Monoidal topology : a categorical approach to order, metric, and topology

Preface 1. Introduction Robert Lowen and Walter Tholen 2. Monoidal structures Gavin J. Seal and Walter Tholen 3. Lax algebras Dirk Hofmann, Gavin J. Seal and Walter Tholen 4. Kleisli monoids Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal 5. Lax algebras as spaces Maria Manuel Clementino, Eva Colebunders and Walter Tholen Bibliography Tables Index.

[1]  S. Eilenberg,et al.  Adjoint functors and triples , 1965 .

[2]  R. Flagg,et al.  Completeness in Continuity Spaces , 2008 .

[3]  Robert Rosebrugh,et al.  A basic distributive law , 2002 .

[4]  E H Moore Definition of Limit in General Integral Analysis. , 1915, Proceedings of the National Academy of Sciences of the United States of America.

[5]  Dirk Hofmann,et al.  On extensions of lax monads , 2004 .

[6]  Jeanne Meisen,et al.  Relations in regular categories , 1974 .

[7]  Dirk Hofmann,et al.  Local homeomorphisms via ultrafilter convergence , 2004 .

[8]  Walter Tholen,et al.  Categorical Structure of Closure Operators: With Applications to Topology, Algebra and Discrete Mathematics , 1995 .

[9]  花井 七郎 可換な closure operators について , 1951 .

[10]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[11]  Pawel Waszkiewicz,et al.  The formal ball model for -categories , 2010, Mathematical Structures in Computer Science.

[12]  Peter J. Huber,et al.  Homotopy theory in general categories , 1961 .

[13]  R. Lowen,et al.  Metrically generated theories , 2005 .

[14]  Oswald Wyler,et al.  Categories of Relations and Functional Relations , 2000, Appl. Categorical Struct..

[15]  M. Barr,et al.  Toposes, Triples and Theories , 1984 .

[16]  Rudolf-E. Hoffmann,et al.  Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications , 1981 .

[17]  Dirk Hofmann,et al.  Approaching Metric Domains , 2013, Appl. Categorical Struct..

[18]  Dirk Hofmann,et al.  Effective Descent Morphisms in Categories of Lax Algebras , 2004, Appl. Categorical Struct..

[19]  Gunther Jäger,et al.  A one-point compactification for lattice-valued convergence spaces , 2012, Fuzzy Sets Syst..

[20]  E. J. McShane Partial Orderings and Moore-Smith Limits , 1952 .

[21]  J. Isbell Six theorems about injective metric spaces , 1964 .

[22]  Claudio Hermida From coherent structures to universal properties , 2000 .

[23]  Franck van Breugel,et al.  An introduction to metric semantics: operational and denotational models for programming and specification languages , 2001, Theor. Comput. Sci..

[24]  Roy Dyckhoff,et al.  Total reflections, partial products, and hereditary factorizations , 1984 .

[25]  Marcel Erné,et al.  Scott convergence and scott topology in partially ordered sets II , 1981 .

[26]  John W. Tukey,et al.  Convergence and uniformity in topology , 1940 .

[27]  Alan Day Filter monads, continuous lattices and closure systems , 1975 .

[28]  S. Mrówka Compactness and product spaces , 1959 .

[29]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[30]  Sabine Fenstermacher,et al.  Sheaves In Geometry And Logic A First Introduction To Topos Theory , 2016 .

[31]  D. V. Thampuran Extended topology: Filters and convergence I , 1965 .

[32]  Hans-Joachim Kowalsky,et al.  Limesräume und Komplettierung , 1954 .

[33]  V. E. Cazanescu Algebraic theories , 2004 .

[34]  Stephen Herman Kamnitzer Protoreflections, relational algebras and topology. , 1974 .

[35]  Ralph Kopperman,et al.  Continuity Spaces: Reconciling Domains and Metric Spaces , 1997, Theor. Comput. Sci..

[36]  Andre Scedrov,et al.  Categories, allegories , 1990, North-Holland mathematical library.

[37]  Marcello M. Bonsangue,et al.  Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding , 1995, Theor. Comput. Sci..

[38]  Roy Dyckhoff,et al.  Exponentiable morphisms, partial products and pullback complements , 1987 .

[39]  Robert D. Rosebrugh,et al.  Boundedness and Complete Distributivity , 2001, Appl. Categorical Struct..

[40]  F. William Lawvere,et al.  Metric spaces, generalized logic, and closed categories , 1973 .

[41]  Ross Street,et al.  Fibrations and Yoneda's lemma in a 2-category , 1974 .

[42]  R. J. Wood Ordered Sets via Adjunctions , 2002 .

[43]  Jimmie Lawson THE ROUND IDEAL COMPLETION VIA SOBRIFICATION , 2008 .

[44]  Walter Tholen,et al.  Universality of Coproducts in Categories of Lax Algebras , 2006, Appl. Categorical Struct..

[45]  Dirk Hofmann,et al.  Descent morphisms and a van Kampen Theorem in categories of lax algebras , 2012 .

[46]  P. Johnstone Sketches of an Elephant: A Topos Theory Compendium Volume 1 , 2002 .

[47]  Paul Brock,et al.  On Convergence Approach Spaces , 1998, Appl. Categorical Struct..

[48]  C. Schubert Lax Algebras: A Scenic Approach , 2006 .

[49]  Walter Tholen,et al.  Factorizations, Localizations, and the Orthogonal Subcategory Problem , 1983 .

[50]  Solomon Feferman,et al.  Categorical Foundations and Foundations of Category Theory , 1977 .

[51]  Walter Tholen,et al.  On the categorical meaning of Hausdorff and Gromov distances, I , 2009, 0901.0618.

[52]  G. M. Kelly,et al.  A $2$-categorical approach to change of base and geometric morphisms I , 1991 .

[53]  Saunders MacLane,et al.  Natural Associativity and Commutativity , 1963 .

[54]  Aaron Klein Relations in categories , 1970 .

[55]  Robert Lowen,et al.  A Kuratowski–Mrówka theorem in approach theory , 2005 .

[56]  Claudio Pisani,et al.  CONVERGENCE IN EXPONENTIABLE SPACES , 2001 .

[57]  G. M. Kelly,et al.  On topological quotient maps preserved by pullbacks or products , 1970, Mathematical Proceedings of the Cambridge Philosophical Society.

[58]  Jan Paseka,et al.  Algebraic and Categorical Aspects of Quantales , 2008 .

[59]  R. Lowen,et al.  Local compactness in approach spaces II , 1998 .

[60]  Rory B. B. Lucyshyn-Wright,et al.  Domains occur among spaces as strict algebras among lax , 2011, Mathematical Structures in Computer Science.

[61]  H. Herrlich,et al.  Improving constructions in topology , 1991 .

[62]  J. Isbell,et al.  General function spaces, products and continuous lattices , 1986, Mathematical Proceedings of the Cambridge Philosophical Society.

[63]  E. Manes Monads in topology , 2010 .

[64]  R. Rosebrugh,et al.  Normed Spaces and the Change of Base for Enriched Categories , 2008 .

[65]  Jiří Adámek,et al.  Continuous categories revisited. , 2003 .

[66]  Marta Bunge Coherent extensions and relational algebras , 1974 .

[67]  Walter Tholen A categorical guide to separation, compactness and perfectness. , 1999 .

[68]  J. Kelley Convergence in topology , 1950 .

[69]  Robert Lowen KURATOWSKI'S MEASURE OF NON-COMPACTNESS REVISITED , 1988 .

[70]  Dirk Hofmann,et al.  Kleisli compositions for topological spaces , 2006 .

[71]  J. de Groot An isomorphism principle in general topology , 1967 .

[72]  Michael B. Smyth,et al.  Effectively given Domains , 1977, Theor. Comput. Sci..

[73]  W. Tholen On Wyler's Taut lift theorem , 1978 .

[74]  U. Höhle Many Valued Topology and its Applications , 2001 .

[75]  A. Bielecki Sur une généralisation d'un théorème de Weierstrass , 1932 .

[76]  Reinhard Börger Coproducts and ultrafilters , 1987 .

[77]  H. L. Smith,et al.  A General Theory of Limits , 1922 .

[78]  Hans-Joachim Kowalsky,et al.  Beiträge zur topologischen Algebra , 1954 .

[79]  Dirk Hofmann,et al.  Approximation in quantale-enriched categories , 2010, ArXiv.

[80]  Walter Tholen,et al.  Categorical Foundations: A Functional Approach to General Topology , 2003 .

[81]  Dirk Hofmann,et al.  Triquotient maps via ultrafilter convergence , 2002 .

[82]  Ralph Kopperman,et al.  All topologies come from generalized metrics , 1988 .

[83]  E. G. Manes Compact Hausdorff objects , 1974 .

[84]  Dirk Hofmann,et al.  One Setting for All: Metric, Topology, Uniformity, Approach Structure , 2004, Appl. Categorical Struct..

[85]  E. Lowen,et al.  A QUASITOPOS CONTAINING CONV AND MET AS FULL SUBCATEGORIES , 1988 .

[86]  G. M. Kelly,et al.  BASIC CONCEPTS OF ENRICHED CATEGORY THEORY , 2022, Elements of ∞-Category Theory.

[87]  Dirk Hofmann,et al.  Exponentiation in V-categories , 2006 .

[88]  Dirk Hofmann,et al.  Axioms for Sequential Convergence , 2007, Appl. Categorical Struct..

[89]  Gavin J. Seal A Kleisli-based Approach to Lax Algebras , 2009, Appl. Categorical Struct..

[90]  R. Lowen,et al.  A new lax algebraic characterization of approach spaces , 2008 .

[91]  Francis Borceux,et al.  A Handbook of Categorical Algebra 3: Categories of sheaves , 1994 .

[92]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[93]  Isar Stubbe,et al.  Categorical structures enriched in a quantaloid: tensored and cotensored categories , 2004, math/0411366.

[94]  Dirk Hofmann,et al.  Probabilistic metric spaces as enriched categories , 2012, Fuzzy Sets Syst..

[95]  Peter T. Johnstone,et al.  On a Topological Topos , 1979 .

[96]  Marcel Erné,et al.  Standard completions for quasiordered sets , 1983 .

[97]  J. Duskin Variations on Beck's tripleability criterion , 1969 .

[98]  R. Lowen Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad , 1997 .

[99]  L. Vietoris,et al.  Bereiche zweiter Ordnung , 1922 .

[100]  F. W. Lawvere,et al.  The Category of Categories as a Foundation for Mathematics , 1966 .

[101]  J. de Groot,et al.  THE COMPACTNESS OPERATOR IN GENERAL TOPOLOGY , 1966 .

[102]  K. I. Rosenthal Quantales and their applications , 1990 .

[103]  Gerhard Grimeisen Gefilterte Summation von Filtern und iterierte Grenzprozesse. I , 1960 .

[104]  Sergey A. Solovyov,et al.  On the category Q-Mod , 2008 .

[105]  R. Lowen Approach Spaces A Common Supercategory of TOP and MET , 1989 .

[106]  Proper maps for lax algebras and the Kuratowski-Mrówka Theorem , 2013 .

[107]  Dirk Hofmann Exponentiation for unitary structures , 2006 .

[108]  R. J. Wood,et al.  Constructive complete distributivity. I , 1990, Mathematical Proceedings of the Cambridge Philosophical Society.

[109]  Ross Street,et al.  Variation through enrichment , 1983 .

[110]  H. Herrlich On the representability of partial morphisms in top and in related constructs , 1988 .

[111]  George Janelidze,et al.  Categorical Galois Theory: Revision and Some Recent Developments , 2004 .

[112]  R. Lowen,et al.  Lax algebras via initial monad morphisms: APP, TOP, MET and ORD , 2011 .

[113]  S. Lack,et al.  The formal theory of monads II , 2002 .

[114]  Walter Tholen,et al.  Effective descent maps of topological spaces , 1994 .

[115]  Paul Brock,et al.  Approach Spaces, Limit Tower Spaces, and Probabilistic Convergence Spaces , 1997, Appl. Categorical Struct..

[116]  Ernest G. Manes,et al.  Taut Monads and T0-spaces , 2002, Theor. Comput. Sci..

[117]  R. Lowen,et al.  A note on separation in AP , 2003 .

[118]  Fuzzy functions and an extension of the category L-Top of Chang-Goguen L-topological spaces , 2002, math/0204139.

[119]  Achim Jung,et al.  Stably Compact Spaces and the Probabilistic Powerspace construction , 2004, DTMPP.

[120]  Dirk Hofmann,et al.  Lax algebra meets topology , 2012 .

[121]  Pawel Waszkiewicz On Domain Theory over Girard Quantales , 2009, Fundam. Informaticae.

[122]  Dirk Hofmann,et al.  Relative injectivity as cocompleteness for a class of distributors , 2008 .

[123]  Dirk Hofmann,et al.  Covering Morphisms in Categories of Relational Algebras , 2014, Appl. Categorical Struct..

[124]  M. Stone Applications of the theory of Boolean rings to general topology , 1937 .

[125]  Marcel Erné Z-Continuous Posets and Their Topological Manifestation , 1999, Appl. Categorical Struct..

[126]  Dirk Hofmann,et al.  Topological Features of Lax Algebras , 2003, Appl. Categorical Struct..

[127]  R. Börger,et al.  Cantors Diagonalprinzip für Kategorien , 1978 .

[128]  Joachim Machner $T$-algebras of the monad $L$-Fuzz , 1985 .

[129]  Melvin Hochster,et al.  Prime ideal structure in commutative rings , 1969 .

[130]  G.S.H. Cruttwell,et al.  A unified framework for generalized multicategories , 2009, 0907.2460.

[131]  J. J. M. M. Rutten Weighted colimits and formal balls in generalized metric spaces , 1997 .

[132]  Isar Stubbe Q-MODULES ARE Q-SUPLATTICES , 2008, 0809.4343.

[133]  Albert Burroni $T$-catégories (catégories dans un triple) , 1971 .

[134]  Jirí Adámek,et al.  Abstract and Concrete Categories - The Joy of Cats , 1990 .

[135]  A. Tychonoff Über die topologische Erweiterung von Räumen , 1930 .

[136]  Horst Herrlich,et al.  Factorizations, denseness, separation, and relatively compact objects , 1987 .

[137]  W. Groß Grundzüge der Mengenlehre , 1915 .

[138]  Martín Hötzel Escardó,et al.  Properly injective spaces and function spaces , 1998 .

[139]  Nicolas Bourbaki,et al.  Eléments de Mathématique , 1964 .

[140]  Casimir Kuratowski Evaluation de la classe borélienne ou projective d'un ensemble de points à l'aide des symboles logiques , 1931 .

[141]  Walter Tholen,et al.  Ordered Topological Structures , 2009 .

[142]  U. Höhle M-valued Sets and Sheaves over Integral Commutative CL-Monoids , 1992 .

[143]  S. Lack,et al.  Introduction to extensive and distributive categories , 1993 .

[144]  Walter Tholen,et al.  Tychonoff’s Theorem in a category , 1996 .

[145]  A. Tarski,et al.  Une contribution à la théorie de la mesure , 1930 .

[146]  Jean Benabou,et al.  Fibered categories and the foundations of naive category theory , 1985, Journal of Symbolic Logic.

[147]  Dirk Hofmann,et al.  Exponentiability in categories of lax algebras , 2003 .

[148]  F. W. Lawvere,et al.  Sets for Mathematics , 2003 .

[149]  E. Kreyszig Interaction between General Topology and Functional Analysis , 1997 .

[150]  Robert Rosebrugh,et al.  Distributive laws and factorization , 2002 .

[151]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

[152]  H. Hahn Sur quelques points du calcul fonctionnel , 1908 .

[153]  J. Lambek Deductive systems and categories II. Standard constructions and closed categories , 1969 .

[154]  A. Kock Monads for which Structures are Adjoint to Units , 1995 .

[155]  Oswald Wyler,et al.  On the categories of general topology and topological algebra , 1971 .

[156]  G. M. Kelly A note on relations relative to a factorization system , 1991 .

[157]  Philip S. Mulry,et al.  MONAD COMPOSITIONS I: GENERAL CONSTRUCTIONS AND RECURSIVE DISTRIBUTIVE LAWS , 2007 .

[158]  A. Gerlo,et al.  Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings , 2006, Appl. Categorical Struct..

[159]  R. Flagg,et al.  Quantales and continuity spaces , 1997 .

[160]  K. I. Rosenthal The Theory of Quantaloids , 1996 .

[161]  Gavin J. Seal,et al.  On the monadic nature of categories of ordered sets , 2011 .

[162]  Solomon Feferman,et al.  Set-Theoretical foundations of category theory , 1969 .

[163]  Christoph Schubert,et al.  Extensions in the theory of lax algebras , 2010 .

[164]  Dirk Hofmann,et al.  Topological theories and closed objects , 2007 .

[165]  Dirk Hofmann An algebraic description of regular epimorphisms in topology , 2005 .

[166]  Gavin J. Seal,et al.  Order-adjoint monads and injective objects , 2010 .

[167]  Jirí Adámek,et al.  Algebraic Theories: A Categorical Introduction to General Algebra , 2010 .

[168]  Gavin J. Seal CANONICAL AND OP-CANONICAL LAX ALGEBRAS , 2005 .

[169]  Saunders Mac Lane,et al.  One universe as a foundation for category theory , 1969 .

[170]  Walter Tholen,et al.  Metric, topology and multicategory—a common approach , 2003 .

[171]  F. Borceux Handbook Of Categorical Algebra 1 Basic Category Theory , 2008 .

[172]  Lars Birkedal,et al.  The category-theoretic solution of recursive metric-space equations , 2010, Theor. Comput. Sci..

[173]  Ernie Manes,et al.  Taut Monads, Dynamic Logic and Determinism , 2007, MFPS.

[174]  E. Čech On Bicompact Spaces , 1937 .

[175]  G. Birkhoff Moore-Smith Convergence in General Topology , 1937 .

[176]  E. Lowen,et al.  Topological quasitopos hulls of categories containing topological and metric objects , 1989 .

[177]  Günther Richter,et al.  An Elementary Approach to Exponential Spaces , 2001, Appl. Categorical Struct..

[178]  S. Niefield,et al.  Cartesianness: topological spaces, uniform spaces, and affine schemes , 1982 .

[179]  Walter Tholen,et al.  Multiplicative structures over sup-lattices , 1989 .

[180]  Oswald Wyler,et al.  Lecture notes on Topoi and Quasitopoi , 1991 .

[181]  Werner Gähler Monads and Convergence , 1988 .

[182]  E. Manes,et al.  A triple theoretic construction of compact algebras , 1969 .

[183]  P. Gabriel,et al.  Lokal α-präsentierbare Kategorien , 1971 .

[184]  Kim R. Wagner,et al.  Solving Recursive Domain Equations with Enriched Categories. , 1994 .

[185]  Hausdorff and Gromov Distances in Quantale−Enriched Categories , 2008 .

[186]  Jimmie D. Lawson,et al.  Stably compact spaces , 2010, Mathematical Structures in Computer Science.

[187]  Dirk Hofmann,et al.  Lawvere Completeness in Topology , 2007, Appl. Categorical Struct..

[188]  Walter Tholen,et al.  TOPOLOGY IN A CATEGORY: COMPACTNESS , 1996 .

[189]  Th. Marny On epireflective subcategories of topological categories , 1979 .

[190]  Axel Möbus,et al.  Alexandrov compactification of relational algebras , 1983 .