Category 2009-04-06 14:27:19 2009-04-06 16:02:57 Definition metagraph metacategory unit law associativity axioms identity composition operations graph dom cod category theory metagraph metacategory category unit law associativity axioms identity composition operations graph dom cod % this is the default PlanetPhysics preamble. as your \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} The concept of category emerged in 1943-1945 from work in Algebraic Topology and Homological Algebra by S. Eilenberg and S. Mac Lane, as a generalization of the algebraic concepts of group, semigroup, groupoid, as well as of the topological concepts and diagrams employed in homological algebra. Thus many properties of mathematical systems can be unified by a presentation with diagrams of arrows that may represent functions, transformations, distributions, operators, etc., and that-- in the case of concrete categories-- may also include objects such as class elements, sets, topological spaces, etc. ; the usefulness of such diagrams comes from the composition of the arrows and the (fundamental) axioms that define any category which allow mathematical constructions to be represented by universal properties of diagrams. To introduce the modern concept of category, according to S. MacLane \cite{MacLane2000} without using any set theory, one needs to introduce first the notions of {\em metagraph} and {\em metacategory}. \begin{definition} A concrete \emph{metagraph} $\mathcal{M}_G$ consists of objects, $A, B, C,$... and arrows $f, g, h,$... between objects, and two operations as follows: \begin{itemize} \item a {\em Domain operation}, $dom$, which assigns to each arrow $f$ an object $A~ =~dom ~f$ \item a {\em Codomain operation}, $cod$, which assigns to each arrow $f$ an object $B~ = ~cod ~f,$ represented as $f: A \to B$ or $A \stackrel{f}{\longrightarrow} B$ \end{itemize} \end{definition} \begin{definition} A \emph{metacategory} $\mathbb{C}$ is a metagraph with two additional operations: \begin{itemize} \item {\em Identity}, $id$ or {\bf 1}, which assigns to each object $A$ a unique arrow $id_A$, or $1_A$; \item {\em Composition}, $\circ$, which assigns to each pair of arrows $<g,f>$ with $dom~ g = cod~ f$ a unique arrow $g \circ f$ called their \emph{composite}, such that $g \circ f : dom f \to cod g,$ \end{itemize} that are subject to two axioms: \begin{itemize} \item {\em c1. (Unit law)}: for all arrows $f: A \to B$ and $g:B \to C$ the composition with the identity arrow $1_B$ results in $ 1_B \circ f = f$ and $g \circ 1_B = g ;$ \item {\em c2. Associativity}: for given objects and arrows in the sequence: $$A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \stackrel{h}{\longrightarrow} D , $$ one always the equality $$ h \circ(g \circ f) = (h \circ g) \circ f , $$ whenever the composition $\circ$ is defined. \end{itemize} \end{definition} \begin{definition} A \emph{category} $\mathcal{C}$ is an interpretation of a metacategory within set theory. Thus, a {\em category} is a {\em graph} defined by a set $Ob \mathcal{C}:=\mathbb{O}$, a set of arrows (called also \emph{morphisms}) $Mor\mathcal{C}:= \mathbb{A}$, and two functions: $$ dom: Mor \mathcal{C} \to Ob \mathcal{C}$$ and $$cod: Mor\mathcal{C} \to Ob \mathcal{C},$$ with two additional functions: $$id: Ob \mathcal{C} \to Mor \mathcal{C}$$ defined by the assignments $\mathbb{A} \times_\mathbb{O} \mathbb{A} \longrightarrow \mathbb{A}$ called \emph{identity}, and a composition $c = \circ$, that is $ c \to id_c$, defined by the assignments $(g,f) \longrightarrow g \circ f$, such that $$ dom(id_A) = A = cod(id_A), dom(g \circ f) = domf, cod (g \circ f)= codg,$$ for all objects $A \in Ob \mathcal{C}$ and all composable pairs of arrows (morphisms) $(g,f) \in \mathbb{A} \times_\mathbb{O} \mathbb{A}, $ and also such that the unit law and associativity axioms {\em c1} and {\em c2} hold. \end{definition} For convenience one also defines a $Hom$ (or $hom$) set as: $$Hom(B,C) = [f|f \in \mathcal{C}, dom f= B, cod f = C]$$ \subsection{Alternative definitions} There are several alternative definitions of a category. Thus, as defined by W.F. Lawvere, a {\em category} is an interpretation of the ETAC axioms of the elementary theory of abstract categories. For small categories-- whose $Ob \mathcal{C}$ is a set and also $Mor\mathcal{C}$ is a set-- one has a \PMlinkexternal{\em dirtect definition}{http://planetmath.org/encyclopedia/AlternativeDefinitionOfSmallCategory.html} \subsection{Applications in Physics and Mathematical Biophysics} A `categorification' of theoretical physics (including quantum field theories) began as early as 1968 \cite{Baianu-Marinescu68,Baianu1971}, whereas categories of sets were introduced in mathematical biophysics in 1958 \cite{Rosen58a,Rosen58b}, followed by the introduction of biotheoretical models in categories with structure in 1968-1971 \cite{Baianu-Marinescu68,Baianu70, Baianu71, Baianu70}. The `categorification' process in physics continues today, especially after 1985 (\cite{Baianu87} and references cited therein). \begin{thebibliography}{99} \bibitem{BAJ-DJ98a} Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: \emph{Advances in Mathematics}, 135, 145--206. \bibitem{BAJ-DJ98B} Baez, J. \& Dolan, J., 1998b, ``Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1--36. \bibitem{BAJ-DJ2k1} Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in \emph{Mathematics Unlimited -- 2001 and Beyond}, Berlin: Springer, 29--50. \bibitem{BAJ-DJ97} Baez, J., 1997, An Introduction to n-Categories, in \emph{Category Theory and Computer Science, Lecture Notes in Computer Science}, 1290, Berlin: Springer-Verlag, 1--33. \bibitem{ICB4} Baianu, I.C.: 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. \emph{Ibid.}, \textbf{33} (3), 339--354. \bibitem{ICB4} Baianu, I.C.: 1971b, 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{Bgg2} Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, \textbf{(M,R)}--Systems and Their Higher Dimensional Algebra, \PMlinkexternal{PDF's of Abstract and Preprint of Report}{\\http://www.ag.uiuc.edu/fs401/QAuto.pdf}. \bibitem{BBGG1} Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, 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. \bibitem{Bggb4} Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225. \section{Topical references for Categories and Algebraic Topology Applications in Theoretical Physics} \begin{thebibliography}{199} \bibitem{AS} Alfsen, E.M. and F. W. Schultz: \emph{Geometry of State Spaces of Operator Algebras}, Birkh\"auser, Boston--Basel--Berlin (2003). \bibitem{AMF56} Atiyah, 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., 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., 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, \emph{History and Philosophy of Logic}, 23, (2): 77--94. \bibitem{AS96} Awodey, S., 1996, Structure in Mathematics and Logic: A Categorical Perspective, \emph{Philosophia Mathematica}, 3: 209--237. \bibitem{AS2k4} Awodey, S., 2004, An Answer to Hellman's Question: Does Category Theory Provide a Framework for Mathematical Structuralism, \emph{Philosophia Mathematica}, 12: 54--64. \bibitem{AS2k6} Awodey, S., 2006, Category Theory, Oxford: Clarendon Press. \bibitem{BAJ-DJ98a} Baez, J. \& Dolan, J., 1998a, Higher-Dimensional Algebra III. n-Categories and the Algebra of Opetopes, in: \emph{Advances in Mathematics}, 135, 145--206. \bibitem{BAJ-DJ98B} Baez, J. \& Dolan, J., 1998b, ``Categorification", Higher Category Theory, Contemporary Mathematics, 230, Providence: AMS, 1--36. \bibitem{BAJ-DJ2k1} Baez, J. \& Dolan, J., 2001, From Finite Sets to Feynman Diagrams, in \emph{Mathematics Unlimited -- 2001 and Beyond}, Berlin: Springer, 29--50. \bibitem{BAJ-DJ97} Baez, J., 1997, An Introduction to n-Categories, in \emph{Category Theory and Computer Science, Lecture Notes in Computer Science}, 1290, Berlin: Springer-Verlag, 1--33. \bibitem{ICB4} Baianu, I.C.: 1971a, Organismic Supercategories and Qualitative Dynamics of Systems. \emph{Ibid.}, \textbf{33} (3), 339--354. \bibitem{ICB4} Baianu, I.C.: 1971b, 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{ICB-HG-EO84} Baianu, I.C., H. S. Gutowsky, and E. Oldfield: 1984, {\em Proc. Natl. Acad. Sci. USA}, \textbf{81}(12): 3713-3717. \bibitem{Bgg2} Baianu, I. C., Glazebrook, J. F. and G. Georgescu: 2004, Categories of Quantum Automata and N-Valued \L ukasiewicz Algebras in Relation to Dynamic Bionetworks, \textbf{(M,R)}--Systems and Their Higher Dimensional Algebra, \PMlinkexternal{PDF's of Abstract and Preprint of Report}{\\http://www.ag.uiuc.edu/fs401/QAuto.pdf}. \bibitem{ICB8} Baianu, I.C.: 2004a, Quantum Nano--Automata (QNA): Microphysical Measurements with Microphysical QNA Instruments, \emph{CERN Preprint EXT--2004--125}. \bibitem{Bgb2} Baianu, I. C., Brown, R. and J. F. Glazebrook: 2006a, {\em Quantum Algebraic Topology and Field Theories}. \PMlinkexternal{Preprint subm.}{http://www.ag.uiuc.edu/fs40l/QAT.pdf}. \bibitem{BBGG1} Baianu I. C., Brown R., Georgescu G. and J. F. Glazebrook: 2006b, 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. \bibitem{Bggb4} Baianu, I.C., R. Brown and J. F. Glazebrook: 2007b, A Non-Abelian, Categorical Ontology of Spacetimes and Quantum Gravity, Axiomathes, 17: 169-225. \bibitem{Ba-We2k} M.~Barr and C.~Wells. {\em Toposes, Triples and Theories}. Montreal: McGill University, 2000. \bibitem{Ba-We85} Barr, M. \& Wells, C., 1985, Toposes, Triples and Theories, New York: Springer-Verlag. \bibitem{BM-CW99} Barr, M. \& Wells, C., 1999, Category Theory for Computing Science, Montreal: CRM. \bibitem{BaM98} Batanin, M., 1998, Monoidal Globular Categories as a Natural Environment for the Theory of Weak n-Categories, \emph{Advances in Mathematics}, 136: 39--103. \bibitem{BJL81} Bell, J. L., 1981, Category Theory and the Foundations of Mathematics, \emph{British Journal for the Philosophy of Science}, 32, 349--358. \bibitem{BJL82} Bell, J. L., 1982, Categories, Toposes and Sets, \emph{Synthese}, 51, 3, 293--337. \bibitem{BJL86} Bell, J. L., 1986, From Absolute to Local Mathematics, \emph{Synthese}, 69, 3, 409--426. \bibitem{BJL88} Bell, J. L., 1988, Toposes and Local Set Theories: An Introduction, Oxford: Oxford University Press. \bibitem{BG-MCLS99} Birkoff, G. \& Mac Lane, S., 1999, Algebra, 3rd ed., Providence: AMS. \bibitem{BDK2k3} Biss, D.K., 2003, Which Functor is the Projective Line?, \emph{American Mathematical Monthly}, 110, 7, 574--592. \bibitem{BA-SA83} Blass, A. \& Scedrov, A., 1983, Classifying Topoi and Finite Forcing , Journal of Pure and Applied Algebra, 28, 111--140. \bibitem{BA-SA89} Blass, A. \& Scedrov, A., 1989, Freyd's Model for the Independence of the Axiom of Choice, Providence: AMS. \bibitem{BASA92} Blass, A. \& Scedrov, A., 1992, Complete Topoi Representing Models of Set Theory, \emph{Annals of Pure and Applied Logic}, 57, no. 1, 1--26. \bibitem{BA84} Blass, A., 1984, The Interaction Between Category Theory and Set Theory., Mathematical Applications of Category Theory, 30, Providence: AMS, 5--29. \bibitem{BR-SP2k4} Blute, R. \& Scott, P., 2004, Category Theory for Linear Logicians., in Linear Logic in Computer Science \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{BrownBook1} R. Brown: \emph{Topology and Groupoids}, BookSurge LLC (2006). \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,: 2007a, \emph{Non-Abelian Algebraic Topology},\PMlinkexternal{Vol.I PDF}{http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf}. \bibitem{BGB2k7b} Brown, R., Glazebrook, J. F. and I.C. Baianu.: 2007b, 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{BPP2k4} Brown, R., Paton, R. and T. Porter.: 2004, Categorical language and hierarchical models for cell systems, in \emph{Computation in Cells and Tissues - Perspectives and Tools of Thought}, Paton, R.; Bolouri, H.; Holcombe, M.; Parish, J.H.; Tateson, R. (Eds.) Natural Computing Series, Springer Verlag, 289-303. \bibitem{BP2k3} Brown R. and T. Porter: 2003, Category theory and higher dimensional algebra: potential descriptive tools in neuroscience, In: Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1, 80-92. \bibitem{Br-Har-Ka-Po2k2} Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., \emph{Theory and Applications of Categories} \textbf{10}, 71-93. \bibitem{Br-Hardy76} Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, \emph{Math. Nachr.}, 71: 273-286. \bibitem{Br-Po-analogy2k6} Brown, R. and T. Porter: 2006, Category Theory: an abstract setting for analogy and comparison, In: What is Category Theory?, \emph{Advanced Studies in Mathematics and Logic, Polimetrica Publisher}, Italy, (2006) 257-274. \bibitem{Br-Sp76} Brown, R. and Spencer, C.B.: 1976, Double groupoids and crossed modules, \emph{Cah. Top. G\'{e}om. Diff.} \textbf{17}, 343-362. \bibitem{BRTPT2k6} Brown R, and Porter T (2006) Category theory: an abstract setting for analogy and comparison. In: What is category theory? {\em Advanced studies in mathematics and logic}. Polimetrica Publisher, Italy, pp. 257-274. \bibitem{BR-SCB76} Brown R, Razak Salleh A (1999) Free crossed resolutions of groups and presentations of modules of identities among relations. {\em LMS J. Comput. Math.}, \textbf{2}: 25--61. \bibitem{BDA55} Buchsbaum, D. A.: 1955, Exact categories and duality., Trans. Amer. Math. Soc. \textbf{80}: 1-34. \bibitem{BDA55} Buchsbaum, D. A.: 1969, A note on homology in categories., Ann. of Math. \textbf{69}: 66-74. \bibitem{BI65} Bucur, I. (1965). {\em Homological Algebra}. (orig. title: ``Algebra Omologica'') Ed. Didactica si Pedagogica: Bucharest. \bibitem{BI-DA68} Bucur, I., and Deleanu A. (1968). {\em Introduction to the Theory of Categories and Functors}. J.Wiley and Sons: London \bibitem{BL2k3} Bunge, M. and S. Lack: 2003, Van Kampen theorems for toposes, \emph{Adv. in Math.} \textbf{179}, 291-317. \bibitem{BM74} Bunge, M., 1974, "Topos Theory and Souslin's Hypothesis", Journal of Pure and Applied Algebra, 4, 159-187. \bibitem{BM84} Bunge, M., 1984, "Toposes in Logic and Logic in Toposes", Topoi, 3, no. 1, 13-22. \bibitem{BM-LS2k3} Bunge M, Lack S (2003) Van Kampen theorems for toposes. {\em Adv Math}, \textbf {179}: 291-317. \bibitem{BJ-ICJ2k1} Butterfield J., Isham C.J. (2001) Spacetime and the philosophical challenges of quantum gravity. In: Callender C, Hugget N (eds) Physics meets philosophy at the Planck scale. Cambridge University Press, pp 33-89. \bibitem{BJ-ICJ98-2k2} Butterfield J., Isham C.J. 1998, 1999, 2000-2002, A topos perspective on the Kochen-Specker theorem I-IV, Int J Theor Phys 37(11):2669-2733; 38(3):827-859; 39(6):1413-1436; 41(4): 613-639. \bibitem{CH-ES56} Cartan, H. and Eilenberg, S. 1956. {\em Homological Algebra}, Princeton Univ. Press: Pinceton. \bibitem{Chaician} M. Chaician and A. Demichev. 1996. Introduction to Quantum Groups, World Scientific . \bibitem{CC46} Chevalley, C. 1946. The theory of Lie groups. Princeton University Press, Princeton NJ \bibitem{CPM65} Cohen, P.M. 1965. {\em Universal Algebra}, Harper and Row: New York, london and Tokyo. \bibitem{CF} M. Crainic and R. Fernandes.2003. Integrability of Lie brackets, {\em Ann.of Math}. \textbf{157}: 575-620. \bibitem{CA94} Connes A 1994. \emph{Noncommutative geometry}. Academic Press: New York. \bibitem{CR-LL63} Croisot, R. and Lesieur, L. 1963. \emph{Alg\`ebre noeth\'erienne non-commutative.}, Gauthier-Villard: Paris. \bibitem{CRL94} Crole, R.L., 1994, {\em Categories for Types}, Cambridge: Cambridge University Press. \bibitem{CJ-LJ91} Couture, J. \& Lambek, J., 1991, {\em Philosophical Reflections on the Foundations of Mathematics}, Erkenntnis, 34, 2, 187--209. \bibitem{DJ-ALEX60-71} Dieudonn\'e, J. \& Grothendieck, A., 1960, [1971], \'El\'ements de G\'eom\'etrie Alg\'ebrique, Berlin: Springer-Verlag. \bibitem{Dirac30} Dirac, P. A. M., 1930, {\em The Principles of Quantum Mechanics}, Oxford: Clarendon Press. \bibitem{Dirac33} Dirac, P. A. M., 1933, {\em The Lagrangian in Quantum Mechanics}, Physikalische Zeitschrift der Sowietunion, \textbf{3}: 64-72. \bibitem{Dirac43} Dirac, P. A. M.,, 1943, {\em Quantum Electrodynamics}, Communications of the Dublin Institute for Advanced Studies, \textbf{A1}: 1-36. \bibitem{Dixmier} Dixmier, J., 1981, Von Neumann Algebras, Amsterdam: North-Holland Publishing Company. [First published in French in 1957: Les Algebres d'Operateurs dans l'Espace Hilbertien, Paris: Gauthier--Villars.] \bibitem{Durdevich1} M. Durdevich : Geometry of quantum principal bundles I, Commun. Math. Phys. \textbf{175} (3) (1996), 457--521. \bibitem{Durdevich2} M. Durdevich : Geometry of quantum principal bundles II, Rev. Math. Phys. \textbf{9} (5) (1997), 531--607. \bibitem{EC} Ehresmann, C.: 1965, \emph{Cat\'egories et Structures}, Dunod, Paris. \bibitem{EC} Ehresmann, C.: 1966, Trends Toward Unity in Mathematics., \emph{Cahiers de Topologie et Geometrie Differentielle} \textbf{8}: 1-7. \bibitem{Eh-pseudo} Ehresmann, C.: 1952, Structures locales et structures infinit\'esimales, \emph{C.R.A.S.} Paris \textbf{274}: 587-589. \bibitem{Eh} Ehresmann, C.: 1959, Cat\'egories topologiques et cat\'egories diff\'erentiables, \emph{Coll. G\'eom. Diff. Glob.} Bruxelles, pp.137-150. \bibitem{Eh-quintettes} Ehresmann, C.:1963, Cat\'egories doubles des quintettes: applications covariantes , \emph{C.R.A.S. Paris}, \textbf{256}: 1891--1894. \bibitem{EA-VJP87} Ehresmann, A. C. \& Vanbremeersch, J-P., 1987, "Hierarchical Evolutive Systems: a Mathematical Model for Complex Systems", Bulletin of Mathematical Biology, 49, no. 1, 13--50. \bibitem{Eh-Oe} Ehresmann, C.: 1984, \emph{Oeuvres compl\`etes et comment\'ees: Amiens, 1980-84}, edited and commented by Andr\'ee Ehresmann. \bibitem{EACV1} Ehresmann, A. C. and J.-P. Vanbremersch: 1987, Hierarchical Evolutive Systems: A mathematical model for complex systems, \emph{Bull. of Math. Biol.} \textbf{49} (1): 13-50. \bibitem{EACV2} Ehresmann, A. C. and J.-P. Vanbremersch: 2006, The Memory Evolutive Systems as a model of Rosen's Organisms, \emph{Axiomathes} \textbf{16} (1--2): 13-50. \bibitem{EML1} Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., \emph{American Mathematical Society 43}: 757-831. \bibitem{EL} Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, \emph{Transactions of the American Mathematical Society} \textbf{58}: 231-294. \bibitem{ES-CH56} Eilenberg, S. \& Cartan, H., 1956, Homological Algebra, Princeton: Princeton University Press. \bibitem{ES-MCLS42} Eilenberg, S. \& MacLane, S., 1942, "Group Extensions and Homology", Annals of Mathematics, 43, 757--831. \bibitem{ES-SN52} Eilenberg, S. \& Steenrod, N., 1952, Foundations of Algebraic Topology, Princeton: Princeton University Press. \bibitem{ES60} Eilenberg, S.: 1960. Abstract description of some basic functors., J. Indian Math.Soc., \textbf{24} :221-234. \bibitem{S.Eilenberg} S.Eilenberg. Relations between Homology and Homotopy Groups. Proc.Natl.Acad.Sci.USA (1966),v:10--14. \bibitem{ED88} Ellerman, D., 1988, "Category Theory and Concrete Universals", Synthese, 28, 409--429. \bibitem{ETH} Z. F. Ezawa, G. Tsitsishvilli and K. Hasebe : Noncommutative geometry, extended $W_{\infty}$ algebra and Grassmannian solitons in multicomponent Hall systems, arXiv:hep--th/0209198. \bibitem{FS77} Feferman, S., 1977. Categorical Foundations and Foundations of Category Theory, in \emph{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, \emph{Transactions of the American Mathematical Society}, 94: 365-403. \bibitem{Feynman} Feynman, R. P., 1948, Space--Time Approach to Non--Relativistic Quantum Mechanics., \emph{Reviews of Modern Physics}, 20: 367--387. [It is reprinted in (Schwinger 1958).] \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", Journal of Pure and Applied Algebra, 19, 103--125. \bibitem{FP87} Freyd, P., 1987, "Choice and Well-Ordering", Annals of Pure and Applied Logic, 35, 2, 149--166. \bibitem{FP90} Freyd, P., 1990, Categories, Allegories, Amsterdam: North Holland. \bibitem{FP2k2} Freyd, P., 2002, "Cartesian Logic", Theoretical Computer Science, 278, no. 1--2, 3--21. \bibitem{FP-FH-SA87} Freyd, P., Friedman, H. \& Scedrov, A., 1987, "Lindembaum Algebras of Intuitionistic Theories and Free Categories", Annals of Pure and Applied Logic, 35, 2, 167--172. \bibitem{Gablot} Gablot, R. 1971. Sur deux classes de cat\'{e}gories de Grothendieck. Thesis.. Univ. de Lille. \bibitem{Gabriel1} Gabriel, P.: 1962, Des cat\'egories ab\'eliennes, \emph{Bull. Soc. Math. France} \textbf{90}: 323-448. \bibitem{Gabriel2} Gabriel, P. and M.Zisman:. 1967: \emph{Category of fractions and homotopy theory}, \emph{Ergebnesse der math.} Springer: Berlin. \bibitem{GabrielNP} Gabriel, P. and N. Popescu: 1964, Caract\'{e}risation des cat\'egories ab\'eliennes avec g\'{e}n\'{e}rateurs et limites inductives. , \emph{CRAS Paris} \textbf{258}: 4188-4191. \bibitem{GA-RG-SM2k} Galli, A. \& Reyes, G. \& Sagastume, M., 2000, "Completeness Theorems via the Double Dual Functor", Studia Logical, 64, no. 1, 61--81. \bibitem{GN} Gelfan'd, I. and Naimark, M., 1943. On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space.,Recueil Math\'ematique [Matematicheskii Sbornik] Nouvelle S\'erie, 12 [54]: 197-213. [Reprinted in C*--algebras: 1943--1993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.] \bibitem{GV70} Georgescu, G. and C. Vraciu 1970. "On the Characterization of \L{}ukasiewicz Algebras." \emph{J Algebra}, \textbf{16} (4), 486-495. \bibitem{GS-ZM2K2} Ghilardi, S. \& Zawadowski, M., 2002, Sheaves, Games \& Model Completions: A Categorical Approach to Nonclassical Porpositional Logics, Dordrecht: Kluwer. \bibitem{gs89} Ghilardi, S., 1989, "Presheaf Semantics and Independence Results for some Non-classical first-order logics", Archive for Mathematical Logic, 29, no. 2, 125--136. \bibitem{Gob68} Goblot, R., 1968, Cat\'egories modulaires , {\em C. R. Acad. Sci. Paris, S\'erie A.}, \textbf{267}: 381--383. \bibitem{Gob71} Goblot, R., 1971, Sur deux classes de cat\'egories de Grothendieck, {\em Th\`ese.}, Univ. Lille, 1971. \bibitem{GR79} Goldblatt, R., 1979, Topoi: The Categorical Analysis of Logic, Studies in logic and the foundations of mathematics, Amsterdam: Elsevier North-Holland Publ. Comp. \bibitem{Goldie} Goldie, A. W., 1964, Localization in non-commutative noetherian rings, {\em J.Algebra}, \textbf{1}: 286-297. \bibitem{Godement} Godement,R. 1958. Th\'{e}orie des faisceaux. Hermann: Paris. \bibitem{GRAY65} Gray, C. W.: 1965. Sheaves with values in a category.,\emph {Topology}, 3: 1-18. \bibitem{Alex1} Grothendieck, A.: 1971, Rev\^{e}tements \'Etales et Groupe Fondamental (SGA1), chapter VI: Cat\'egories fibr\'ees et descente, \emph{Lecture Notes in Math.} \textbf{224}, Springer--Verlag: Berlin. \bibitem{Alex2} Grothendieck, A.: 1957, Sur quelque point d-alg\'{e}bre homologique. , \emph{Tohoku Math. J.}, \textbf{9:} 119-121. \bibitem{Alex3} Grothendieck, A. and J. Dieudon\'{e}.: 1960, El\'{e}ments de geometrie alg\'{e}brique., \emph{Publ. Inst. des Hautes Etudes de Science}, \textbf{4}. \bibitem{ALEXsem} Grothendieck, A. et al., S\'eminaire de G\'eom\'etrie Alg\'ebrique, Vol. 1--7, Berlin: Springer-Verlag. \bibitem{ALEX57} Grothendieck, A., 1957, "Sur Quelques Points d'alg\`ebre homologique", Tohoku Mathematics Journal, 9, 119--221. \bibitem{TMMFJ84} Groups Authors: Jo\~ao Faria Martins, Timothy Porter., On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical $math.QA/0608484 [abs, ps, pdf, other]$. \bibitem{GL66} Gruson, L, 1966, Compl\'etion ab\'elienne. {\em Bull. Math.Soc. France}, \textbf{90}: 17-40. \bibitem{HKK} K.A. Hardie, K.H. Kamps and R.W. Kieboom. 2000. A homotopy 2-groupoid of a Hausdorff space, \emph{Applied Cat. Structures} 8: 209--234. \bibitem{HWS82} Hatcher, W. S. 1982. {\em The Logical Foundations of Mathematics}, Oxford: Pergamon Press. \bibitem{Heller58} Heller, A. :1958, Homological algebra in Abelian categories., \emph{Ann. of Math.} \textbf{68}: 484-525. \bibitem{HellerRowe62} Heller, A. and K. A. Rowe.:1962, On the category of sheaves., \emph{Amer J. Math.} \textbf{84}: 205-216. \bibitem{HG2k3} Hellman, G., 2003, "Does Category Theory Provide a Framework for Mathematical Structuralism?", Philosophia Mathematica, 11, 2, 129--157. \bibitem{HC-MM-PJ2K} Hermida, C. \& Makkai, M. \& Power, J., 2000, On Weak Higher-dimensional Categories. I, Journal of Pure and Applied Algebra, 154, no. 1-3, 221--246. \bibitem{HC-MM-PI2K1} Hermida, C. \& Makkai, M. \& Power, J., 2001, On Weak Higher-dimensional Categories. II, Journal of Pure and Applied Algebra, 157, no. 2-3, 247--277. \bibitem{HC-MM-PI2K2} Hermida, C. \& Makkai, M. \& Power, J., 2002, On Weak Higher-dimensional Categories. III, Journal of Pure and Applied Algebra, 166, no. 1-2, 83--104. \bibitem{HPJbook} Higgins, P. J.: 2005, \emph{Categories and groupoids}, Van Nostrand Mathematical Studies: 32, (1971); \emph{Reprints in Theory and Applications of Categories}, No. 7: 1-195. \bibitem{HPJ2k5} Higgins, Philip J. Thin elements and commutative shells in cubical $\omega$-categories. Theory Appl. Categ. 14 (2005), No. 4, 60--74 (electronic). (Reviewer: Timothy Porter) 18D05. \bibitem{HJ-RE-RG90} Hyland, J.M.E. \& Robinson, E.P. \& Rosolini, G., 1990, "The Discrete Objects in the Effective Topos", Proceedings of the London Mathematical Society (3), 60, no. 1, 1--36. \bibitem{HJME82} Hyland, J.M.E., 1982, "The Effective Topos", Studies in Logic and the Foundations of Mathematics, 110, Amsterdam: North Holland, 165--216. \bibitem{HJME88} Hyland, J. M..E., 1988, "A Small Complete Category", Annals of Pure and Applied Logic, 40, no. 2, 135--165. \bibitem{HJME91} Hyland, J. M .E., 1991, "First Steps in Synthetic Domain Theory", Category Theory (Como 1990), Lecture Notes in Mathematics, 1488, Berlin: Springer, 131-156. \bibitem{HJME2K2} Hyland, J. M.E., 2002, "Proof Theory in the Abstract", Annals of Pure and Applied Logic, 114, no. 1--3, 43--78. \bibitem{E.Hurewicz} E.Hurewicz. CW Complexes.Trans AMS.1955. \bibitem{IT-PR-IB70} Ionescu, Th., R. Parvan and I. Baianu, 1970, {\em C. R. Acad. Sci. Paris, S\'erie A.}, \textbf{269}: 112-116, {\em communiqu\'ee par Louis N\'eel}. \bibitem{Isham1} C. J. Isham : A new approach to quantising space--time: I. quantising on a general category, \emph{Adv. Theor. Math. Phys.} \textbf{7} (2003), 331--367. \bibitem{JB99} Jacobs, B., 1999, Categorical Logic and Type Theory, Amsterdam: North Holland. \bibitem{JPT77} Johnstone, P. T., 1977, Topos Theory, New York: Academic Press. \bibitem{JPT79A} Johnstone, P. T., 1979a, "Conditions Related to De Morgan's Law", Applications of Sheaves, Lecture Notes in Mathematics, 753, Berlin: Springer, 479--491. \bibitem{JPT79B} Johnstone, P.T., 1979b, "Another Condition Equivalent to De Morgan's Law", Communications in Algebra, 7, no. 12, 1309--1312. \bibitem{JPT81} Johnstone, P. T., 1981, "Tychonoff's Theorem without the Axiom of Choice", Fundamenta Mathematicae, 113, no. 1, 21--35. \bibitem{JPT52} Johnstone, P. T., 1982, Stone Spaces, Cambridge:Cambridge University Press. \bibitem{JPT85} Johnstone, P. T., 1985, "How General is a Generalized Space?", Aspects of Topology, Cambridge: Cambridge University Press, 77--111. \bibitem{JPT2K2A} Johnstone, P. T., 2002a, Sketches of an Elephant: a Topos Theory Compendium. Vol. 1, Oxford Logic Guides, 43, Oxford: Oxford University Press. \bibitem{JAMI95} Joyal, A. \& Moerdijk, I., 1995, Algebraic Set Theory, Cambridge: Cambridge University Press. \bibitem{kampen1-1933} Van Kampen, E. H.: 1933, On the Connection Between the Fundamental Groups of some Related Spaces, \emph{Amer. J. Math.} \textbf{55}: 261-267 \bibitem{KDM58} Kan, D. M., 1958, "Adjoint Functors", Transactions of the American Mathematical Society, 87, 294-329. \bibitem{Kleisli62} Kleisli, H.: 1962, Homotopy theory in Abelian categories.,{\em Can. J. Math.}, \textbf{14}: 139-169. \bibitem{KJT70} Knight, J.T., 1970, On epimorphisms of non-commutative rings., {\em Proc. Cambridge Phil. Soc.}, \textbf{25}: 266-271. \bibitem{KA81} Kock, A., 1981, Synthetic Differential Geometry, London Mathematical Society Lecture Note Series, 51, Cambridge: Cambridge University Press. \bibitem{KN1} S. Kobayashi and K. Nomizu : Foundations of Differential Geometry Vol I., Wiley Interscience, New York--London 1963. \bibitem{Krips} H. Krips : Measurement in Quantum Theory, \emph{The Stanford Encyclopedia of Philosophy } ({Winter 1999 Edition}), Edward N. Zalta (ed.), $URL=<http://plato.stanford.edu/archives/win1999/entries/qt--measurement/>$ \bibitem{LTY} Lam, T. Y., 1966, The category of noetherian modules, {\em Proc. Natl. Acad. Sci. USA}, \textbf{55}: 1038-104. \bibitem{LJ-SPJ81} Lambek, J. \& Scott, P. J., 1981, "Intuitionistic Type Theory and Foundations", Journal of Philosophical Logic, 10, 1, 101--115. \bibitem{LJ-SPJ86} Lambek, J. \& Scott, P.J., 1986, Introduction to Higher Order Categorical Logic, Cambridge: Cambridge University Press. \bibitem{LJ68} Lambek, J., 1968, "Deductive Systems and Categories I. Syntactic Calculus and Residuated Categories", Mathematical Systems Theory, 2, 287--318. \bibitem{LJ69} Lambek, J., 1969, "Deductive Systems and Categories II. Standard Constructions and Closed Categories", Category Theory, Homology Theory and their Applications I, Berlin: Springer, 76--122. \bibitem{LJ72} Lambek, J., 1972, "Deductive Systems and Categories III. Cartesian Closed Categories, Intuitionistic Propositional Calculus, and Combinatory Logic", Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics, 274, Berlin: Springer, 57--82. \bibitem{LJ82} Lambek, J., 1982, "The Influence of Heraclitus on Modern Mathematics", Scientific Philosophy Today, J. Agassi and R.S. Cohen, eds., Dordrecht, Reidel, 111--122. \bibitem{LJ86} Lambek, J., 1986, "Cartesian Closed Categories and Typed lambda calculi", Combinators and Functional Programming Languages, Lecture Notes in Computer Science, 242, Berlin: Springer, 136--175. \bibitem{LT89A} Lambek, J., 1989A, "On Some Connections Between Logic and Category Theory", Studia Logica, 48, 3, 269--278. \bibitem{LJ89B} Lambek, J., 1989B, "On the Sheaf of Possible Worlds", Categorical Topology and its relation to Analysis, Algebra and Combinatorics, Teaneck: World Scientific Publishing, 36--53. \bibitem{LJ94a} Lambek, J., 1994a, "Some Aspects of Categorical Logic", Logic, Methodology and Philosophy of Science IX, Studies in Logic and the Foundations of Mathematics 134, Amsterdam: North Holland, 69--89. \bibitem{LJ94b} Lambek, J., 1994b, "What is a Deductive System?", What is a Logical System?, Studies in Logic and Computation, 4, Oxford: Oxford University Press, 141--159. \bibitem{LJ2k4} Lambek, J., 2004, "What is the world of Mathematics? Provinces of Logic Determined", Annals of Pure and Applied Logic, 126(1-3), 149--158. \bibitem{LaSc} Lambek, J. and P.~J.~Scott. {\em Introduction to higher order categorical logic}. Cambridge University Press, 1986. \bibitem{Lance} E. C. Lance : Hilbert C*--Modules. \emph{London Math. Soc. Lect. Notes} \textbf{210}, \emph{Cambridge Univ. Press.} 1995. \bibitem{LE-MJP2k5} Landry, E. \& Marquis, J.-P., 2005, "Categories in Context: Historical, Foundational and philosophical", Philosophia Mathematica, 13, 1--43. \bibitem{LE99} Landry, E., 1999, "Category Theory: the Language of Mathematics", Philosophy of Science, 66, 3: supplement, S14--S27. \bibitem{LE99} Landry, E., 2001, "Logicism, Structuralism and Objectivity", Topoi, 20, 1, 79--95. \bibitem{LandNP98} Landsman, N. P.: 1998, \emph{Mathematical Topics between Classical and Quantum Mechanics}, Springer Verlag: New York. \bibitem{Land} N. P. Landsman : Mathematical topics between classical and quantum mechanics. \emph{Springer Verlag}, New York, 1998. \bibitem{Land1} N. P. Landsman : Compact quantum groupoids, arXiv:math\^a~@~Tph/9912006 \bibitem{LPRM94} La Palme Reyes, M., et. al., 1994, "The non-Boolean Logic of Natural Language Negation", Philosophia Mathematica, 2, no. 1, 45--68. \bibitem{LPRM99} La Palme Reyes, M., et. al., 1999, "Count Nouns, Mass Nouns, and their Transformations: a Unified Category-theoretic Semantics", Language, Logic and Concepts, Cambridge: MIT Press, 427--452. \bibitem{LFW64} Lawvere, F. W., 1964, "An Elementary Theory of the Category of Sets", Proceedings of the National Academy of Sciences U.S.A., 52, 1506--1511. \bibitem{LFW65} Lawvere, F. W., 1965, "Algebraic Theories, Algebraic Categories, and Algebraic Functors", Theory of Models, Amsterdam: North Holland, 413--418. \bibitem{LFW66} Lawvere, F. W., 1966, "The Category of Categories as a Foundation for Mathematics", Proceedings of the Conference on Categorical Algebra, La Jolla, New York: Springer-Verlag, 1--21. \bibitem{MCLS98} MacLane, S., 1997, Categories for the Working Mathematician, 2nd edition, New York: Springer-Verlag. \bibitem{EML1} Eilenberg, S. and S. Mac Lane.: 1942, Natural Isomorphisms in Group Theory., \emph{American Mathematical Society 43}: 757-831. \bibitem{EL} Eilenberg, S. and S. Mac Lane: 1945, The General Theory of Natural Equivalences, \emph{Transactions of the American Mathematical Society} \textbf{58}: 231-294. \bibitem{Other} See also an extensive \PMlinkname{category theory bibliography}{BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics} \end{thebibliography}