Internal and external aspects of continuous logic and categorical logic for sheaves over quantales

In this text we explore and propose notions of sheaves over commutative, integral quantales, which are based on extensions of results of the theory of sheaves over locales: the interplay of sheaves as valued-sets and the analogy of sheaves as enriched categories. Over these proposals, we define logi...

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Autores:: Reyes Gaona, David

Tipo de recurso:

Fecha de publicación:: 2023

Institución:: Universidad Nacional de Colombia

Repositorio:: Universidad Nacional de Colombia

Idioma:: eng

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repository_id_str
dc.title.eng.fl_str_mv	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales
dc.title.translated.spa.fl_str_mv	Aspectos internos y externos de lógica continua y lógica categórica para haces sobre cuantales
title	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales
spellingShingle	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales 510 - Matemáticas::514 - Topología 510 - Matemáticas::512 - Álgebra 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::515 - Análisis 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::514 - Topología 510 - Matemáticas::512 - Álgebra 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::515 - Análisis 510 - Matemáticas::511 - Principios generales de las matemáticas Algebra-métodos gráficos Lógica Algebra - Graphic methods Logic Sheaves Quantales Enriched categories Metric spaces Quantale valued logic Haces Cuantales Categorías enriquecidas Espacios métricos Lógica cuantal valuada
title_short	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales
title_full	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales
title_fullStr	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales
title_full_unstemmed	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales
title_sort	Internal and external aspects of continuous logic and categorical logic for sheaves over quantales
dc.creator.fl_str_mv	Reyes Gaona, David
dc.contributor.advisor.none.fl_str_mv	Mariano, Hugo Luiz Zambrano Ramírez, Pedro Hernán
dc.contributor.author.none.fl_str_mv	Reyes Gaona, David
dc.contributor.researchgroup.spa.fl_str_mv	Interacciones Entre Teoría de Modelos, Teoría de Conjuntos, Categorías, Análisis y Geometría
dc.contributor.researchgate.spa.fl_str_mv	Reyes, David
dc.subject.ddc.spa.fl_str_mv	510 - Matemáticas::514 - Topología 510 - Matemáticas::512 - Álgebra 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::515 - Análisis 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::514 - Topología 510 - Matemáticas::512 - Álgebra 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::515 - Análisis 510 - Matemáticas::511 - Principios generales de las matemáticas
topic	510 - Matemáticas::514 - Topología 510 - Matemáticas::512 - Álgebra 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::515 - Análisis 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::514 - Topología 510 - Matemáticas::512 - Álgebra 510 - Matemáticas::511 - Principios generales de las matemáticas 510 - Matemáticas::515 - Análisis 510 - Matemáticas::511 - Principios generales de las matemáticas Algebra-métodos gráficos Lógica Algebra - Graphic methods Logic Sheaves Quantales Enriched categories Metric spaces Quantale valued logic Haces Cuantales Categorías enriquecidas Espacios métricos Lógica cuantal valuada
dc.subject.lemb.spa.fl_str_mv	Algebra-métodos gráficos Lógica
dc.subject.lemb.eng.fl_str_mv	Algebra - Graphic methods Logic
dc.subject.proposal.eng.fl_str_mv	Sheaves Quantales Enriched categories Metric spaces Quantale valued logic
dc.subject.proposal.spa.fl_str_mv	Haces Cuantales Categorías enriquecidas Espacios métricos Lógica cuantal valuada
description	In this text we explore and propose notions of sheaves over commutative, integral quantales, which are based on extensions of results of the theory of sheaves over locales: the interplay of sheaves as valued-sets and the analogy of sheaves as enriched categories. Over these proposals, we define logics that find semantics in these sheaf-like objects, on the one hand, a categorical logic that characterize the notion of sheaves associated to complete valued sets as a model of certain internal construction, and in contrast an externally defined logic whose nature is based on continuous logic for metric spaces which finds in the proposal of sheaves as enriched categories an structure for interpret the semantic. (Texto tomado de la fuente)
publishDate	2023
dc.date.accessioned.none.fl_str_mv	2023-11-30T14:23:58Z
dc.date.available.none.fl_str_mv	2023-11-30T14:23:58Z
dc.date.issued.none.fl_str_mv	2023
dc.type.spa.fl_str_mv	Trabajo de grado - Maestría
dc.type.driver.spa.fl_str_mv	info:eu-repo/semantics/masterThesis
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dc.identifier.instname.spa.fl_str_mv	Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv	Repositorio Institucional Universidad Nacional de Colombia
dc.identifier.repourl.spa.fl_str_mv	https://repositorio.unal.edu.co/
url	https://repositorio.unal.edu.co/handle/unal/85026 https://repositorio.unal.edu.co/
identifier_str_mv	Universidad Nacional de Colombia Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv	eng
language	eng
dc.relation.references.spa.fl_str_mv	Kopperman, R. (1988). All topologies come from generalized metrics. American Mathematical Monthly, 95(2), 89-97. Walters, R. F. C. (1981). "Sheaves and Cauchy-Complete categories." Cahiers de topologie et géométrie différentielle catégoriques, 22(3), 283-286. Walters, R. F. C. (1982). "Sheaves on sites as Cauchy-Complete categories." Journal of pure and applied Algebra, 24, 95-102. Benabou, J. (1973). "Les distributeurs." Inst. Math. Pure Appl Univ. Louvain-la-Neuve, 33, 161-189. Lawvere, F. W. (1973). "Metric spaces, generalized logic, and closed categories." Rendiconti del seminario matématico e fisico de Milano, 43, 135-166. Kopperman, R. (1981). "First order topological axioms." The Journal of Symbolic Logic, 46, 475-489. Flagg, R. (1992). "Completeness In Continuity Spaces." Canadian Mathematical Society, 13, 183-199. Flagg, R., & Kopperman, R. (1997). "Continuity spaces: Reconciling domains and metric spaces." Theoretical Computer Science, 177, 111-138. Flagg, R. (1997). "Quantales and continuity spaces." Algebra Universalis, 37, 257-276. Stubbe, I. (2005). "Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid." Applied Categorical Structures, 13(3), 235-255. Stubbe, I. (2005). "Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories." Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46, 99-121. Stubbe, I. (2005). "Categorical structures enriched in a quantaloid: categories, distributors and functors." Theory Appl. Categ, 14, 1-45. Borceux, F., van de Bossche (1986). "Quantales and their sheaves." Order, 3, 61-87. Miraglia, F., & Solitro, U. (1998). "Sheaves over right sided idempotent quantales." Logic Journal of IGPL, 6(4), 545-600. Höhle, U. (1998). "GL-quantales: Q-valued sets and their singletons." Studia logica, 61, 123-148. Resende, P. (2011). "Grupoid sheaves as quantale sheaves." J. Pure Appl. Algebra, 216, 41-70. Bénabou, J. (1967). "Introduction to bicategories." Lecture Notes in Math, 47, 1-77. Mulvey, C. (1986). "J. 1986." Suppl. Rend. Circ. Mat. Palermo Ser, 2, 99-104. Hyland, J., Johnstone, P., & Pitts, A. (1980). "Tripos Theory." Mathematical Proceedings of the Cambridge Philosophical Society, 88 (2), 205-232. Pitts, A. M. (1999). "Tripos Theory in Retrospect." Electronic Notes in Theoretical Computer Science, 23, 111-127. Weiss, I. (2018). "Value semigroups, values quantales, and positivity domains." 27. Lieberman, M., Rosicky, J., & Zambrano, P. (2018). "Tameness in generalized metric structures." 22. [Preprint]. https://arxiv.org/abs/1810.02317 Shulman, M. A. (2010). "Stack semantics and the comparison of material and structural set theories." [Preprint]. https://arxiv.org/abs/arXiv:1004.3802 Reyes, D., & Zambrano, P. (2021). "Co-quantale valued logics." [Preprint]. https://arxiv.org/abs/arXiv:2102.06067 Hofman, D., & Reis, C. (2017). "Convergence and quantale-enriched categories." [Preprint]. https://arxiv.org/abs/arXiv:1705.08671 Alvim, J. G., Mendes, C. A., & Mariano, H. L. (2023). "{$Q$}-Sets and Friends: Categorical Constructions and Categorical Properties." [Preprint]. https://arxiv.org/abs/arXiv:2302.03123 Alvim, J. G., Mendes, C. A., & Mariano, H. L. (2023). "{$Q$}-Sets and Friends: Regarding Singleton and Gluing Completeness." [Preprint]. https://arxiv.org/abs/arXiv:2302.03691 Tenório, A. L., Mendes, C. A., & Mariano, H. L. (2022). "Introducing sheaves over commutative semicartesian quantales." [Preprint]. https://arxiv.org/abs/arXiv:2204.08351 Ben-Yaacov, I., Berenstein, A., Henson, C. W., & Usvyatsov, A. (2008). "Model theory for metric structures." In Chatzidakis, Z., Macpherson, D., Pillay, A., Wilkie, A. (Eds.), Model Theory with Applications to Algebra and Analysis (Vol. 2, pp. 315–427). Cambridge: Cambridge University Press. DOI: 10.1017/CBO9780511735219.011 Bell, J. (2005). "Set Theory: Boolean-Valued Models and Independence Proofs" (3rd ed.). Oxford: Oxford University Press. Schweizer, B., & Sklar, A. (1983). "Probabilistic Metric Spaces." Amsterdam: North Holland. Borceux, F. (1994). "Handbook of Categorical Algebra, Volume 3, Sheaf Theory." Cambridge: Cambridge University Press. McLarty, C. (1992). "Elementary Categories, Elementary Toposes." Oxford: Clarendon Press. Mac Lane, S., & Moerdijk, I. (1992). "Sheaves in Geometry and Logic: A First Introduction to Topos Theory." Springer. Johnstone, P. T. (2002). "Sketches of an Elephant: Topos Theory Compendium." Oxford: Oxford University Press. A. L. da Conceição Tenório, C. de Andrade Mendes, J. Goudet Alvim, H.L. Mariano. "Sheaves over quantales and Grothendieck L-topoi." Work in progress, Hugo Mariano students in IME-USP, 202X. de Andrade Mendes C., Mariano H.L. "Sheaf-like categories over semicartesian quantales and applications." PhD Thesis, Work in progress, Hugo Mariano student in IME-USP, 202X. Moncayo V. J. R., Zambrano P.H. "Constructible sets in lattice-valued models." Master Thesis, Pedro Zambrano student in UNAL (Bog), 2023. K.I. Rosenthal. "Quantales and Their Applications." Pitman Research Notes in Mathematics Series, Harlow, UK, 1990. M.P. Fourman, D.S. Scott. "Sheaves and Logic." Lectures Notes in Mathematics, Springer 753, 1979. L.M. Acosta. "Temas de teoría de retículos." Universidad Nacional De Colombia, Bogotá, Colombia, 2015. G.M. Kelly. "Basic Concepts of Enriched Category Theory." Theory and Applications of Categories, 2005. G. Gierz, K. H. Hofmann, K. K., J. Lawson, M. Mislove, D. Scott. "A Compendium of Continuous Lattices." Springer-Verlag Berlin Heidelberg, 1980. M. Goldstern, H. Judah. "The Incompleteness Phenomenon: A New Course in Mathematical Logic." A K Peters, 1998. C. C. Chang, H. J. Keisler. "Continuous Model Theory." Princeton University Press, 1966. Hausdorff F. "Grundzüge der Mengenlehre." Cambridge University Press, Veit, Leipzig, 1914. Hofmann D., Seal G., Tholen W. "Monoidal Topology: A Categorical Approach to Order, Metric and Topology." New York: Cambridge University Press, 2014. John L. Bell. "Set Theory: Boolean-valued Models and Independence Proofs." Oxford Logic Guides, Clarendon Press, volume 47, Oxford, United Kingdom, 2005. D. Scott. (1972). "Continuous lattices." Lecture Notes in Mathematics - Springer-verlag-, 274, 97-136. DOI: 10.1007/BFb0073967. José Goudet Alvim, Arthur Francisco Schwerz Cahali, Hugo Luiz Mariano. (2022). "Induced Morphisms between Heyting-valued Models." Journal of Applied Logics, 9, 5-40. nLab. (2023). "Hyperdoctrine." Recuperado de https://ncatlab.org/nlab/show/hyperdoctrine. nLab. (2023). "Karoubi envelope." Recuperado de https://ncatlab.org/nlab/show/Karoubi+envelope. Iovino, J. (1995). Stable Banach Spaces and Banach Space Structures, I: Fundamentals. En C. Raymond (Ed.), Handbook of Metric Fixed Point Theory (pp. 329-386). Taylor & Francis. DOI: 10.1201/9780429332890-10 Henson, C. W., & Iovino, J. (2003). Ultraproducts in Analysis. En Editores del libro (Eds.), Analysis and Logic (pp. xi-xiv). Cambridge University Press. DOI: https://doi.org/10.1017/CBO9781107360006.002
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spelling	Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Mariano, Hugo Luizecebb601694b29c422dc1f913ed72b23Zambrano Ramírez, Pedro Hernán81ad894ee4d503e5a9459e9ca6218ef3Reyes Gaona, David57da3ccaacb8455a32f46e12096de01aInteracciones Entre Teoría de Modelos, Teoría de Conjuntos, Categorías, Análisis y GeometríaReyes, David2023-11-30T14:23:58Z2023-11-30T14:23:58Z2023https://repositorio.unal.edu.co/handle/unal/85026Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/In this text we explore and propose notions of sheaves over commutative, integral quantales, which are based on extensions of results of the theory of sheaves over locales: the interplay of sheaves as valued-sets and the analogy of sheaves as enriched categories. Over these proposals, we define logics that find semantics in these sheaf-like objects, on the one hand, a categorical logic that characterize the notion of sheaves associated to complete valued sets as a model of certain internal construction, and in contrast an externally defined logic whose nature is based on continuous logic for metric spaces which finds in the proposal of sheaves as enriched categories an structure for interpret the semantic. (Texto tomado de la fuente)En este texto exploramos y proponemos nociones de haces sobre cuantales conmutativos e integrales, basadas en extensiones de resultados de la teoría de haces sobre locales: la interacción de los haces como conjuntos valuados y la analogía de los haces como categorías enriquecidas. Sobre estas propuestas, definimos lógicas que encuentran su semántica en estos objetos tipo haz; por un lado, una lógica categórica que caracteriza la noción de haces asociada a conjuntos valuados completos como un modelo de cierta construcción interna, y en contraste, una lógica definida externamente cuya naturaleza se basa en la lógica continua para espacios métricos, la cual encuentra en la propuesta de haces como categorías enriquecidas una estructura para interpretar su semántica.MaestríaMaestría en MatemáticasLógica matemáticax, 112 páginasapplication/pdfengUniversidad Nacional de ColombiaBogotá - Ciencias - Maestría en Ciencias - MatemáticasFacultad de CienciasBogotá, ColombiaUniversidad Nacional de Colombia - Sede Bogotá510 - Matemáticas::514 - Topología510 - Matemáticas::512 - Álgebra510 - Matemáticas::511 - Principios generales de las matemáticas510 - Matemáticas::515 - Análisis510 - Matemáticas::511 - Principios generales de las matemáticas510 - Matemáticas::514 - Topología510 - Matemáticas::512 - Álgebra510 - Matemáticas::511 - Principios generales de las matemáticas510 - Matemáticas::515 - Análisis510 - Matemáticas::511 - Principios generales de las matemáticasAlgebra-métodos gráficosLógicaAlgebra - Graphic methodsLogicSheavesQuantalesEnriched categoriesMetric spacesQuantale valued logicHacesCuantalesCategorías enriquecidasEspacios métricosLógica cuantal valuadaInternal and external aspects of continuous logic and categorical logic for sheaves over quantalesAspectos internos y externos de lógica continua y lógica categórica para haces sobre cuantalesTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMKopperman, R. (1988). All topologies come from generalized metrics. American Mathematical Monthly, 95(2), 89-97.Walters, R. F. C. (1981). "Sheaves and Cauchy-Complete categories." Cahiers de topologie et géométrie différentielle catégoriques, 22(3), 283-286.Walters, R. F. C. (1982). "Sheaves on sites as Cauchy-Complete categories." Journal of pure and applied Algebra, 24, 95-102.Benabou, J. (1973). "Les distributeurs." Inst. Math. Pure Appl Univ. Louvain-la-Neuve, 33, 161-189.Lawvere, F. W. (1973). "Metric spaces, generalized logic, and closed categories." Rendiconti del seminario matématico e fisico de Milano, 43, 135-166.Kopperman, R. (1981). "First order topological axioms." The Journal of Symbolic Logic, 46, 475-489.Flagg, R. (1992). "Completeness In Continuity Spaces." Canadian Mathematical Society, 13, 183-199.Flagg, R., & Kopperman, R. (1997). "Continuity spaces: Reconciling domains and metric spaces." Theoretical Computer Science, 177, 111-138.Flagg, R. (1997). "Quantales and continuity spaces." Algebra Universalis, 37, 257-276.Stubbe, I. (2005). "Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid." Applied Categorical Structures, 13(3), 235-255.Stubbe, I. (2005). "Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories." Cahiers de Topologie et Géométrie Différentielle Catégoriques, 46, 99-121.Stubbe, I. (2005). "Categorical structures enriched in a quantaloid: categories, distributors and functors." Theory Appl. Categ, 14, 1-45.Borceux, F., van de Bossche (1986). "Quantales and their sheaves." Order, 3, 61-87.Miraglia, F., & Solitro, U. (1998). "Sheaves over right sided idempotent quantales." Logic Journal of IGPL, 6(4), 545-600.Höhle, U. (1998). "GL-quantales: Q-valued sets and their singletons." Studia logica, 61, 123-148.Resende, P. (2011). "Grupoid sheaves as quantale sheaves." J. Pure Appl. Algebra, 216, 41-70.Bénabou, J. (1967). "Introduction to bicategories." Lecture Notes in Math, 47, 1-77.Mulvey, C. (1986). "J. 1986." Suppl. Rend. Circ. Mat. Palermo Ser, 2, 99-104.Hyland, J., Johnstone, P., & Pitts, A. (1980). "Tripos Theory." Mathematical Proceedings of the Cambridge Philosophical Society, 88 (2), 205-232.Pitts, A. M. (1999). "Tripos Theory in Retrospect." Electronic Notes in Theoretical Computer Science, 23, 111-127.Weiss, I. (2018). "Value semigroups, values quantales, and positivity domains." 27.Lieberman, M., Rosicky, J., & Zambrano, P. (2018). "Tameness in generalized metric structures." 22. [Preprint]. https://arxiv.org/abs/1810.02317Shulman, M. A. (2010). "Stack semantics and the comparison of material and structural set theories." [Preprint]. https://arxiv.org/abs/arXiv:1004.3802Reyes, D., & Zambrano, P. (2021). "Co-quantale valued logics." [Preprint]. https://arxiv.org/abs/arXiv:2102.06067Hofman, D., & Reis, C. (2017). "Convergence and quantale-enriched categories." [Preprint]. https://arxiv.org/abs/arXiv:1705.08671Alvim, J. G., Mendes, C. A., & Mariano, H. L. (2023). "{$Q$}-Sets and Friends: Categorical Constructions and Categorical Properties." [Preprint]. https://arxiv.org/abs/arXiv:2302.03123Alvim, J. G., Mendes, C. A., & Mariano, H. L. (2023). "{$Q$}-Sets and Friends: Regarding Singleton and Gluing Completeness." [Preprint]. https://arxiv.org/abs/arXiv:2302.03691Tenório, A. L., Mendes, C. A., & Mariano, H. L. (2022). "Introducing sheaves over commutative semicartesian quantales." [Preprint]. https://arxiv.org/abs/arXiv:2204.08351Ben-Yaacov, I., Berenstein, A., Henson, C. W., & Usvyatsov, A. (2008). "Model theory for metric structures." In Chatzidakis, Z., Macpherson, D., Pillay, A., Wilkie, A. (Eds.), Model Theory with Applications to Algebra and Analysis (Vol. 2, pp. 315–427). Cambridge: Cambridge University Press. DOI: 10.1017/CBO9780511735219.011Bell, J. (2005). "Set Theory: Boolean-Valued Models and Independence Proofs" (3rd ed.). Oxford: Oxford University Press.Schweizer, B., & Sklar, A. (1983). "Probabilistic Metric Spaces." Amsterdam: North Holland.Borceux, F. (1994). "Handbook of Categorical Algebra, Volume 3, Sheaf Theory." Cambridge: Cambridge University Press.McLarty, C. (1992). "Elementary Categories, Elementary Toposes." Oxford: Clarendon Press.Mac Lane, S., & Moerdijk, I. (1992). "Sheaves in Geometry and Logic: A First Introduction to Topos Theory." Springer.Johnstone, P. T. (2002). "Sketches of an Elephant: Topos Theory Compendium." Oxford: Oxford University Press.A. L. da Conceição Tenório, C. de Andrade Mendes, J. Goudet Alvim, H.L. Mariano. "Sheaves over quantales and Grothendieck L-topoi." Work in progress, Hugo Mariano students in IME-USP, 202X.de Andrade Mendes C., Mariano H.L. "Sheaf-like categories over semicartesian quantales and applications." PhD Thesis, Work in progress, Hugo Mariano student in IME-USP, 202X.Moncayo V. J. R., Zambrano P.H. "Constructible sets in lattice-valued models." Master Thesis, Pedro Zambrano student in UNAL (Bog), 2023.K.I. Rosenthal. "Quantales and Their Applications." Pitman Research Notes in Mathematics Series, Harlow, UK, 1990.M.P. Fourman, D.S. Scott. "Sheaves and Logic." Lectures Notes in Mathematics, Springer 753, 1979.L.M. Acosta. "Temas de teoría de retículos." Universidad Nacional De Colombia, Bogotá, Colombia, 2015.G.M. Kelly. "Basic Concepts of Enriched Category Theory." Theory and Applications of Categories, 2005.G. Gierz, K. H. Hofmann, K. K., J. Lawson, M. Mislove, D. Scott. "A Compendium of Continuous Lattices." Springer-Verlag Berlin Heidelberg, 1980.M. Goldstern, H. Judah. "The Incompleteness Phenomenon: A New Course in Mathematical Logic." A K Peters, 1998.C. C. Chang, H. J. Keisler. "Continuous Model Theory." Princeton University Press, 1966.Hausdorff F. "Grundzüge der Mengenlehre." Cambridge University Press, Veit, Leipzig, 1914.Hofmann D., Seal G., Tholen W. "Monoidal Topology: A Categorical Approach to Order, Metric and Topology." New York: Cambridge University Press, 2014.John L. Bell. "Set Theory: Boolean-valued Models and Independence Proofs." Oxford Logic Guides, Clarendon Press, volume 47, Oxford, United Kingdom, 2005.D. Scott. (1972). "Continuous lattices." Lecture Notes in Mathematics - Springer-verlag-, 274, 97-136. DOI: 10.1007/BFb0073967.José Goudet Alvim, Arthur Francisco Schwerz Cahali, Hugo Luiz Mariano. (2022). "Induced Morphisms between Heyting-valued Models." Journal of Applied Logics, 9, 5-40.nLab. (2023). "Hyperdoctrine." Recuperado de https://ncatlab.org/nlab/show/hyperdoctrine.nLab. (2023). "Karoubi envelope." Recuperado de https://ncatlab.org/nlab/show/Karoubi+envelope.Iovino, J. (1995). Stable Banach Spaces and Banach Space Structures, I: Fundamentals. En C. Raymond (Ed.), Handbook of Metric Fixed Point Theory (pp. 329-386). Taylor & Francis. DOI: 10.1201/9780429332890-10Henson, C. W., & Iovino, J. (2003). Ultraproducts in Analysis. En Editores del libro (Eds.), Analysis and Logic (pp. xi-xiv). Cambridge University Press. 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Internal and external aspects of continuous logic and categorical logic for sheaves over quantales

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