U contains V). The corresponding direct system is(F(U), rU,V) where r is the restriction map. The direct limit of this system is called the stalk of F at x, denotedFx. For each neighborhood U of x, the canonical morphism F(U) Fx associates to a section s of F over Uan element sx of the stalk Fx called the germ of s at x.Direct limits in the category of topological spaces are given by placing the nal topology on the underlyingset-theoretic direct limit.Direct limits are linked to inverse limits viaHom(limXi, Y ) = limHom(Xi, Y ).1.3. RELATED CONSTRUCTIONS AND GENERALIZATIONS 3Consider a sequence {An, n} where An is a C*-algebra and n : An An is a *-homomorphism. TheC*-analog of the direct limit construction gives a C*-algebra satisfying the universal property above.1.3 Related constructions and generalizationsThe categorical dual of the direct limit is called the inverse limit (or projective limit). More general concepts arethe limits and colimits of category theory. The terminology is somewhat confusing: direct limits are colimits whileinverse limits are limits.1.4 See alsoInverse, or projective limit1.5 ReferencesBourbaki, Nicolas (1968), Elements of mathematics. Theory of sets, Translated from the French, Paris: Her-mann, MR 0237342.Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nded.), Springer-Verlag.Chapter 2Disjoint unionIn set theory, the disjoint union (or discriminated union) of a family of sets is a modied union operation thatindexes the elements according to which set they originated in. Or slightly dierent from this, the disjoint union ofa family of subsets is the usual union of the subsets which are pairwise disjoint disjoint sets means they have noelement in common.Note that these two concepts are dierent but strongly related. Moreover, it seems that they are essentially identicalto each other in category theory. That is, both are realizations of the coproduct of category of sets.2.1 ExampleDisjoint union of sets A0 = {1, 2, 3} and A1 = {1, 2} can be computed by nding:A0= {(1, 0), (2, 0), (3, 0)}A1= {(1, 1), (2, 1)}soA0 A1= A0 A1= {(1, 0), (2, 0), (3, 0), (1, 1), (2, 1)}2.2 Set theory denitionFormally, let {Ai : i I} be a family of sets indexed by i. The disjoint union of this family is the setiIAi=iI{(x, i) : x Ai}.The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which Aithe element x came from.Each of the sets Ai is canonically isomorphic to the setAi= {(x, i) : x Ai}.Through this isomorphism, one may consider that Ai is canonically embedded in the disjoint union. For i j, the setsAi* and Aj* are disjoint even if the sets Ai and Aj are not.In the extreme case where each of the Ai is equal to some xed set A for each i I, the disjoint union is the Cartesianproduct of A and I:42.3. CATEGORY THEORY POINT OF VIEW 5iIAi= AI.One may occasionally see the notationiIAifor the disjoint union of a family of sets, or the notation A + B for the disjoint union of two sets. This notation ismeant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the termsin the family. Compare this to the notation for the Cartesian product of a family of sets.Disjoint unions are also sometimes writteniI Ai or iI Ai .In the language of category theory, the disjoint union is the coproduct in the category of sets. It therefore satises theassociated universal property. This also means that the disjoint union is the categorical dual of the Cartesian productconstruction. See coproduct for more details.For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation,the indexed family can be treated simply as a collection of sets. In this case Aiis referred to as a copy of Ai and thenotation ACA is sometimes used.2.3 Category theory point of viewIn category theory the disjoint union is dened as a coproduct in the category of sets.As such, the disjoint union is dened up to an isomorphism, and the above denition is just one realization of thecoproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct.This justies the second denition in the lead.This categorical aspect of the disjoint union explains whyis frequently used, instead of, to denote coproduct.2.4 See alsoCoproductDisjoint union (topology)Disjoint union of graphsPartition of a setSum typeTagged unionUnion (computer science)2.5 ReferencesLang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised thirded.), New York: Springer-Verlag, p. 60, ISBN 978-0-387-95385-4Weisstein, Eric W., Disjoint Union, MathWorld.Chapter 3FunctorThis article is about the mathematical concept. For other uses, see Functor (disambiguation).In mathematics, a functor is a type of mapping between categories which is applied in category theory. Functors canbe thought of as homomorphisms between categories. In the category of small categories, functors can be thought ofmore generally as morphisms.Functors were rst considered in algebraic topology, where algebraic objects (like the fundamental group) are asso-ciated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functorsare used throughout modern mathematics to relate various categories. Thus, functors are generally applicable in areaswithin mathematics that category theory can make an abstraction of.The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap,[1] who used the term in alinguistic context:[2] see function word.3.1 DenitionLet C and D be categories. A functor F from C to D is a mapping that[3]associates to each object X C an object F(X) D ,associates to each morphismf : X YCa morphismF(f) : F(X) F(Y ) D such that thefollowing two conditions hold: F(idX) = idF(X) for every object X C F(g f) = F(g) F(f) for all morphisms f: X Yand g: Y Z.That is, functors must preserve identity morphisms and composition of morphisms.3.1.1 Covariance and contravarianceThere are many constructions in mathematics that would be functors but for the fact that they turn morphisms aroundand reverse composition. We then dene a contravariant functor F from C to D as a mapping thatassociates to each object X C an object F(X) D,associates to each morphism f: X Y C a morphism F(f) : F(Y ) F(X) D such that F(idX) = idF(X) for every object X C , F(g f) = F(f) F(g) for all morphisms f: X Yand g: Y Z.63.2. EXAMPLES 7Note that contravariant functors reverse the direction of composition.Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Notethat one can also dene a contravariant functor as a covariant functor on the opposite category Cop .[4] Some authorsprefer to write all expressions covariantly. That is, instead of saying F: C D is a contravariant functor, theysimply write F: Cop D (or sometimes F: C Dop ) and call it a functor.Contravariant functors are also occasionally called cofunctors.3.1.2 Opposite functorEvery functor F:C D induces the opposite functor Fop:Cop Dop , where Cop and Dop are the oppositecategories toC andD .[5] By denition,Fop maps objects and morphisms identically toF. SinceCop does notcoincide with C as a category, and similarly for D , Fop is distinguished from F. For example, when composingF: C0 C1 with G : Cop1 C2 , one should use either G Fop or Gop F . Note that, following the property ofopposite category, (Fop)op= F .3.1.3 Bifunctors and multifunctorsA bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, theHom functor is of the type Cop C Set. It can be seen as a functor in two arguments. The Hom functor is a naturalexample; it is contravariant in one argument, covariant in the other.Amultifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctorwith n = 2.3.2 ExamplesDiagram: For categories C and J, a diagram of type J in C is a covariant functor D : J C .(Category theoretical) presheaf: For categories C and J, a J-presheaf on C is a contravariant functor D : C J .Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open(X) under inclusion.Like every partially ordered set, Open(X) forms a small category by adding a single arrow U Vif and only ifU V. Contravariant functors on Open(X) are called presheaves on X. For instance, by assigning to every open setU the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.Constant functor: The functor C D which maps every object of C to a xed object X in D and every morphismin C to the identity morphism on X. Such a functor is called a constant or selection functor.Endofunctor: A functor that maps a category to itself.Identity functor in category C, written 1C or idC, maps an object to itself and a morphism to itself. Identity functoris an endofunctor.Diagonal functor: The diagonal functor is dened as the functor from D to the functor category DCwhich sendseach object in D to the constant functor at that object.Limit functor: For a xed index category J, if every functor JC has a limit (for instance if C is complete), thenthe limit functor CJC assigns to each functor its limit. The existence of this functor can be proved by realizing thatit is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitableversion of the axiom of choice. Similar remarks apply to the colimit functor (which is covariant).Power sets: The power set functor P : Set Set maps each set to its power set and each function f: X Yto themap which sends U X to its image f(U) Y. One can also consider the contravariant power set functor whichsends f: X Yto the map which sends V Yto its inverse image f1(V ) X.Dual vector space: The map which assigns to every vector space its dual space and to every linear map its dual ortranspose is a contravariant functor from the category of all vector spaces over a xed eld to itself.Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguishedpoints. The objects are pairs (X, x0), where X is a topological space and x0 is a point in X. A morphism from (X, x0)to (Y, y0) is given by a continuous map f : X Y with f(x0) = y0.8 CHAPTER 3. FUNCTORTo every topological space X with distinguished point x0, one can dene the fundamental group based at x0, denoted1(X, x0). This is the group of homotopy classes of loops based at x0. If f : X Y morphism of pointed spaces, thenevery loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation iscompatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphismfrom (X, x0) to (Y, y0). We thus obtain a functor from the category of pointed topological spaces to the categoryof groups.In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves,but they cannot be composed unless they share an endpoint. Thus one has the fundamental groupoid instead of thefundamental group, and this construction is functorial.Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuousmaps as morphisms) to the category of real associative algebras is given by assigning to every topological space Xthe algebra C(X) of all real-valued continuous functions on that space. Every continuous map f : X Y induces analgebra homomorphism C(f) : C(Y) C(X) by the rule C(f)() = o f for every in C(Y).Tangent and cotangent bundles: The map which sends every dierentiable manifold to its tangent bundle and everysmooth map to its derivative is a covariant functor from the category of dierentiable manifolds to the category ofvector bundles.Doing this constructions pointwise gives the tangent space, a covariant functor from the category of pointed dieren-tiable manifolds to the category of real vector spaces. Likewise, cotangent space is a contravariant functor, essentiallythe composition of the tangent space with the dual space above.Group actions/representations:Every group G can be considered as a category with a single object whose mor-phisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set,i.e. a G-set. Likewise, a functor from G to the category of vector spaces, VectK, is a linear representation of G. Ingeneral, a functor G C can be considered as an action of G on an object in the category C. If C is a group, thenthis action is a group homomorphism.Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra denes a functor.Tensor products: If C denotes the category of vector spaces over a xed eld, with linear maps as morphisms, thenthe tensor product V W denes a functor C C C which is covariant in both arguments.[6]Forgetful functors: The functor U : Grp Set which maps a group to its underlying set and a group homomorphismto its underlying function of sets is a functor.[7] Functors like these, which forget some structure, are termed forgetfulfunctors. Another example is the functor Rng Ab which maps a ring to its underlying additive abelian group.Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms).Free functors:Going in the opposite direction of forgetful functors are free functors. The free functor F : Set Grp sends every set X to the free group generated by X. Functions get mapped to group homomorphisms betweenfree groups. Free constructions exist for many categories based on structured sets. See free object.Homomorphismgroups: To every pair A, B of abelian groups one can assign the abelian group Hom(A,B) consistingof all group homomorphisms from A to B. This is a functor which is contravariant in the rst and covariant in thesecond argument, i.e. it is a functor Abop Ab Ab (where Ab denotes the category of abelian groups with grouphomomorphisms). If f : A1 A2 and g : B1 B2 are morphisms in Ab, then the group homomorphism Hom(f,g): Hom(A2,B1) Hom(A1,B2) is given by g f. See Hom functor.Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objectsin C one can assign the set Hom(X,Y) of morphisms from X to Y. This denes a functor to Set which is contravariantin the rst argument and covariant in the second, i.e. it is a functor Cop C Set. If f : X1 X2 and g : Y1 Y2are morphisms in C, then the group homomorphism Hom(f,g) : Hom(X2,Y1) Hom(X1,Y2) is given by g f.Functors like these are called representable functors. An important goal in many settings is to determine whether agiven functor is representable.3.3 PropertiesTwo important consequences of the functor axioms are:F transforms each commutative diagram in C into a commutative diagram in D;3.4. RELATION TO OTHER CATEGORICAL CONCEPTS 9if f is an isomorphism in C, then F(f) is an isomorphism in D.One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form thecomposite functor GF from A to C. Composition of functors is associative where dened. Identity of compositionof functors is identity functor. This shows that functors can be considered as morphisms in categories of categories,for example in the category of small categories.A small category with a single object is the same thing as a monoid: the morphisms of a one-object category canbe thought of as elements of the monoid, and composition in the category is thought of as the monoid operation.Functors between one-object categories correspond to monoid homomorphisms. So in a sense, functors betweenarbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.3.4 Relation to other categorical conceptsLet C and D be categories. The collection of all functors C D form the objects of a category: the functor category.Morphisms in this category are natural transformations between functors.Functors are often dened by universal properties; examples are the tensor product, the direct sum and direct productof groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limitand colimit generalize several of the above.Universal constructions often give rise to pairs of adjoint functors.3.5 Computer implementationsFunctors sometimes appear in functional programming. For instance, the programming language Haskell has a classFunctor where fmap is a polytypic function used to map functions (morphisms on Hask, the category of Haskell types)between existing types to functions between some new types.3.6 See alsoFunctor categoryKan extensionPseudofunctor3.7 Notes[1] Mac Lane, Saunders (1971), Categories for the Working Mathematician, Springer-Verlag: New York, p. 30, ISBN 978-3-540-90035-1[2] Carnap, The Logical Syntax of Language, p. 1314, 1937, Routledge & Kegan Paul[3] Jacobson (2009), p. 19, def. 1.2.[4] Jacobson (2009), p. 1920.[5] Mac Lane, Saunders; Moerdijk, Ieke (1992), Sheaves in geometry and logic:a rst introduction to topos theory, Springer,ISBN 978-0-387-97710-2[6] Hazewinkel, Michiel; Gubareni, Nadezhda Mikhalovna; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004), Algebras,rings and modules, Springer, ISBN 978-1-4020-2690-4[7] Jacobson (2009), p. 20, ex. 2.10 CHAPTER 3. FUNCTOR3.8 ReferencesJacobson, Nathan (2009), Basic algebra 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7.3.9 External linksHazewinkel, Michiel, ed. (2001), Functor, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4see functor in nLab and the variations discussed and linked to there.Andr Joyal, CatLab, a wiki project dedicated to the exposition of categorical mathematicsHillman, Chris. A Categorical Primer. CiteSeerX: 10 .1 .1 .24 .3264: formal introduction to category theory.J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of CatsStanford Encyclopedia of Philosophy: "Category Theory" by Jean-Pierre Marquis. Extensive bibliography.List of academic conferences on category theoryBaez, John, 1996,The Tale of n-categories." An informal introduction to higher order categories.WildCats is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms,categories, functors, natural transformations, universal properties.The catsters, a YouTube channel about category theory.Category Theory at PlanetMath.org.Video archive of recorded talks relevant to categories, logic and the foundations of physics.Interactive Web page which generates examples of categorical constructions in the category of nite sets.Chapter 4Limit (category theory)In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universalconstructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructionssuch as disjoint unions, direct sums, coproducts, pushouts and direct limits.Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high levelof abstraction. In order to understand them, it is helpful to rst study the specic examples these concepts are meantto generalize.4.1 DenitionLimits and colimits in a category C are dened by means of diagrams in C. Formally, a diagram of type J in C is afunctor from J to C:F : J C.The category J is thought of as index category, and the diagram F is thought of as indexing a collection of objectsand morphisms in C patterned on J.One is most often interested in the case where the category J is a small or even nite category. A diagram is said tobe small or nite whenever J is.4.1.1 LimitsLet F : J C be a diagram of type J in a category C. A cone to F is an object N of C together with a family X : N F(X) of morphisms indexed by the objects X of J, such that for every morphism f : X Y in J, we have F(f) oX = Y.A limit of the diagram F : J C is a cone (L, ) to F such that for any other cone (N, ) to F there exists a uniquemorphism u : N L such that X o u = X for all X in J.One says that the cone (N, ) factors through the cone (L, ) with the unique factorization u. The morphism u issometimes called the mediating morphism.Limits are also referred to as universal cones, since they are characterized by a universal property (see below for moreinformation). As with every universal property, the above denition describes a balanced state of generality: Thelimit object L has to be general enough to allow any other cone to factor through it; on the other hand, L has to besuciently specic, so that only one such factorization is possible for every cone.Limits may also be characterized as terminal objects in the category of cones to F.It is possible that a diagram does not have a limit at all. However, if a diagram does have a limit then this limit isessentially unique: it is unique up to a unique isomorphism. For this reason one often speaks of the limit of F.1112 CHAPTER 4. LIMIT (CATEGORY THEORY)A universal cone4.1.2 ColimitsSee also: Direct limitThe dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the denitionsof these by inverting all morphisms in the above denitions, we will explicitly state them here:A co-cone of a diagram F : J C is an object N of C together with a family of morphismsX : F(X) Nfor every object X of J, such that for every morphism f : X Y in J, we have Y o F(f)= X.A colimit of a diagram F : J C is a co-cone (L, ) of F such that for any other co-cone (N, ) of F there exists aunique morphism u : L N such that u o X = X for all X in J.Colimits are also referred to as universal co-cones. They can be characterized as initial objects in the category ofco-cones from F.As with limits, if a diagram F has a colimit then this colimit is unique up to a unique isomorphism.4.1.3 VariationsLimits and colimits can also be dened for collections of objects and morphisms without the use of diagrams. Thedenitions are the same (note that in denitions above we never needed to use composition of morphisms in J). Thisvariation, however, adds no new information. Any collection of objects and morphisms denes a (possibly large)4.2. EXAMPLES 13A universal co-conedirected graph G. If we let J be the free category generated by G, there is a universal diagram F : J C whose imagecontains G. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection ofobjects and morphisms.Weak limit and weak colimits are dened like limits and colimits, except that the uniqueness property of the me-diating morphism is dropped.4.2 Examples4.2.1 LimitsThe denition of limits is general enough to subsume several constructions useful in practical settings. In the followingwe will consider the limit (L, ) of a diagram F : J C.Terminal objects. If J is the empty category there is only one diagram of type J: the empty one (similar tothe empty function in set theory). A cone to the empty diagram is essentially just an object of C. The limit ofF is any object that is uniquely factored through by every other object. This is just the denition of a terminalobject.Products. If J is a discrete category then a diagramF is essentially nothing but a family of objects of C, indexedby J. The limit L of F is called the product of these objects. The cone consists of a family of morphisms X: L F(X) called the projections of the product. In the category of sets, for instance, the products are givenby Cartesian products and the projections are just the natural projections onto the various factors.14 CHAPTER 4. LIMIT (CATEGORY THEORY)Powers. A special case of a product is when the diagram F is a constant functor to an object X of C. Thelimit of this diagram is called the Jthpower of X and denoted XJ.Equalizers. If J is a category with two objects and two parallel morphisms from object 1 to object 2 then adiagram of type J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer ofthose morphisms.Kernels. A kernel is a special case of an equalizer where one of the morphisms is a zero morphism.Pullbacks. Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identitymorphisms are f : X Z and g : Y Z. The limit L of F is called a pullback or a ber product. It can nicelybe visualized as a commutative square:Inverse limits. Let J be a directed poset (considered as a small category by adding arrows i j if and only ifi j) and let F : Jop C be a diagram. The limit of F is called (confusingly) an inverse limit or projective limit.If J = 1, the category with a single object and morphism, then a diagram of type J is essentially just an objectX of C. A cone to an object X is just a morphism with codomain X. A morphism f : Y X is a limit of thediagram X if and only if f is an isomorphism. More generally, if J is any category with an initial object i,then any diagram of type J has a limit, namely any object isomorphic to F(i). Such an isomorphism uniquelydetermines a universal cone to F.Topological limits. Limits of functions are a special case of limits of lters, which are related to categoricallimits as follows. Given a topological space X, denote F the set of lters on X, x X a point, V(x) F the4.3. PROPERTIES 15neighborhood lter of x, A F a particular lter and Fx,A= {G F | V (x) A G} the set of lters nerthan A and that converge to x. The lters F are given a small and thin category structure by adding an arrowA B if and only if A B. The injection Ix,A: Fx,A F becomes a functor and the following equivalenceholds :x is a topological limit of A if and only if A is a categorical limit of Ix,A4.2.2 ColimitsExamples of colimits are given by the dual versions of the examples above:Initial objects are colimits of empty diagrams.Coproducts are colimits of diagrams indexed by discrete categories.Copowers are colimits of constant diagrams from discrete categories.Coequalizers are colimits of a parallel pair of morphisms.Cokernels are coequalizers of a morphism and a parallel zero morphism.Pushouts are colimits of a pair of morphisms with common domain.Direct limits are colimits of diagrams indexed by directed sets.4.3 Properties4.3.1 Existence of limitsA given diagram F : J C may or may not have a limit (or colimit) in C. Indeed, there may not even be a cone to F,let alone a universal cone.A category C is said to have limits of type J if every diagram of type J has a limit in C. Specically, a category Cis said tohave products if it has limits of type J for every small discrete category J (it need not have large products),have equalizers if it has limits of type (i.e. every parallel pair of morphisms has an equalizer),have pullbacks if it has limits of type (i.e. every pair of morphisms with common codomain hasa pullback).A complete category is a category that has all small limits (i.e. all limits of type J for every small category J).One can also make the dual denitions. A category has colimits of type J if every diagram of type J has a colimitin C. A cocomplete category is one that has all small colimits.The existence theorem for limits states that if a category C has equalizers and all products indexed by the classesOb(J) and Hom(J), then C has all limits of type J. In this case, the limit of a diagram F : J C can be constructedas the equalizer of the two morphismss, t :iOb(J)F(i) fHom(J)F(cod(f))given (in component form) bys =(F(f) F(dom(f)))fHom(J)t =(F(cod(f)))fHom(J).There is a dual existence theorem for colimits in terms of coequalizers and coproducts. Both of these theoremsgive sucient and necessary conditions for the existence of all (co)limits of type J.16 CHAPTER 4. LIMIT (CATEGORY THEORY)4.3.2 Universal propertyLimits and colimits are important special cases of universal constructions.Let C be a category and let J be a small index category. The functor category CJmay be thought of the category ofall diagrams of type J in C. The diagonal functor : C CJis the functor that maps each object N in C to the constant functor (N) : J C to N. That is, (N)(X) = N for eachobject X in J and (N)(f) = idN for each morphism f in J.Given a diagram F: J C (thought of as an object in CJ), a natural transformation : (N) F (which is just amorphism in the category CJ) is the same thing as a cone from N to F. To see this, rst note that (N)(X) = N forall X implies that the components of are morphisms X : N F(X), which all share the domain N. Moreover therequirement that the cones diagrams commute is true simply because this is a natural transformation. (Dually, anatural transformation : F (N) is the same thing as a co-cone from F to N.)Therefore, the denitions of limits and colimits can then be restated in the form:A limit of F is a universal morphism from to F.A colimit of F is a universal morphism from F to .4.3.3 AdjunctionsLike all universal constructions, the formation of limits and colimits is functorial in nature. In other words, if everydiagram of type J has a limit in C (for J small) there exists a limit functorlim : CJ Cwhich assigns each diagram its limit and each natural transformation : F G the unique morphism lim : lim F lim G commuting with the corresponding universal cones. This functor is right adjoint to the diagonal functor :C CJ. This adjunction gives a bijection between the set of all morphisms from N to lim F and the set of all conesfrom N to FHom(N, limF) = Cone(N, F)which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F.If the index category J is connected (and nonempty) then the unit of the adjunction is an isomorphism so that lim isa left inverse of . This fails if J is not connected. For example, if J is a discrete category, the components of theunit are the diagonal morphisms : N NJ.Dually, if every diagram of type J has a colimit in C (for J small) there exists a colimit functorcolim : CJ Cwhich assigns each diagram its colimit. This functor is left adjoint to the diagonal functor : C CJ, and one has anatural isomorphismHom(colimF, N) = Cocone(F, N).The unit of this adjunction is the universal cocone from F to colim F. If J is connected (and nonempty) then thecounit is an isomorphism, so that colim is a left inverse of .Note that both the limit and the colimit functors are covariant functors.4.4. FUNCTORS AND LIMITS 174.3.4 As representations of functorsOne can use Hom functors to relate limits and colimits in a category C to limits in Set, the category of sets. Thisfollows, in part, from the fact the covariant Hom functor Hom(N, ) : C Set preserves all limits in C. By duality,the contravariant Hom functor must take colimits to limits.If a diagram F : J C has a limit in C, denoted by lim F, there is a canonical isomorphismHom(N, limF) = limHom(N, F)which is natural in the variable N. Here the functor Hom(N, F) is the composition of the Hom functor Hom(N, )with F. This isomorphism is the unique one which respects the limiting cones.One can use the above relationship to dene the limit of F in C. The rst step is to observe that the limit of the functorHom(N, F) can be identied with the set of all cones from N to F:limHom(N, F) = Cone(N, F).The limiting cone is given by the family of maps X : Cone(N, F) Hom(N, FX) where X() = X. If one is givenan object L of C together with a natural isomorphism : Hom(, L) Cone(, F), the object L will be a limit of Fwith the limiting cone given by L(idL). In fancy language, this amounts to saying that a limit of F is a representationof the functor Cone(, F) : C Set.Dually, if a diagram F : J C has a colimit in C, denoted colim F, there is a unique canonical isomorphismHom(colimF, N) = limHom(F, N)which is natural in the variable N and respects the colimiting cones. Identifying the limit of Hom(F, N) with the setCocone(F, N), this relationship can be used to dene the colimit of the diagram F as a representation of the functorCocone(F, ).4.3.5 Interchange of limits and colimits of setsLet I be a nite category and J be a small ltered category. For any bifunctorF : I J Setthere is a natural isomorphismcolimJ limIF(i, j) limI colimJF(i, j).In words, ltered colimits in Set commute with nite limits.4.4 Functors and limitsIf F : J C is a diagram in C and G : C D is a functor then by composition (recall that a diagram is just a functor)one obtains a diagram GF : J D. A natural question is then:How are the limits of GF related to those of F?18 CHAPTER 4. LIMIT (CATEGORY THEORY)4.4.1 Preservation of limitsA functor G : C D induces a map from Cone(F) to Cone(GF): if is a cone from N to F then G is a cone fromGN to GF. The functor G is said to preserve the limits of F if (GL, G) is a limit of GF whenever (L, ) is a limitof F. (Note that if the limit of F does not exist, then G vacuously preserves the limits of F.)A functor G is said to preserve all limits of type J if it preserves the limits of all diagrams F : J C. For example,one can say that G preserves products, equalizers, pullbacks, etc. A continuous functor is one that preserves allsmall limits.One can make analogous denitions for colimits. For instance, a functor G preserves the colimits of F if G(L, ) isa colimit of GF whenever (L, ) is a colimit of F. A cocontinuous functor is one that preserves all small colimits.If C is a complete category, then, by the above existence theorem for limits, a functor G : C D is continuous ifand only if it preserves (small) products and equalizers. Dually, G is cocontinuous if and only if it preserves (small)coproducts and coequalizers.An important property of adjoint functors is that every right adjoint functor is continuous and every left adjointfunctor is cocontinuous. Since adjoint functors exist in abundance, this gives numerous examples of continuous andcocontinuous functors.For a given diagram F : J C and functor G : C D, if both F and GF have specied limits there is a uniquecanonical morphismF : G lim F lim GFwhich respects the corresponding limit cones. The functor G preserves the limits of F if and only this map is anisomorphism. If the categories C and D have all limits of type J then lim is a functor and the morphisms F form thecomponents of a natural transformation : G lim lim GJ.The functor G preserves all limits of type J if and only if is a natural isomorphism. In this sense, the functor G canbe said to commute with limits (up to a canonical natural isomorphism).Preservation of limits and colimits is a concept that only applies to covariant functors. For contravariant functors thecorresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.4.4.2 Lifting of limitsA functor G : C D is said to lift limits for a diagram F : J C if whenever (L, ) is a limit of GF there exists alimit (L, ) of F such that G(L, ) = (L, ). A functor G lifts limits of type J if it lifts limits for all diagrams oftype J. One can therefore talk about lifting products, equalizers, pullbacks, etc. Finally, one says that G lifts limitsif it lifts all limits. There are dual denitions for the lifting of colimits.A functor G lifts limits uniquely for a diagram F if there is a unique preimage cone (L, ) such that (L, ) is alimit of F and G(L, ) = (L, ). One can show that G lifts limits uniquely if and only if it lifts limits and is amnestic.Lifting of limits is clearly related to preservation of limits. If G lifts limits for a diagram F and GF has a limit, thenF also has a limit and G preserves the limits of F. It follows that:If G lifts limits of all type J and D has all limits of type J, then C also has all limits of type J and G preservesthese limits.If G lifts all small limits and D is complete, then C is also complete and G is continuous.The dual statements for colimits are equally valid.4.4.3 Creation and reection of limitsLet F : J C be a diagram. A functor G : C D is said to4.5. A NOTE ON TERMINOLOGY 19create limits for F if whenever (L, ) is a limit of GF there exists a unique cone (L, ) to F such that G(L,) = (L, ), and furthermore, this cone is a limit of F.reect limits for F if each cone to F whose image under G is a limit of GF is already a limit of F.Dually, one can dene creation and reection of colimits.The following statements are easily seen to be equivalent:The functor G creates limits.The functor G lifts limits uniquely and reects limits.There are examples of functors which lift limits uniquely but neither create nor reect them.4.4.4 ExamplesFor any category C and object A of C the covariant Hom functor Hom(A,) : C Set preserves all limits inC. In particular, Hom functors are continuous. Hom functors need not preserve colimits.Every representable functor C Set preserves limits (but not necessarily colimits).The forgetful functor U : Grp Set creates (and preserves) all small limits and ltered colimits; however, Udoes not preserve coproducts. This situation is typical of algebraic forgetful functors.The free functor F : Set Grp (which assigns to every set S the free group over S) is left adjoint to forgetfulfunctor U and is, therefore, cocontinuous. This explains why the free product of two free groups G and H isthe free group generated by the disjoint union of the generators of G and H.The inclusion functor Ab Grp creates limits but does not preserve coproducts (the coproduct of two abeliangroups being the direct sum).The forgetful functor Top Set lifts limits and colimits uniquely but creates neither.Let Metc be the category of metric spaces with continuous functions for morphisms. The forgetful functorMetc Set lifts nite limits but does not lift them uniquely.4.5 A note on terminologyOlder terminology referred to limits as inverse limits or projective limits, and to colimits as direct limits orinductive limits. This has been the source of a lot of confusion.There are several ways to remember the modern terminology. First of all,cokernels,coproducts,coequalizers, andcodomainsare types of colimits, whereaskernels,productsequalizers, anddomainsare types of limits. Second, the prex co implies rst variable of the Hom ". Terms like cohomology andcobration all have a slightly stronger association with the rst variable, i.e., the contravariant variable, of the Hombifunctor.20 CHAPTER 4. LIMIT (CATEGORY THEORY)4.6 ReferencesAdmek, Ji; Horst Herrlich; George E. Strecker (1990). Abstract and Concrete Categories (PDF). John Wiley& Sons. ISBN 0-471-60922-6.Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.4.7 External linksInteractive Web page which generates examples of limits and colimits in the category of nite sets. Writtenby Jocelyn Paine.4.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 214.8 Text and image sources, contributors, and licenses4.8.1 Text Direct limit Source: https://en.wikipedia.org/wiki/Direct_limit?oldid=674575168 Contributors: AxelBoldt, Magnus~enwiki, TakuyaMu-rata, Charles Matthews, Giftlite, Fropu, Rich Farmbrough, Gauge, Oleg Alexandrov, Marudubshinki, Salix alba, Masnevets, Shell Kin-ney, Prime Entelechy, SmackBot, RDBury, Turms, Tesseran, Mets501, P199, A. Pichler, CRGreathouse, Mct mht, Cydebot, Headbomb,RobHar, Jakob.scholbach, VolkovBot, Kyle the bot, Kkilger, DesolateReality, Rswarbrick, Beroal, Sandrobt, Marc van Leeuwen, Wik-Head, Addbot, Fryed-peach, Luckas-bot, Point-set topologist, Ringspectrum, Erik9bot, Sawomir Biay, Tiled, WikitanvirBot, ZroBot,Quasihuman, Solomon7968, IkamusumeFan, ChrisGualtieri, Makecat-bot, Mark viking and Anonymous: 24 Disjoint union Source: https://en.wikipedia.org/wiki/Disjoint_union?oldid=664597191 Contributors: Mav, Zundark, Tarquin, Takuya-Murata, Charles Matthews, Dcoetzee, Altenmann, MathMartin, Giftlite, Fropu, CyborgTosser, Abdull, Paul August, Oleg Alexandrov,Linas, Jacobolus, Salix alba, Mathbot, YurikBot, Hairy Dude, Bhny, Tong~enwiki, David Pierce, SmackBot, RDBury, Melchoir, Alan Mc-Beth, Nbarth, Wdvorak, Mets501, Thijs!bot, JAnDbot, Albmont, Americanhero, Pavel Jelnek, The enemies of god, PatHayes, Jamelan,Niceguyedc, Addbot, Arketyp, Gtgith, Vorbeigehende, Yodigo, Erik9bot, Stpasha, WBielas, WikitanvirBot, D.Lazard, ChuispastonBot,Akseli.palen, 10k, Freeze S, Deltahedron, Ktzell29 and Anonymous: 19 Functor Source: https://en.wikipedia.org/wiki/Functor?oldid=672478641 Contributors: AxelBoldt, Jan Hidders, XJaM, Toby Bartels,SimonP, Smelialichu, Michael Hardy, TakuyaMurata, Minesweeper, Charles Matthews, Dysprosia, Jitse Niesen, Greenrd, Tobias Berge-mann, Giftlite, Lethe, Fropu, Pascal666, DefLog~enwiki, Cypa, Smimram, AlexG, Jkl, Rec syn, Gauge, Blotwell, 3mta3, Msh210,Kenyon, Oleg Alexandrov, Robert K S, Clotito, Mathbot, Chobot, YurikBot, Hairy Dude, Grafen, Yahya Abdal-Aziz, Voidxor, Robert-byrne, SmackBot, Mmernex, Incnis Mrsi, McGeddon, Kmarinas86, Nbarth, Adamarthurryan, Lambiam, Antonielly, 16@r, Myasuda,Sam Staton, Julian Mendez, Kilva, Oerjan, Nick Number, JAnDbot, Magioladitis, STBot, R'n'B, Policron, STBotD, Anonymous Dis-sident, JhsBot, YonaBot, Wing gundam, Classicalecon, He7d3r, Cenarium, Alexey Muranov, Beroal, Iq1001, EndlessWorld, Addbot,, Coreyoconnor, Yobot, TaBOT-zerem, Legobot II, FUZxxl, KamikazeBot, FrescoBot, Sawomir Biay, Magmalex, StephanSpahn, ZroBot, Let4time, CocuBot, Helpful Pixie Bot, ChrisGualtieri, Spectral sequence, Lambda Fairy, Mark viking, Cldorian, Chngr,Baum42, AlielJorax and Anonymous: 50 Limit(categorytheory)Source: https://en.wikipedia.org/wiki/Limit_(category_theory)?oldid=671556587Contributors: AxelBoldt,Bryan Derksen, Andre Engels, Michael Hardy, TakuyaMurata, AugPi, Mydogategodshat, Charles Matthews, Itsmeo o, Giftlite, MarkusKrtzsch, Lethe, Fropu, Waltpohl, Zhen Lin, Smimram, Guanabot, Elwikipedista~enwiki, Gauge, Chtito, 3mta3, Ynhockey, Linas,Grammarbot, Rjwilmsi, Bgohla, Tillmo, Chobot, Hairy Dude, Dmharvey, Michael Slone, Grubber, Gaius Cornelius, Buster79, Trova-tore, Yahya Abdal-Aziz, SmackBot, TimBentley, Go for it!, Breno, Mets501, Cydebot, Sam Staton, RobHar, JAnDbot, SwiftBot, R'n'B,Popx, Fwehrung, Tlepp, Alexey Muranov, Addbot, , Yobot, Magog the Ogre, Citation bot, Omnipaedista, RibotBOT, Victor-Porton, Raulshc, Kaoru Itou, FrescoBot, ComputScientist, Negi(afk), RedBot, Magmalex, EmausBot, Vincent Semeria, MerlIwBot,IkamusumeFan, Deltahedron, Mark viking, Noix07, Monkbot and Anonymous: 354.8.2 Images File:Direct_limit_category.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7c/Direct_limit_category.svg License: CCBY-SA 4.0 Contributors: Own work Original artist: IkamusumeFan File:Functor_co-cone_(extended).svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5a/Functor_co-cone_%28extended%29.svg License: Public domain Contributors: Own work Original artist: Ryan Reich File:Functor_cone_(extended).svg Source: https://upload.wikimedia.org/wikipedia/commons/8/81/Functor_cone_%28extended%29.svg License: Public domain Contributors: Own work Original artist: Ryan Reich File:Portal-puzzle.svg Source: https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg License: Public domain Contributors:? 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