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The [[../PreciseIdea/|concept]] of category emerged in 1943-1945 from [[../Work/|work]] in [[../CubicalHigherHomotopyGroupoid/|algebraic topology]] and Homological Algebra by S. Eilenberg and S. Mac Lane [1], as a generalization of the [[../CoIntersections/|algebraic]] concepts of [[../TrivialGroupoid/|semigroup]], [[../TrivialGroupoid/|monoid]], [[../TrivialGroupoid/|group]], [[../GroupoidHomomorphism2/|groupoid]], etc., as well as an extension of [[../CoIntersections/|topological]] concepts and [[../TrivialGroupoid/|diagrams]] employed in algebraic topology and homological algebra. Thus many properties of mathematical [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] can be unified by a presentation with [[../CommutativeSquareDiagram/|diagrams]] of arrows that may represent [[../Bijective/|functions]], transformations, distributions, [[../QuantumOperatorAlgebra4/|operators]], etc., and that-- in the case of concrete categories-- may also include [[../TrivialGroupoid/|objects]] such as class elements, sets, topological spaces, etc. ; the usefulness of such diagrams comes from the composition of the arrows and the (fundamental) axioms that define any category which allow mathematical constructions to be represented by universal properties of diagrams.

Definitions

To introduce the modern concept of category, according to S. MacLane [2] without using any set theory, one needs to introduce first the notions of metagraph and metacategory .

A concrete metagraph β„³G consists of objects, A,B,C,... and arrows f,g,h,... between objects, and two operations as follows:

A metacategory β„‚ is a metagraph with two additional operations:

  • Identity , id or {\mathbf 1}, which assigns to each object A a unique arrow idA, or 1A;
  • Composition , ∘, which assigns to each pair of arrows <g,f> with domg=codf a unique arrow g∘f called their composite , such that g∘f:domf→codg,

that are subject to two axioms:

  • c1. (Unit law) : for all arrows f:A→B and g:B→C the composition with the identity arrow 1B results in 1B∘f=f and g∘1B=g;
  • c2. Associativity : for given objects and arrows in the [[../CategoricalSequence/|(categorical) sequence]]: A⟶fB⟶gC⟶hD, one always the equality h∘(g∘f)=(h∘g)∘f, whenever the composition ∘ is defined.

A category π’ž is an interpretation of a metacategory within set theory. Thus, a category is a graph -- defined by a set Obπ’ž:=𝕆, a set of arrows* (called also [[../TrivialGroupoid/|morphisms]]) Morπ’ž:=𝔸, and two functions:

dom:Morπ’ž→Obπ’ž and

cod:Morπ’ž→Obπ’ž, -- that also has two additional functions: id:Obπ’ž→Morπ’ž defined by the assignments 𝔸×𝕆𝔸⟶𝔸 called identity , and a composition Failed to parse (syntax error): {\displaystyle c = "\circ"} , that is, c⟼idc, defined by the assignments (g,f)⟼g∘f, such that: dom(idA)=A=cod(idA),dom(g∘f)=domf,cod(g∘f)=codg, for all objects A∈Obπ’ž and all composable pairs of arrows (morphisms) (g,f)∈𝔸×𝕆𝔸, and also such that the unit law and associativity axioms c1 and c2 hold.

  • Note that the set of all morphisms Morπ’ž of a category π’ž is sometimes denoted as β„³, or in French publications as Flπ’ž.

For convenience one also defines a Hom (or hom) set as: Hom(B,C):=[f|f∈π’ž,domf=B,codf=C], which is also denoted as [B,C]π’ž, or simply [B,C].

Alternative definitions

There are several alternative definitions of a category. Thus, as defined by W.F. Lawvere, a category is an interpretation of the [[../Formula/|ETAC axioms]] from his [[../Formula/|elementary theory of abstract categories]] [3]. For small categories-- whose Obπ’ž is a set and also Morπ’ž is a set-- one has a \htmladdnormallinkdirect definition. {http://planetmath.org/encyclopedia/AlternativeDefinitionOfSmallCategory.html}

If, on the other hand, Homπ’ž(X,Y) is a class rather than a set then the category π’ž is called large .

All Sources

[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [3] [2] [22] [1] [29] [30] [31] [32]

References

  1. ↑ 1.0 1.1 Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 : 231-294.
  2. ↑ 2.0 2.1 MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.
  3. ↑ 3.0 3.1 Lawvere, F. W., 1966, "The Category of Categories as a Foundation for Mathematics", Proceedings of the Conference on Categorical Algebra, La Jolla, New York: Springer-Verlag, 1--21.
  4. ↑ Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: Advances in Mathematics , 135, 145--206.
  5. ↑ Baez, J. \& Dolan, J., 1998b, ``Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1--36.
  6. ↑ Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited -- 2001 and Beyond , Berlin: Springer, 29--50.
  7. ↑ Baez, J., 1997, An Introduction to n-Categories, in Category Theory and Computer Science, Lecture Notes in Computer Science , 1290, Berlin: Springer-Verlag, 1--33.
  8. ↑ Baianu, I. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. Bulletin of Mathematical Biophysics 30 , 148-159.
  9. ↑ Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics , 32 : 539-561.
  10. ↑ Baianu,I.C. : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Ibid. , 33 (3), 339--354.
  11. ↑ I.C. Baianu: 1971b, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science , September 1--4, 1971, University of Bucharest.
  12. ↑ I.C. Baianu: \L ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
  13. ↑ I.C. Baianu and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin of Mathematical Biophysics , 35 (4), 475--486.
  14. ↑ I.C. Baianu: 1973, Some Algebraic Properties of (M,R) -- Systems. Bulletin of Mathematical Biophysics 35 , 213-217.
  15. ↑ I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of (M,R) -- Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19 : 388-391.
  16. ↑ I.C. Baianu: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics , 39 : 249-258.
  17. ↑ I.C. Baianu: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42 : 431-446.
  18. ↑ Baianu, I.C.: 1987. Mathematical Models in Medicine , vol. 7., Ch.11 Pergamon Press, New York, 1513 -1577; URLs: CERN Preprint No. EXT-2004-072: , available here as PDF, or as as an archived html document.
  19. ↑ Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R) --Systems and Their Higher Dimensional Algebra, [\\http://www.ag.uiuc.edu/fs401/QAuto.pdf PDF's of Abstract and Preprint of Report].
  20. ↑ Baianu, I.C. Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic networks, Axiomathes 16 Nos. 1--2, 65--122.
  21. ↑ Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
  22. ↑ 22.0 22.1 Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43 : 757-831.
  23. ↑ Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58 : 231-294.
  24. ↑ Eilenberg, S. \& Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press.
  25. ↑ Grothendieck, A. et al., S\'eminaire de G\'eom\'etrie Alg\'ebrique, Vol. 1--7, Berlin: Springer-Verlag.
  26. ↑ Grothendieck, A., 1957, "Sur Quelques Points d'alg\`ebre homologique", Tohoku Mathematics Journal, 9, 119--221.
  27. ↑ Lawvere, F. W., 1964, "An Elementary Theory of the Category of Sets", Proceedings of the National Academy of Sciences U.S.A., 52, 1506--1511.
  28. ↑ Lawvere, F. W., 1965, "Algebraic Theories, Algebraic Categories, and Algebraic Functors", Theory of Models, Amsterdam: North Holland, 413--418.
  29. ↑ Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and \L ukasiewicz--Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes , 16 Nos. 1--2: 65--122.
  30. ↑ Rosen, R.: 1958a, A Relational Theory of Biological Systems., Bulletin of Mathematical Biophysics 20 : 245-260.
  31. ↑ Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., Bulletin of Mathematical Biophysics 20 : 317-341.
  32. ↑ See also a more extensive [[../BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics/|category theory bibliography]]

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