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 adjointness, equivalence relations, 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, Łukasiewicz and logics
6. Axioms of algebraic topology and algebraic geometry
7. Axioms of abstract and universal algebras
8. Abstract Relational Theories, algebraic systems and relational structures
9. Axioms of homological algebra
10. Axioms of ETAC and category theory
11. Axioms of 2-categories and n-categories
12. Axioms of Abelian structures and theories
13. Axioms of Abelian categories ( to , incl. axioms)
14. Categories of logic algebras
15. functor categories and super-categories
16. index of category theory
17. axioms of topoi and extended toposes
18. Axioms of ETAS, supercategories and higher dimensional algebra
19. Axioms for non-Abelian structures and theories
20. Axioms of non-Abelian algebraic topology
21. Axioms of algebraic quantum field theories
22. Topic entry on real numbers
23. Classical and categorical Galois theories
24. Axioms of model theory
25. Axioms for symbolic and categorical computations
26. Axioms of measure theory
27. Axioms of representation theory (e.g., algebra, group, groupoid representations, and so on)
28. new contributed additions

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

Bibliography

1

Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France, 84: 307–317.

1

Auslander, M. 1965. Coherent Functors. Proc. Conf. Cat. Algebra, La Jolla, 189–231.

2

Awodey, S. & Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic, 65, 3, 1168–1182.

3

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.

3

Awodey, S. & Reck, E. R., 2002, Completeness and Categoricity II. Twentieth-Century Metalogic to Twenty-first-Century Semantics, History and Philosophy of Logic, 23, 2, 77–94.

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: Algèbre commutative., in Èléments de Mathématique., Chs. 1–6., Hermann: Paris.

11

BJk4) Brown, R. and G. Janelidze: 2004, Galois theory and a new homotopy double groupoid of a map of spaces, Applied Categorical Structures 12: 63-80.

12

Brown, R., Higgins, P. J. and R. Sivera,: 2007, Non-Abelian Algebraic Topology, vol. I pdf doc.

13

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.

14

Feferman, S., 1977, Categorical Foundations and Foundations of Category Theory, in Logic, Foundations of Mathematics and Computability, R. Butts (ed.), Reidel, 149-169.

15

Fell, J. M. G., 1960, The Dual Spaces of C*-Algebras, Transactions of the American Mathematical Society, 94: 365-403.

16

Freyd, P., 1960. Functor Theory (Dissertation). Princeton University, Princeton, New Jersey.

17

Freyd, P., 1963, Relative homological algebra made absolute. , Proc. Natl. Acad. USA, 49:19-20.

18

Freyd, P., 1964, Abelian Categories. An Introduction to the Theory of Functors, New York and London: Harper and Row.

19

Freyd, P., 1965, The Theories of Functors and Models., Theories of Models, Amsterdam: North Holland, 107–120.

20

Freyd, P., 1966, Algebra-valued Functors in general categories and tensor product in particular., Colloq. Mat. 14: 89–105.

21

Freyd, P., 1972, Aspects of Topoi, Bulletin of the Australian Mathematical Society, 7: 1–76.

22

Freyd, P., 1980, The Axiom of Choice, Journal of Pure and Applied Algebra, 19, 103–125.

23

Lawvere, F. W., 1965, Algebraic Theories, Algebraic Categories, and Algebraic Functors, Theory of Models, Amsterdam: North Holland, 413–418.

24

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.

25

Lawvere, F. W., 1969a, Diagonal Arguments and Cartesian Closed Categories, in Category Theory, Homology Theory, and their Applications II, Berlin: Springer, 134–145.

26

Lawvere, F. W., 1969b, Adjointness in Foundations, Dialectica, 23: 281–295.

27

Lawvere, F. W., 1970, Equality in Hyper doctrines and Comprehension Schema as an Adjoint Functor, Applications of Categorical Algebra, Providence: AMS, 1-14.

28

Lawvere, F. W., 1971, Quantifiers and Sheaves, Actes du Congrés International des Mathématiciens, Tome 1, Paris: Gauthier-Villars, 329–334.

29

Mac Lane, S., 1969, Foundations for Categories and Sets, in Category Theory, Homology Theory and their Applications II, Berlin: Springer, 146–164.

30

Mac Lane, S., 1971, Categorical algebra and Set-Theoretic Foundations, in Axiomatic Set Theory, Providence: AMS, 231–240.

31

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.