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56 pages, year 2007

Abstract:

Lawvere's Elementary Theory of Abstract Categories (ETAC) provides an axiomatic construction of the theory of categories and functors. Intuitively, with this terminology and axioms, a category is meant to be any structure which is a direct interpretation of ETAC. A functor is then understood to be a triple consisting of two such categories and of a rule F (`the functor') which assigns to each arrow or morphism x of the first category, a unique morphism, written as `F(x)' of the second category, in such a way that the usual two conditions on both objects and arrows in the standard functor definition are fulfilled --the functor is well behaved, i.e., it carries object identities to image object identities, and commutative diagrams to image commmutative diagrams of the corresponding image objects and image morphisms. At the next level, one then defines natural transformations or functorial morphisms between functors as meta-level abbreviated formulas and equations pertaining to commutative diagrams of the distinct images of two functors acting on both objects and morphisms. As the name indicates natural transformations are also well--behaved in terms of the ETAC equations that are satisfied by natural transformations. REFERENCES: W. F. Lawvere: 1966. The Category of Categories as a Foundation for Mathematics. , In Proc. Conf. Categorical Algebra--La Jolla, 1965, Eilenberg, S et al., eds. Springer --Verlag: Berlin, Heidelberg and New York, pp. 1--20. Other Relevant Publications by Willaim F. Lawvere: 1. Functorial Semantics of Algebraic Theories, Proceedings of the National Academy of Science 50, No. 5 (November 1963), 869-872. (W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories. Proc. Natl. Acad. Sci. USA, 50: 869--872;) 2. Elementary Theory of the Category of Sets, Proceedings of the National Academy of Science 52, No. 6 (December 1964), 1506-1511. 3. Algebraic Theories, Algebraic Categories, and Algebraic Functors, Theory of Models; North-Holland, Amsterdam (1965), 413-418. 4. Functorial Semantics of Elementary Theories, Journal of Symbolic Logic, Abstract, Vol. 31 (1966), 294-295. 5. The Category of Categories as a Foundation for Mathematics, La Jolla Conference on Categorical Algebra, Springer-Verlag (1966), 1-20. 6. Some Algebraic Problems in the Context of Functorial Semantics of Algebraic Theories, Springer Lecture Notes in Mathematics No. 61, Springer-Verlag (1968), 41-61. 7. Ordinal Sums and Equational Doctrines, Springer Lecture Notes in Mathematics No. 80, Springer-Verlag (1969), 141-155. 8. Diagonal Arguments and Cartesian Closed Categories, Springer Lecture Notes in Mathematics No. 92, Springer-Verlag (1969), 134-145. 9. Adjointness in Foundations, Dialectica 23 (1969), 281-296. 10. Equality in Hyperdoctrines and Comprehension Schema as an Adjoint Functor, Proceedings of the American Mathematical Society Symposium on Pure Mathematics XVII (1970), 1-14. . . . 29. Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes, Category Theory, Proceedings Como 1990. Springer Lecture Notes in Mathematics 1488, Springer-Verlag (1991) 279-281. 30. Some Thoughts on the Future of Category Theory, Category Theory, Proceedings Como 1990. Springer Lecture Notes in Mathematics 1488, Springer-Verlag (1991) 1-13. 31. Categories of Space and of Quantity, International Symposium on Structures in Mathematical Theories (1990), San Sebastian, Spain; The Space of Mathematics: Philosophical, Epistemological and Historical Explorations, DeGruyter, Berlin (1992), 14-30. 32. Tools for the Advancement of Objective Logic: Closed Categories and Toposes, J. Macnamara & G. E. Reyes (Eds). The Logical Foundations of Cognition, Oxford University Press (1994), 43-56. 33. Cohesive Toposes and Cantor's "lauter Einsen", Philosophia Mathematica, The Canadian Society for History and Philosophy of Mathematics, Series III, Vol. 2 (1994), 5-15. 34. Theory of Mathematical Categories, (1995) Encyclopedia article, publication delayed. 35. Adjoints in and among Bicategories, Proceedings of the 1994 Siena Conference in Memory of Roberto Magari. Logic & Algebra, Lecture Notes in Pure and Applied Algebra 180: 181-189, Ed. Ursini/Aglian, Marcel Dekker, Inc.: Basel, New York, 1996. I.C. Baianu: 1970, Organismic Supercategories: II. On Multistable Systems. \emph{Bulletin of Mathematical Biophysics}, \textbf{32}: 539-561. I.C. Baianu : 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. \emph{Ibid.}, \textbf{33} (3), 339--354. I.C. Baianu: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory. \emph{Bulletin of Mathematical Biophysics}, \textbf{39}: 249-258. I.C. Baianu: 1980, Natural Transformations of Organismic Structures. \emph{Bulletin of Mathematical Biophysics} \textbf{42}: 431-446. I.C. Baianu, 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, \emph{Axiomathes} \textbf{16} Nos. 1--2, 65--122. R. Brown R, P.J. Higgins, and R. Sivera.: \textit{``Non--Abelian Algebraic Topology''}, . http://www. bangor.ac.uk/mas010/nonab--a--t.html; http://www.bangor.ac.uk/mas010/nonab--t/partI010604.pdf (\textit{in preparation}).(2008). R. Brown, J. F. Glazebrook and I. C. Baianu: A categorical and higher dimensional algebra framework for complex systems and spacetime structures, \emph{Axiomathes} \textbf{17}:409--493. (2007). R. Brown and C.B. Spencer: Double groupoids and crossed modules, \emph{Cahiers Top. G\'eom.Diff.} \textbf{17} (1976), 343--362. L. L\"ofgren: 1968. On Axiomatic Explanation of Complete Self--Reproduction. \emph{Bull. Math. Biophysics}, \textbf{30}: 317--348. S. MacLane. 2000. Ch.1: Axioms for Categories, in \emph{Categories for the Working Mathematician}. Springer: Berlin, 2nd Edition. =================== AMS MSC: 18-00 (Category theory; homological algebra :: General reference works ) 55N40 (Algebraic topology :: Homology and cohomology theories :: Axioms for homology theory and uniqueness theorems) 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories) http://mathforum.org/library/view/4348.html http://www.bangor.ac.uk/~mas010/ http://mathforum.org/library/topics/alg_topol/ http://www.bangor.ac.uk/~mas010/pdffiles/BBG-158-NAC0STQG.pdf http://www.bangor.ac.uk/~mas010/publicfull.htm

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