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category (Definition)

The concept of category emerged in 1943-1945 from work in algebraic topology and Homological Algebra by S. Eilenberg and S. Mac Lane [29], as a generalization of the algebraic concepts of semigroup, monoid, group, groupoid, etc., as well as an extension of topological concepts and diagrams employed in algebraic topology and homological algebra. Thus many properties of mathematical systems can be unified by a presentation with diagrams of arrows that may represent functions, transformations, distributions, operators, etc., and that– in the case of concrete categories– may also include 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 [27] without using any set theory, one needs to introduce first the notions of metagraph and metacategory.

Definition 0.1   A concrete metagraph $\mathcal{M}_G$ consists of objects, $A, B, C,$... and arrows $f, g, h,$... between objects, and two operations as follows:
  • a domain operation, $dom$, which assigns to each arrow $f$ an object $A~ =~dom ~f$
  • a codomain operation, $cod$, which assigns to each arrow $f$ an object $B~ = ~cod ~f,$ represented as $f: A \to B$ or $A \stackrel{f}{\longrightarrow} B$
Definition 0.2   A metacategory $\mathbb{C}$ is a metagraph with two additional operations:
  • Identity, $id$ or 1, which assigns to each object $A$ a unique arrow $id_A$, or $1_A$;
  • Composition, $\circ$, which assigns to each pair of arrows $<g,f>$ with $dom~ g = cod~ f$ a unique arrow $g \circ f$ called their composite, such that $g \circ f : dom f \to cod g,$

that are subject to two axioms:

  • c1. (Unit law): for all arrows $f: A \to B$ and $g:B \to C$ the composition with the identity arrow $1_B$ results in

    $1_B \circ f = f$ and $g \circ 1_B = g ;$

  • c2. Associativity: for given objects and arrows in the (categorical) sequence:

    $\displaystyle A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \stackrel{h}{\longrightarrow} D , $
    one always the equality

    $\displaystyle h \circ(g \circ f) = (h \circ g) \circ f , $
    whenever the composition $\circ$ is defined.
Definition 0.3   A category $\mathcal{C}$ is an interpretation of a metacategory within set theory. Thus, a category is a graph – defined by a set $Ob \mathcal{C}:=\mathbb{O}$, a set of arrows* (called also morphisms) $Mor\mathcal{C}:= \mathbb{A}$, and two functions:

$\displaystyle dom: Mor \mathcal{C} \to Ob \mathcal{C}$
and

$\displaystyle cod: Mor\mathcal{C} \to Ob \mathcal{C},~$
– that also has two additional functions:

$\displaystyle id: Ob \mathcal{C} \to Mor \mathcal{C}$
defined by the assignments

$\displaystyle \mathbb{A} \times_\mathbb{O} \mathbb{A} \longrightarrow \mathbb{A}$
called identity, and a composition $c = \lq\lq \circ''$, that is,

$\displaystyle c \longmapsto id_c$
, defined by the assignments $(g,f) \longmapsto g \circ f$, such that:

$\displaystyle dom(id_A) = A = cod(id_A), dom(g \circ f) = domf, cod (g \circ f)= codg,$
for all objects $A \in Ob \mathcal{C}$ and all composable pairs of arrows (morphisms) $(g,f) \in \mathbb{A} \times_\mathbb{O} \mathbb{A}, $ and also such that the unit law and associativity axioms c1 and c2 hold.

*Note that the set of all morphisms $Mor~ \mathcal{C}$ of a category $\mathcal{C}$ is sometimes denoted as $\mathcal{M}$, or in French publications as $Fl ~ \mathcal{C}$.

For convenience one also defines a $Hom$ (or $hom$) set as:

$\displaystyle Hom(B,C) := [f\vert f \in \mathcal{C}, dom f= B, cod f = C],$
which is also denoted as $[B,C]_\mathcal{C}$, 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 ETAC axioms from his elementary theory of abstract categories [26]. For small categories– whose $Ob \mathcal{C}$ is a set and also $Mor\mathcal{C}$ is a set– one has a direct definition.

If, on the other hand, $Hom_\mathcal{C} (X,Y)$ is a class rather than a set then the category $\mathcal{C}$ is called large.

Bibliography

1
Baez, J. & Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: Advances in Mathematics, 135, 145–206.
2
Baez, J. & Dolan, J., 1998b, “Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1–36.
3
Baez, J. & Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in Mathematics Unlimited – 2001 and Beyond, Berlin: Springer, 29–50.
4
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.
5
Baianu, I. and M. Marinescu: 1968, Organismic Supercategories: Towards a Unitary Theory of Systems. Bulletin of Mathematical Biophysics 30, 148-159.
6
Baianu, I.C.: 1970, Organismic Supercategories: II. On Multistable Systems. Bulletin of Mathematical Biophysics, 32: 539-561.
7
Baianu,I.C. : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. Ibid., 33 (3), 339–354.
8
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.
9
I.C. Baianu: Łukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059. Health Physics and Radiation Effects (June 29, 2004).
10
I.C. Baianu and D. Scripcariu: 1973, On Adjoint Dynamical Systems. The Bulletin of Mathematical Biophysics, 35(4), 475–486.
11
I.C. Baianu: 1973, Some Algebraic Properties of (M,R) – Systems. Bulletin of Mathematical Biophysics 35, 213-217.
12
I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of (M,R)– Systems. Revue Roumaine de Mathematiques Pures et Appliquees 19: 388-391.
13
I.C. Baianu: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory. Bulletin of Mathematical Biophysics, 39: 249-258.
14
I.C. Baianu: 1980, Natural Transformations of Organismic Structures. Bulletin of Mathematical Biophysics 42: 431-446.
15
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.
16
Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued Łukasiewicz Algebras in Relation to Dynamic Bionetworks, (M,R)–Systems and Their Higher Dimensional Algebra, PDF's of Abstract and Preprint of Report.
17
Baianu, I.C. Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic networks, Axiomathes 16 Nos. 1–2, 65–122.
18
Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225.
28
Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831.
20
Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.
21
Eilenberg, S. & Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press.
22
Grothendieck, A. et al., Séminaire de Géométrie Algébrique, Vol. 1–7, Berlin: Springer-Verlag.
23
Grothendieck, A., 1957, "Sur Quelques Points d'algèbre homologique", Tohoku Mathematics Journal, 9, 119–221.
24
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.
25
Lawvere, F. W., 1965, "Algebraic Theories, Algebraic Categories, and Algebraic Functors", Theory of Models, Amsterdam: North Holland, 413–418.
26
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.
27
MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag.
28
Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., American Mathematical Society 43: 757-831.
29
Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, Transactions of the American Mathematical Society 58: 231-294.
30
Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, Complex Nonlinear Biodynamics in Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks., Axiomathes, 16 Nos. 1–2: 65–122.
31
Rosen, R.: 1958a, A Relational Theory of Biological Systems., Bulletin of Mathematical Biophysics 20: 245-260.
32
Rosen, R.: 1958b, The Representation of Biological Systems from the Standpoint of the Theory of Categories., Bulletin of Mathematical Biophysics 20: 317-341.
33
See also a more extensive category theory bibliography



"category" is owned by bci1.

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See Also: groupoid, categorical sequence, square diagram, commutative square diagram, graph, supercategory, category, categorical diagrams defined as functors, categorical physics, index of algebraic topology, category theory, axiomatic theories of metacategories and supercategories, algebraic topology

Also defines:  metagraph, metacategory, category axioms, large category, small category, 4Hom (X, Y)$, unit law, associativity axioms, identity, composition, operations, categorification, groupoidification, graph, dom, cod
Keywords:  category theory, metagraph, metacategory, algebraic topology, homology theory, category, unit law, associativity axioms, identity, composition, operations, graph, dom, cod

Attachments:
alternative definition of category (Definition) by rspuzio
alternative definition of small category (Definition) by bci1
category theory applications (Application) by bci1

Cross-references: elementary theory of abstract categories, ETAC axioms, morphisms, codomain, domain, objects, operators, functions, systems, diagrams, topological, groupoid, group, monoid, semigroup, algebraic, algebraic topology, work, concept
There are 228 references to this object.

This is version 44 of category, born on 2009-04-06, modified 2009-05-02.
Object id is 634, canonical name is Category.
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Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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