PlanetPhysics/Axiomatic Theories and Categorical Foundations of Mathematics

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This is a contributed topic entry on the axiomatic foundations of mathematics.

Axiomatic Theories and Categorical Foundations of Mathematical Physics and Mathematics

  1. Axiomatic foundations of [[../DualityAndTriality/|adjointness]], [[../GroupoidHomomorphism2/|equivalence relations]], [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]] and abstract mathematics
  2. Syntax, semantics and structures
  3. Axioms of set theory and theories of classes
  4. Axiomatics and logics
  5. Axioms of logic algebras and lattices: Post, \htmladdnormallink{\L{}ukasiewicz}{http://planetphysics.us/encyclopedia/AlgebraicCategoryOfLMnLogicAlgebras.html} and MV logics
  6. Axioms of [[../AlgebraicTopology/|algebraic topology]] and [[../CoIntersections/|algebraic]] geometry
  7. Axioms of abstract and universal algebras
  8. Abstract [[../RSystemsCategory/|Relational Theories]], algebraic [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|systems]] and relational structures
  9. Axioms of homological algebra
  10. Axioms of [[../IndexOfCategories/|ETAC and category theory]]
  11. Axioms of [[../2Category/|2-categories]] and [[../InfinityGroupoid/|n-categories]] #Axioms of Abelian structures and theories
  12. Axioms of [[../AbelianCategory/|Abelian categories]] (Ab1 to Ab6, incl. * axioms)
  13. [[../AlgebraicCategoryOfLMnLogicAlgebras/|Categories of logic algebras]]
  14. [[../TrivialGroupoid/|functor categories]] and [[../ETAS/|super-categories]]
  15. [[../IndexOfCategoryTheory/|index of category theory]] #[[../GrothendieckTopos/|axioms of topoi]] and extended toposes
  16. Axioms of [[../ETACAxioms/|ETAS]], [[../SuperCategory6/|supercategories]] and [[../HigherDimensionalAlgebra2/|higher dimensional algebra]] #Axioms for [[../AbelianCategory3/|non-Abelian]] structures and theories
  17. Axioms of non-Abelian [[../AlgebraicTopology/|algebraic topology]]
  18. Axioms of [[../AlgebraicQuantumFieldTheoriesAQFT/|algebraic quantum field theories]]
  19. Topic entry on real numbers
  20. Classical and categorical Galois theories
  21. Axioms of model theory
  22. Axioms for symbolic and categorical [[../LQG2/|computations]] #Axioms of measure theory
  23. Axioms of [[../CategoricalGroupRepresentation/|representation]] theory (e.g., algebra, [[../TrivialGroupoid/|group]], [[../GroupRepresentations/|groupoid representations]],

and so on)

  1. new contributed additions

Note The following page is only a short list of relevant papers. A more substantial bibliography is now being compiled separately.

\begin{thebibliography} {99}

</ref>[1][1][2][3][3][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][29][30]</references>

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  1. ↑ 1.0 1.1 Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France , 84 : 307--317. Cite error: Invalid <ref> tag; name "AMF56" defined multiple times with different content
  2. ↑ Awodey, S. \& Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic , 65, 3, 1168--1182.
  3. ↑ 3.0 3.1 Awodey, S. \& Reck, E. R., 2002, Completeness and Categoricity I. Nineteen-Century Axiomatics to Twentieth-Century Metalogic., History and Philosophy of Logic , 23, 1, 1--30. Cite error: Invalid <ref> tag; name "AS-RER2k2" defined multiple times with different content
  4. ↑ Baez, J., 1997, An Introduction to n-Categories, Category Theory and Computer Science , Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.
  5. ↑ Baianu, I.C.: 1971, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science , September 1-4, 1971, Bucharest.
  6. ↑ Bell, J. L., 1986, From Absolute to Local Mathematics, Synthese , 69 (3): 409--426.
  7. ↑ Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction , Oxford: Oxford University Press.
  8. ↑ Birkoff, G. and Mac Lane, S., 1999, Algebra , 3rd ed., Providence: AMS.
  9. ↑ Borceux, F.: 1994, Handbook of Categorical Algebra , vols: 1--3, in Encyclopedia of Mathematics and its Applications 50 to 52 , Cambridge University Press.
  10. ↑ Bourbaki, N. 1961 and 1964: Alg\`{e bre commutative.}, in \`{E}l\'{e}ments de Math\'{e}matique., Chs. 1--6., Hermann: Paris. \bibitem (BJk4) Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, \emph{Applied Categorical Structures} 12 : 63-80.
  11. ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2007, \emph{Non-Abelian Algebraic Topology}, vol. I pdf doc.
  12. ↑ Brown, R., Glazebrook, J. F. and I.C. Baianu.: 2007, A Conceptual, Categorical and Higher Dimensional Algebra Framework of Universal Ontology and the Theory of Levels for Highly Complex Structures and Dynamics., Axiomathes (17): 321--379.
  13. ↑ Feferman, S., 1977, Categorical Foundations and Foundations of Category Theory, in Logic, Foundations of Mathematics and Computability , R. Butts (ed.), Reidel, 149-169.
  14. ↑ Fell, J. M. G., 1960, The Dual Spaces of C*-Algebras, Transactions of the American Mathematical Society , 94: 365-403.
  15. ↑ Freyd, P., 1960. Functor Theory (Dissertation). Princeton University, Princeton, New Jersey.
  16. ↑ Freyd, P., 1963, Relative homological algebra made absolute. , Proc. Natl. Acad. USA , 49 :19-20.
  17. ↑ Freyd, P., 1964, Abelian Categories. An Introduction to the Theory of Functors, New York and London: Harper and Row.
  18. ↑ Freyd, P., 1965, The Theories of Functors and Models., Theories of Models , Amsterdam: North Holland, 107--120.
  19. ↑ Freyd, P., 1966, Algebra-valued Functors in general categories and tensor product in particular., Colloq. Mat . {14}: 89--105.
  20. ↑ Freyd, P., 1972, Aspects of Topoi, Bulletin of the Australian Mathematical Society , 7 : 1--76.
  21. ↑ Freyd, P., 1980, The Axiom of Choice, Journal of Pure and Applied Algebra , 19, 103--125.
  22. ↑ Lawvere, F. W., 1965, Algebraic Theories, Algebraic Categories, and Algebraic Functors, Theory of Models , Amsterdam: North Holland, 413--418.
  23. ↑ Lawvere, F. W.: 1966, The Category of Categories as a Foundation for Mathematics., in Proc. Conf. Categorical Algebra- La Jolla ., Eilenberg, S. et al., eds. Springer--Verlag: Berlin, Heidelberg and New York., pp. 1-20.
  24. ↑ Lawvere, F. W., 1969a, Diagonal Arguments and Cartesian Closed Categories, in Category Theory, Homology Theory, and their Applications II , Berlin: Springer, 134--145.
  25. ↑ Lawvere, F. W., 1969b, Adjointness in Foundations, Dialectica , 23 : 281--295.
  26. ↑ Lawvere, F. W., 1970, Equality in Hyper doctrines and Comprehension Schema as an Adjoint Functor, Applications of Categorical Algebra , Providence: AMS, 1-14.
  27. ↑ Lawvere, F. W., 1971, Quantifiers and Sheaves, Actes du Congr\'es International des Math\'ematiciens , Tome 1, Paris: Gauthier-Villars, 329--334.
  28. ↑ Mac Lane, S., 1969, Foundations for Categories and Sets, in Category Theory, Homology Theory and their Applications II , Berlin: Springer, 146--164.
  29. ↑ Mac Lane, S., 1971, Categorical algebra and Set-Theoretic Foundations, in Axiomatic Set Theory , Providence: AMS, 231--240.
  30. ↑ Mac Lane, S., 1975, Sets, Topoi, and Internal Logic in Categories, Studies in Logic and the Foundations of Mathematics , 80, Amsterdam: North Holland, 119--134.
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