glossary of category theory

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Glossary of category theory Wikipedia This is a glossary of properties and concepts in category theory in mathematics. 1 Categories A category A is said to be: small if the class of all morphisms is a set (i.e., not a proper class); otherwise large. locally small if the morphisms between every pair of objects A and B form a set. Some authors assume a foundation in which the collection of all classes forms a “conglomerate”, in which case a quasicategory is a category whose objects and morphisms merely form a conglomerate. [1] (NB other authors use the term “quasicategory” with a different meaning. [2] ) isomorphic to a category B if there is an isomorphism between them. equivalent to a category B if there is an equivalence between them. concrete if there is a faithful functor from A to Set; e.g., Vec, Grp and Top. discrete if each morphism is an identity morphism (of some object). thin category if there is at most one morphism between any pair of objects. a subcategory of a category B if there is an inclusion functor given from A to B. a full subcategory of a category B if the inclusion functor is full. wellpowered if for each object A there is only a set of pairwise non-isomorphic subobjects. complete if all small limits exist. cartesian closed if it has a terminal object and that any two objects have a product and exponential. abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. normal if every monic is normal. [3] balanced if every bimorphism is an isomorphism. preadditive if it is enriched over the monoidal category of abelian groups. More generally, it is R-linear if it is enriched over the monoidal category of R-modules, for R a commutative ring. additive if it is preadditive and admits all finitary biproducts. skeletal if isomorphic objects are necessarily identical. 1

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  • Glossary of category theoryWikipedia

    This is a glossary of properties and concepts in category theory in mathematics.

    1 CategoriesA category A is said to be:

    small if the class of all morphisms is a set (i.e., not a proper class); otherwise large.

    locally small if the morphisms between every pair of objects A and B form a set.

    Some authors assume a foundation in which the collection of all classes forms a conglomerate, in which casea quasicategory is a category whose objects and morphisms merely form a conglomerate.[1] (NB other authorsuse the term quasicategory with a dierent meaning.[2])

    isomorphic to a category B if there is an isomorphism between them.

    equivalent to a category B if there is an equivalence between them.

    concrete if there is a faithful functor from A to Set; e.g., Vec, Grp and Top.

    discrete if each morphism is an identity morphism (of some object).

    thin category if there is at most one morphism between any pair of objects.

    a subcategory of a category B if there is an inclusion functor given from A to B.

    a full subcategory of a category B if the inclusion functor is full.

    wellpowered if for each object A there is only a set of pairwise non-isomorphic subobjects.

    complete if all small limits exist.

    cartesian closed if it has a terminal object and that any two objects have a product and exponential.

    abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphismsare normal.

    normal if every monic is normal.[3]

    balanced if every bimorphism is an isomorphism.

    preadditive if it is enriched over the monoidal category of abelian groups. More generally, it is R-linear if itis enriched over the monoidal category of R-modules, for R a commutative ring.

    additive if it is preadditive and admits all nitary biproducts.

    skeletal if isomorphic objects are necessarily identical.

    1

  • 2 4 OBJECTS

    2 MorphismsA morphism f in a category is called:

    an epimorphism if g = h whenever g f = h f . In other words, f is the dual of a monomorphism. an identity if f maps an object A to A and for any morphisms g with domain A and h with codomain A,g f = g and f h = h .

    an inverse to a morphism g if g f is dened and is equal to the identity morphism on the codomain of g,and f g is dened and equal to the identity morphism on the domain of g. The inverse of g is unique and isdenoted by g1. f is a left inverse to g if f g is dened and is equal to the identity morphism on the domainof g, and similarly for a right inverse.

    an isomorphism if there exists an inverse of f. a monomorphism (also called monic) if g = h whenever f g = f h ; e.g., an injection in Set. In otherwords, f is the dual of an epimorphism.

    a bimorphism is a morphism that is both an epimorphism and a monomorphism. a retraction if it has a right inverse. a coretraction if it has a left inverse.

    3 FunctorsA functor F is said to be:

    a constant if F maps every object in a category to the same object A and every morphism to the identity on A. faithful if F is injective when restricted to each hom-set. full if F is surjective when restricted to each hom-set. isomorphism-dense (sometimes called essentially surjective) if for every B there exists A such that F(A) isisomorphic to B.

    an equivalence if F is faithful, full and isomorphism-dense. amnestic provided that if k is an isomorphism and F(k) is an identity, then k is an identity. reect identities provided that if F(k) is an identity then k is an identity as well. reect isomorphisms provided that if F(k) is an isomorphism then k is an isomorphism as well.

    4 ObjectsAn object A in a category is said to be:

    isomorphic to an object B if there is an isomorphism between A and B. initial if there is exactly one morphism from A to each object B; e.g., empty set in Set. terminal if there is exactly one morphism from each object B to A; e.g., singletons in Set. a zero object if it is both initial and terminal, such as a trivial group in Grp.

    An object A in an abelian category is:

    simple if it is not isomorphic to the zero object and any subobject of A is isomorphic to zero or to A. nite length if it has a composition series. Themaximumnumber of proper subobjects in any such compositionseries is called the length of A.[4]

  • 35 Notes[1] Admek, Ji; Herrlich, Horst; Strecker, George E (2004) [1990]. Abstract and Concrete Categories (The Joy of Cats)

    (PDF). New York: Wiley & Sons. p. 40. ISBN 0-471-60922-6.

    [2] Joyal, A. (2002). Quasi-categories and Kan complexes. Journal of Pure and Applied Algebra 175 (1-3): 207222.doi:10.1016/S0022-4049(02)00135-4.

    [3] http://planetmath.org/encyclopedia/NormalCategory.html

    [4] Kashiwara & Schapira 2006, exercise 8.20

    6 References Kashiwara, Masaki; Schapira, Pierre (2006). Categories and sheaves Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). New York, NY: Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.

    Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topol-ogy, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications 97. Cambridge: CambridgeUniversity Press. ISBN 0-521-83414-7. Zbl 1034.18001.

  • 4 7 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

    7 Text and image sources, contributors, and licenses7.1 Text

    Glossary of category theory Source: http://en.wikipedia.org/wiki/Glossary_of_category_theory?oldid=663243528 Contributors: Zun-dark, B4hand, TakuyaMurata, Ciphergoth, Fropu, DemonThing, Zzo38, Paul August, Spayrard, Ntmatter, Kierano, RuudKoot, SixWinged-Seraph, Salix alba, Hairy Dude, Michael Slone, Benja, Kompik, Benandorsqueaks, Unint, Xyzzyplugh, Dr Greg, CmdrObot, Cydebot,Sam Staton, Ameliorate!, RobHar, Conniption, Isilanes, CopyToWiktionaryBot, R'n'B, Squids and Chips, Cenarium, Addbot, Zulon,FarmerDavid, Yobot, Citation bot, A Hubery, Citation bot 1, Summsumm, Trappist the monk, EmausBot, Rfs2, Deltahedron, Cwobeeland Anonymous: 9

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