# PlanetPhysics/Axiomatic Theories and Categorical Foundations of Mathematics

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

### Axiomatic Theories and Categorical Foundations of Mathematical Physics and Mathematics

* Axiomatic foundations of adjointness, equivalence relations, isomorphism and abstract mathematics
* Syntax, semantics and structures
* Axioms of set theory and theories of classes
* Axiomatics and logics
* Axioms of logic algebras and lattices: Post, \htmladdnormallink{\Lukasiewicz}{http://planetphysics.us/encyclopedia/AlgebraicCategoryOfLMnLogicAlgebras.html} and $MV$ logics
* Axioms of algebraic topology and algebraic geometry
* Axioms of abstract and universal algebras
* Abstract Relational Theories, algebraic systems and relational structures
* Axioms of homological algebra
* Axioms of ETAC and category theory
* Axioms of 2-categories and n-categories #Axioms of Abelian structures and theories
* Axioms of Abelian categories ($Ab1$ to $Ab6$, incl. $*$ axioms)
* Categories of logic algebras
* functor categories and super-categories
* index of category theory #axioms of topoi and extended toposes
* Axioms of ETAS, supercategories and higher dimensional algebra #Axioms for non-Abelian structures and theories
* Axioms of non-Abelian algebraic topology
* Axioms of algebraic quantum field theories
* Topic entry on real numbers
* Classical and categorical Galois theories
* Axioms of model theory
* Axioms for symbolic and categorical computations #Axioms of measure theory
* Axioms of representation theory (e.g., algebra, group, groupoid representations,
and so on)
* new contributed additions
Note The following page is only a short list of relevant papers. A more substantial bibliography is now being compiled separately.
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* 1 2 Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. Bull. Soc. Math. France , 84 : 307--317.
* ↑ Awodey, S. \& Butz, C., 2000, Topological Completeness for Higher Order Logic., Journal of Symbolic Logic , 65, 3, 1168--1182.
* 1 2 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.
* ↑ Baez, J., 1997, An Introduction to n-Categories, Category Theory and Computer Science , Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.
* ↑ 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.
* ↑ Bell, J. L., 1986, From Absolute to Local Mathematics, Synthese , 69 (3): 409--426.
* ↑ Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction , Oxford: Oxford University Press.
* ↑ Birkoff, G. and Mac Lane, S., 1999, Algebra , 3rd ed., Providence: AMS.
* ↑ Borceux, F.: 1994, Handbook of Categorical Algebra , vols: 1--3, in Encyclopedia of Mathematics and its Applications 50 to 52 , Cambridge University Press.
* ↑ 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.
* ↑ Brown, R., Higgins, P. J. and R. Sivera,: 2007, \emph{Non-Abelian Algebraic Topology}, vol. I pdf doc.
* ↑ 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.
* ↑ Feferman, S., 1977, Categorical Foundations and Foundations of Category Theory, in Logic, Foundations of Mathematics and Computability , R. Butts (ed.), Reidel, 149-169.
* ↑ Fell, J. M. G., 1960, The Dual Spaces of C*-Algebras, Transactions of the American Mathematical Society , 94: 365-403.
* ↑ Freyd, P., 1960. Functor Theory (Dissertation). Princeton University, Princeton, New Jersey.
* ↑ Freyd, P., 1963, Relative homological algebra made absolute. , Proc. Natl. Acad. USA , 49 :19-20.
* ↑ Freyd, P., 1964, Abelian Categories. An Introduction to the Theory of Functors, New York and London: Harper and Row.
* ↑ Freyd, P., 1965, The Theories of Functors and Models., Theories of Models , Amsterdam: North Holland, 107--120.
* ↑ Freyd, P., 1966, Algebra-valued Functors in general categories and tensor product in particular., Colloq. Mat . {14}: 89--105.
* ↑ Freyd, P., 1972, Aspects of Topoi, Bulletin of the Australian Mathematical Society , 7 : 1--76.
* ↑ Freyd, P., 1980, The Axiom of Choice, Journal of Pure and Applied Algebra , 19, 103--125.
* ↑ Lawvere, F. W., 1965, Algebraic Theories, Algebraic Categories, and Algebraic Functors, Theory of Models , Amsterdam: North Holland, 413--418.
* ↑ 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.
* ↑ Lawvere, F. W., 1969a, Diagonal Arguments and Cartesian Closed Categories, in Category Theory, Homology Theory, and their Applications II , Berlin: Springer, 134--145.
* ↑ Lawvere, F. W., 1969b, Adjointness in Foundations, Dialectica , 23 : 281--295.
* ↑ Lawvere, F. W., 1970, Equality in Hyper doctrines and Comprehension Schema as an Adjoint Functor, Applications of Categorical Algebra , Providence: AMS, 1-14.
* ↑ Lawvere, F. W., 1971, Quantifiers and Sheaves, Actes du Congr\'es International des Math\'ematiciens , Tome 1, Paris: Gauthier-Villars, 329--334.
* ↑ Mac Lane, S., 1969, Foundations for Categories and Sets, in Category Theory, Homology Theory and their Applications II , Berlin: Springer, 146--164.
* ↑ Mac Lane, S., 1971, Categorical algebra and Set-Theoretic Foundations, in Axiomatic Set Theory , Providence: AMS, 231--240.
* ↑ 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.