Talk:PlanetPhysics/Axiomatic Theories and Categorical Foundations of Mathematics

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

\subsection{Axiomatic Theories and Categorical Foundations of Mathematical Physics and Mathematics}

\begin{enumerate} \item Axiomatic foundations of \htmladdnormallink{adjointness}{http://planetphysics.us/encyclopedia/DualityAndTriality.html}, \htmladdnormallink{equivalence relations}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}, \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} and abstract mathematics \item Syntax, semantics and structures \item Axioms of set theory and theories of classes \item Axiomatics and logics \item Axioms of logic algebras and lattices: Post, \htmladdnormallink{\L{}ukasiewicz}{http://planetphysics.us/encyclopedia/AlgebraicCategoryOfLMnLogicAlgebras.html} and $MV$ logics \item Axioms of \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/AlgebraicTopology.html} and \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} geometry \item Axioms of abstract and universal algebras \item Abstract \htmladdnormallink{Relational Theories}{http://planetphysics.us/encyclopedia/RSystemsCategory.html}, algebraic \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} and relational structures \item Axioms of homological algebra \item Axioms of \htmladdnormallink{ETAC and category theory}{http://planetphysics.us/encyclopedia/IndexOfCategories.html} \item Axioms of \htmladdnormallink{2-categories}{http://planetphysics.us/encyclopedia/2Category.html} and \htmladdnormallink{n-categories}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} \item Axioms of Abelian structures and theories \item Axioms of \htmladdnormallink{Abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory.html} ($Ab1$ to $Ab6$, incl. $*$ axioms) \item \htmladdnormallink{Categories of logic algebras}{http://planetphysics.us/encyclopedia/AlgebraicCategoryOfLMnLogicAlgebras.html} \item \htmladdnormallink{functor categories}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and \htmladdnormallink{super-categories}{http://planetphysics.us/encyclopedia/ETAS.html} \item \htmladdnormallink{index of category theory}{http://planetphysics.us/encyclopedia/IndexOfCategoryTheory.html} \item \htmladdnormallink{axioms of topoi}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html} and extended toposes \item Axioms of \htmladdnormallink{ETAS}{http://planetphysics.us/encyclopedia/ETACAxioms.html}, \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html} and \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/HigherDimensionalAlgebra2.html} \item Axioms for \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} structures and theories \item Axioms of non-Abelian \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/AlgebraicTopology.html} \item Axioms of \htmladdnormallink{algebraic quantum field theories}{http://planetphysics.us/encyclopedia/AlgebraicQuantumFieldTheoriesAQFT.html} \item Topic entry on real numbers \item Classical and categorical Galois theories \item Axioms of model theory \item Axioms for symbolic and categorical \htmladdnormallink{computations}{http://planetphysics.us/encyclopedia/LQG2.html} \item Axioms of measure theory \item Axioms of \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} theory (e.g., algebra, \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, \htmladdnormallink{groupoid representations}{http://planetphysics.us/encyclopedia/GroupRepresentations.html}, and so on)

\item new contributed additions \end{enumerate}

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

\begin{thebibliography} {99}

\bibitem{AMF56} Atyiah, M.F. 1956. On the Krull-Schmidt theorem with applications to sheaves. \emph{Bull. Soc. Math. France}, \textbf{84}: 307--317.

\bibitem{AMF56} Auslander, M. 1965. Coherent Functors. \emph{Proc. Conf. Cat. Algebra, La Jolla}, 189--231.

\bibitem{AS-BC2k} Awodey, S. \& Butz, C., 2000, Topological Completeness for Higher Order Logic., {\em Journal of Symbolic Logic}, 65, 3, 1168--1182.

\bibitem{AS-RER2k2} Awodey, S. \& Reck, E. R., 2002, Completeness and Categoricity I. Nineteen-Century Axiomatics to Twentieth-Century Metalogic., {\em History and Philosophy of Logic}, 23, 1, 1--30.

\bibitem{AS-RER2k2} Awodey, S. \& Reck, E. R., 2002, Completeness and Categoricity II. Twentieth-Century Metalogic to Twenty-first-Century Semantics, {\em History and Philosophy of Logic}, 23, 2, 77--94.

\bibitem{BAJ-DJ97} Baez, J., 1997, An Introduction to n-Categories, {\em Category Theory and Computer Science}, Lecture Notes in Computer Science, 1290, Berlin: Springer-Verlag, 1--33.

\bibitem{ICB4} Baianu, I.C.: 1971, Categories, Functors and Quantum Algebraic Computations, in P. Suppes (ed.), \emph{Proceed. Fourth Intl. Congress Logic-Mathematics-Philosophy of Science}, September 1-4, 1971, Bucharest.

\bibitem{BJL86} Bell, J. L., 1986, From Absolute to Local Mathematics, \emph{Synthese}, 69 (3): 409--426.

\bibitem{BJL88} Bell, J. L., 1988, \emph{Toposes and Local Set Theories: An Introduction}, Oxford: Oxford University Press.

\bibitem{BG-MCLS99} Birkoff, G. and Mac Lane, S., 1999, \emph{Algebra}, 3rd ed., Providence: AMS.

\bibitem{Borceux94} Borceux, F.: 1994, \emph{Handbook of Categorical Algebra}, vols: 1--3, in {\em Encyclopedia of Mathematics and its Applications} \textbf{50} to \textbf{52}, Cambridge University Press.

\bibitem{Bourbaki1} Bourbaki, N. 1961 and 1964: \emph{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} \textbf{12}: 63-80.

\bibitem{BHR2} Brown, R., Higgins, P. J. and R. Sivera,: 2007, \emph{Non-Abelian Algebraic Topology}, \htmladdnormallink{vol. I pdf doc.}{http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf}

\bibitem{BGB2k7b} 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., \emph{Axiomathes} (17): 321--379. \bibitem{FS77} Feferman, S., 1977, Categorical Foundations and Foundations of Category Theory, in {\em Logic, Foundations of Mathematics and Computability}, R. Butts (ed.), Reidel, 149-169.

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

\bibitem{FP60} Freyd, P., 1960. Functor Theory (Dissertation). Princeton University, Princeton, New Jersey.

\bibitem{FP63} Freyd, P., 1963, Relative homological algebra made absolute. , {\em Proc. Natl. Acad. USA}, \textbf{49}:19-20.

\bibitem{FP64} Freyd, P., 1964, Abelian Categories. An Introduction to the Theory of Functors, New York and London: Harper and Row.

\bibitem{FP65} Freyd, P., 1965, The Theories of Functors and Models., {\em Theories of Models}, Amsterdam: North Holland, 107--120.

\bibitem{FP66} Freyd, P., 1966, Algebra-valued Functors in general categories and tensor product in particular., {\em Colloq. Mat}. {14}: 89--105.

\bibitem{FP72} Freyd, P., 1972, Aspects of Topoi, {\em Bulletin of the Australian Mathematical Society}, \textbf{7}: 1--76.

\bibitem{FP80} Freyd, P., 1980, The Axiom of Choice, {\em Journal of Pure and Applied Algebra}, 19, 103--125.

\bibitem{LFW65} Lawvere, F. W., 1965, Algebraic Theories, Algebraic Categories, and Algebraic Functors, {\em Theory of Models}, Amsterdam: North Holland, 413--418.

\bibitem{LFW66} Lawvere, F. W.: 1966, The Category of Categories as a Foundation for Mathematics., in \emph{Proc. Conf. Categorical Algebra- La Jolla}., Eilenberg, S. et al., eds. Springer--Verlag: Berlin, Heidelberg and New York., pp. 1-20.

\bibitem{LFW69a} Lawvere, F. W., 1969a, Diagonal Arguments and Cartesian Closed Categories, in {\em Category Theory, Homology Theory, and their Applications II}, Berlin: Springer, 134--145.

\bibitem{LFW69b} Lawvere, F. W., 1969b, Adjointness in Foundations, {\em Dialectica}, \textbf{23}: 281--295.

\bibitem{LFW70} Lawvere, F. W., 1970, Equality in Hyper doctrines and Comprehension Schema as an Adjoint Functor, {\em Applications of Categorical Algebra}, Providence: AMS, 1-14.

\bibitem{LT271} Lawvere, F. W., 1971, Quantifiers and Sheaves, {\em Actes du Congr\'es International des Math\'ematiciens}, Tome 1, Paris: Gauthier-Villars, 329--334.

\bibitem{MCLSS69} Mac Lane, S., 1969, Foundations for Categories and Sets, in {\em Category Theory, Homology Theory and their Applications II}, Berlin: Springer, 146--164.

\bibitem{MCLS71}Mac Lane, S., 1971, Categorical algebra and Set-Theoretic Foundations, in {\em Axiomatic Set Theory}, Providence: AMS, 231--240.

\bibitem{MCLS75} Mac Lane, S., 1975, Sets, Topoi, and Internal Logic in Categories, {\em Studies in Logic and the Foundations of Mathematics}, 80, Amsterdam: North Holland, 119--134.

\end{thebibliography}

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