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References

Published online by Cambridge University Press:  16 March 2018

Marta Bunge
Affiliation:
McGill University, Montréal
Felipe Gago
Affiliation:
Universidade de Santiago de Compostela, Spain
Ana María San Luis
Affiliation:
Universidad de Oviedo, Spain
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References

[1] W., Ambrose, R. S., Palais, and I. M., Singer, Sprays, Anais da Acad. Bras. de Ciências 32-2 (1960) 163–178.Google Scholar
[2] V. I., Arnold, Normal forms of functions in the neighbourhood of degenerate critical points, Uspekhi Matematicheskykh Nauk 29 (1974) 11–49, Russian Mathematical Surveys 29 19–48.Google Scholar
[3] V. I., Arnold, Singularity Theory, in: Jean-Paul, Pier (ed.), Development of Mathematics: 1950 - 2000, Birkhäuser, Basel-Boston-New York (2000) 127–152.Google Scholar
[4] M., Artin, A., Grothendieck, J. L., Verdier, Theórie des Topos et Cohomologie Étale des Schémas Lect. Notes in Math. 269 Exposé IV. Springer-Verlag, Berlin, 1972.
[5] M., Barr, Toposes without points, J. Pure Appl. Algebra 5 (1974) 265–280.Google Scholar
[6] M., Barr and C., Wells, Toposes, Triples and Theories, Springer-Verlag, New York-Berlin-Heilderberg-Tokyo, 1985.Google Scholar
[7] A., Bauer, Five stages of accepting constructive mathematics, Bull. Amer. Math. Soc. 54-3 (2017) 481–498.Google Scholar
[8] L., Belair, Calcul Infinitesimal en Géométrie Différentielle Synthétique, M.Sc. Thesis, Université de Montréal, 1981.Google Scholar
[9] M., Berger and B., Gostiaux, Differential Geometry: Manifolds, Curves, and Surfaces, Springer-Verlag, Berlin-Heidelberg-New York, 1988.Google Scholar
[10] F., Bergeron, Objets infinitésimalment linéaires dans un modele bien adapté de la G.D.S., in: G., E (ed.), Géométrie Différentielle Synthétique, Rapports de Recherche, DMS 80-12, Université de Montréal, 1980.Google Scholar
[11] J. M., Boardman, Singularities of Differentiable Maps, Pub. Math. I. H. E. S. 33 (1967) 21–57.Google Scholar
[12] F., Borceux, Handbook of Categorical Algebra 3, Cambridge University Press, 1994.
[13] Th., Bröcker, Differentiable Germs and Catastrophes, Cambridge University Press, 1982.Google Scholar
[14] O., Bruno, Logical opens of exponential objects, Cahiers de Top. et Géom. Diff. Catégoriques 26 (1985) 311–323.Google Scholar
[15] O., Bruno, Vector fields on RR in well adapted models of synthetic differential geometry, J. Pure Appl. Algebra 45 (1987) 1–14.Google Scholar
[16] M. C., Bunge, Categories of Set-Valued Functors, Ph.D. Thesis, University of Pennsylvania, 1966.Google Scholar
[17] M. C., Bunge, Relative functor categories and categories of algebras, J. Algebra 11 (1969) 64–101.Google Scholar
[18] M. C., Bunge, Internal presheaf toposes, Cahiers de Top. et Géom. Diff. Catégoriques 18-3 (1977) 291–330.Google Scholar
[19] M. C., Bunge, Sheaves and prime model extensions, J. Algebra 68 (1981) 79–96.Google Scholar
[20] M. C., Bunge, Synthetic aspects of C∞-mappings, J. Pure Appl. Algebra 28 (1983) 41–63.Google Scholar
[21] M. C., Bunge, Toposes in logic and logic in toposes, Topoi 3 (1984) 13–22.Google Scholar
[22] M. C., Bunge, On a synthetic proof of the Ambrose-Palais-Singer theorem for infinitesimally linear spaces, Cahiers de Top. et Géom. Diff. Catégoriques 28-2 (1987) 127–142.Google Scholar
[23] M. C., Bunge, Cosheaves and distributions on toposes, Algebra Universalis 34 (1995) 469–484.Google Scholar
[24] M., C and E. J., Dubuc, Archimedian local C∞-rings and models of Synthetic Differential Geometry, Cahiers de Top. et Géom. Diff. Catégoriques 27-3 (1986) 3–22.Google Scholar
[25] M., C and E. J., Dubuc, Local concepts in SDG and germ representability, in: D., Kueker et al. (eds) Mathematical Logic and Theoretical Computer Science, Marcel Dekker, New York and Basel, 1987, 93–159.Google Scholar
[26] M., C and F., Gago Couso, Synthetic aspects of C∞-mappings II : Mather's theorem for infinitesimally represented germs, J. Pure Appl. Algebra 55 (1988) 213–250.Google Scholar
[27] M., C and M., Heggie, Synthetic calculus of variations, Contemporary Mathematics 30 (1984) 30–62.Google Scholar
[28] M., C and P., Sawyer, On connections, geodesics and sprays in synthetic differential geometry, Cahiers de Top. et Géom. Diff. Catégoriques 25 (1984) 221–258.Google Scholar
[29] K. T., Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831–879.Google Scholar
[30] K. T., Chen, The Euler operator, Archive for Rational Mechanics and Analysis 75 (1981) 175–191.Google Scholar
[31] E. J., Dubuc, Adjoint triangles, in: Reports of the Midwest Category Seminar II, Lect. Notes in Math. 69 (1968) 69–91.Google Scholar
[32] E. J., Dubuc, Sur les modèles de la géométrie differentielle synthétique, Cahiers de Top. et Géom. Diff. Catégoriques 20 (1979) 231–279.Google Scholar
[33] E. J., Dubuc, Schémas C∞, in [100], (1981) Exposé 3, pp 16–41.
[34] E. J., Dubuc, C∞-schemes, American J. Math. 103–104 (1981) 683–690.
[35] E. J., Dubuc, Germ representability and local integration of vector fields in a well adapted model of SDG, J. Pure Appl. Algebra 64 (1990) 131–144.Google Scholar
[36] E., J and J., Penon, Objects compacts dans les topos, J. Austral. Math. Soc. A-40 (1986) 203–217.Google Scholar
[37] E., J and G. E., Reyes, Subtoposes of the ring classifier, in: A., Kock (ed.), Topos Theoretic Methods in Geometry, Various Publications Series 30, Matematisk Institut, Aarhus Universitet, 1979, 101–122.Google Scholar
[38] E., J and G., Taubin, Analytic rings, Cahiers de Top. et Géom. Diff. Catégoriques 24 (1983) 225–265.Google Scholar
[39] M., Dummet, Elements of Intuitionism, Clarendon Press, Oxford, 1977.Google Scholar
[40] C., Ehresmann, Les prolongements d'une variété différentiable I, II, III, C. R. Acad. Sc. Paris (1951), Reprinted in Charles Ehresmann : Oeuvres Complêtes et Commentées. Part I, Amiens, 1984.
[41] M., Fourman and M., Hyland, Sheaf models for analysis, in: Applications of Sheaves, Lect. Notes in Math. 753 (1979) 280–301.Google Scholar
[42] P. J., Freyd, Intrinsic Differential Geometry, Unpublished lecture, Philadelphia, 1983.
[43] A., Frölicher, Applications lisses entre espaces et variétés de Frechet, C. R. Acad. Sci. Paris Ser. I Math 293 (1981) 125–127.Google Scholar
[44] F., Gago Couso, Internal Weak Opens, Internal Stability and Morse Theory for Synthetic Germs, Ph.D. Thesis, McGill University, 1988.Google Scholar
[45] F., Gago Couso, Morse Germs in SDG, in: Categorical Algebra and its Applications, Lect. Notes in Math. 1348 (1988) 125–129.Google Scholar
[46] F., Gago Couso, Singularités dans la Géométrie Différentielle Synthétique, Bull. Soc. Math. Belgique 41 (1989) 279–287.Google Scholar
[47] M., Golubitsky and V., Guillemin, Stable Mappings and their Singularities, Springer-Verlag, New York-Heidelberg-Berlin, 1973.Google Scholar
[48] M., Goretsky and R., MacPherson, Stratified Morse Theory, Springer-Verlag, New York-Heidelberg-Berlin, 1987.Google Scholar
[49] I., M and S. V., Fomin, Calculus of Variations, Prentice Hall, Inc., London, 1963.Google Scholar
[50] R., Grayson, Concepts of general topology in constructive mathematics and sheaves, Annals of Math, Logic. 20 (1981) 1–41.Google Scholar
[51] V., Guillemin and A., Pollack, Differential Topology, Prentice Hall Inc, London, 1974.Google Scholar
[52] M., Heggie, An Approach to Synthetic Variational Theory, M.Sci. Thesis, McGill University, 1982.Google Scholar
[53] A., Heyting, Intuitionism. An Introduction, North Holland, 1956.Google Scholar
[54] M. W., Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer-Verlag, 1976.Google Scholar
[55] P. T., Johnstone, Topos Theory, Academic Press, Inc., London, 1977.Google Scholar
[56] P. T., Johnstone, Sketches of an Elephant. A Topos Theory Companion, Volumes 1 and 2, Oxford Science Publications, Oxford Logic Guides 43 44, Clarendon Press, Oxford 2002.Google Scholar
[57] A., Joyal, Les théorèmes de Chevalley-Tarski el remarques sur l'Algébre constructive, Cahiers de Top. et Géom. Diff. Catégoriques 16 (1975) 256–258.Google Scholar
[58] S. C., Kleene, Introduction to Metamathematics, North Holland, 1962.Google Scholar
[59] A., Kock, Universal projective geometry via topos theory, J. Pure Appl. Algebra 9 (1976) 1–24.Google Scholar
[60] A., Kock, Formal manifolds and synthetic theory of jet bundles, Cahiers de Top. et Géom. Diff. Catégoriques 21 (1980) 227–246.Google Scholar
[61] A., Kock, Synthetic Differential Geometry (First and Second Editions), Cambridge University Press, 1981 and 2006.Google Scholar
[62] A., Kock, Synthetic characterization of reduced algebras, J. Pure Appl. Algebra 36 (1985) 273–279.Google Scholar
[63] A., Kock and R., Lavendhomme, Strong infinitesimal linearity, with applications to strong difference and affine connections, Cahiers de Top. et Géom. Diff. Catégoriques 25-3 (1984) 311–324.Google Scholar
[64] A., Kock and G. E., Reyes, Manifolds in formal differential geometry, in: Applications of Sheaves. Proceedings, Durham 1977, Lect. Notes in Math. 753 (1979) 514–533.Google Scholar
[65] A., Kock and G. E., Reyes, Connections in Formal Differential Geometry, in: A., Kock (ed.), Topos Theoretic Methods in Geometry, Various Publications Series 30, Matematisk Institut, Aarhus Universitet, 1979, 158–195.Google Scholar
[66] A., Kock and G. E., Reyes, Aspects of fractional exponent functors, Theory and Applications of Categories 5 (1999) 251–265.Google Scholar
[67] S. A., Kripke, Semantical analysis of intuitionistic logic, in: J.N., Crossley and M., Dummett (eds.), Formal Systems and Recursive Functions, North-Holland, Amsterdam, 1965, 92–130.Google Scholar
[68] S., Lang, Introduction to Differentiable Manifolds, Interscience Publishers, New York London Sydney, 1967.Google Scholar
[69] R., Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Kluwer Academic Publishers, 1996.Google Scholar
[70] F. W., Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. U.S.A. 50 (1963) 869–872.Google Scholar
[71] F. W., Lawvere, An elementary theory of the category of sets, Proc. Nat. Acad. Sci. U.S.A. 52 (1964) 1506–1511.Google Scholar
[72] F. W., Lawvere, Categorical Dynamics, Talk at the University of Chicago, May 1967, in: A., Kock (ed.), Topos Theoretic Methods in Geometry, Various Publications Series 30, Matematisk Institut, Aarhus Universitet, 1979, 1–28.Google Scholar
[73] F. W., Lawvere, Quantifiers and sheaves, Actes du Congrès International des Mathématiciens, Nice 1970, 329–334.
[74] F. W., Lawvere, Towards the description in the smooth topos of the dynamically possible notions and deformations in a continuous body, Cahiers de Top. et Géom. Diff. Catégoriques 21 (1980) 377–392.Google Scholar
[75] F. W., Lawvere, Tiny objects, Unpublished talk at Aarhus University, June 1983.
[76] F. W., Lawvere, Outline of Synthetic Differential Geometry, Unpublished 1988. Available from http://www.acsu.buffalo.edu/wlawvere/SDG Outline.pdf. Accessed 8 March 2017.
[77] F., W, Comments on the development of topos theory, in: J. P., Pier (ed.), Development of Mathematics 1950-2000, Birkhäuser, Basel-Boston-Berlin, 2000, 715–734.Google Scholar
[78] F. W., Lawvere, Categorical algebra for continuum microphysics, J. Pure Appl. Algebra 175 (2002) 267–287.Google Scholar
[79] C., McLarty, Elementary Categories, Elementary Toposes, Oxford Logic Guides 21, Oxford University Press, 1995.
[80] M., Makkai and G. E., Reyes, First-order Categorical Logic, Lect. Notes in Math. 611, Springer-Verlag, 1977.
[81] B., Malgrange, Ideals of Differentiable Functions, Oxford University Press, 1966.Google Scholar
[82] J., Martinet, Singularities of Smooth Functions and Maps, Cambridge University Press, Cambridge, 1982.Google Scholar
[83] J. N., Mather, Stability of C∞-mappings I : The division theorem, Annals of Mathematics 87-1 (1969) 89–104.Google Scholar
[84] J. N., Mather, Stability of C∞-mappings II : Infinitesimal stability implies stability, Annals of Mathematics 89-2 (1969) 254–291.Google Scholar
[85] J. N., Mather, Stability of C∞-mappings III : Finitely determined map germs, Pub. Sci. I.H.E.S. 35 (1969) 127–156.Google Scholar
[86] S., Mac Lane, Categories for the Working Mathematician (Second edition), Springer-Verlag, Berlin-Heidelberg-New York, 1997.Google Scholar
[87] J. M., Milnor, Morse Theory, Annals of Mathematical Studies 51, Princeton University Press, Princeton, N.J., 1969.Google Scholar
[88] J., M and J. D., Stasheff, Characteristic Classes, Annals of Mathematical Studies 76, Princeton University Press, Princeton, N.J., 1974.Google Scholar
[89] I., Moerdijk and G. E., Reyes, Smooth spaces versus continuous spaces in models for Synthetic Differential Geometry, Report 83-02, University of Amsterdam, 1983.Google Scholar
[90] I., Moerdijk and G. E., Reyes, Models for Smooth Infinitesimal Analysis, Springer- Verlag, 1991.Google Scholar
[91] J. R., Munkres, Topology from a Differentiable Viewpoint, University of Virginia Press, 1966.Google Scholar
[92] R., Paré, Colimits in topoi, Bull. Amer. Math. Soc. 80 (1974), no. 3, 556–561.Google Scholar
[93] L. N., Patterson, Connections and prolongations, Canad. J. Math. XXVII-4 (1975) 766–791.Google Scholar
[94] J., Penon, Infinitésimaux et intuitionisme, Cahiers de Top. et Géom. Diff. Catégoriques 22 (1980) 67–72.Google Scholar
[95] J., Penon, Le théoreme d'inversion locale en géométrie algébrique, Lecture given at Louvain-la-Neuve, Journées de faisceaux et logique, May 1982.
[96] J., Penon, De l'Infinitésimal au Local, Thése de Doctorat d'Etât, Université Paris VII, 1985, Diagrammes, Paris VII, 1985.Google Scholar
[97] V., Poénaru, Analyse Différentielle, Lect. Notes in Math. 371, Springer-Verlag, 1970.Google Scholar
[98] W. A., Poor, Differential Geometric Structures, McGraw Hill, 1981.Google Scholar
[99] N., Van Quê and G. E., Reyes, Smooth functors and synthetic calculus, in: A. S., Troelstra and D. van, Dalen (eds.), The L.E.J. Brower Centenary Symposium, North-Holland, Amsterdam-New York-Oxford, 1982, 377–396.Google Scholar
[100] G. E., Reyes, (ed.), Géométrie Différentielle Synthétique, Rapports de Recherche du Département des Mathématiques et Statistiques 80-11, 80–12, Université de Montréal, 1980.
[101] G. E., Reyes, A remark of M. Makkai on Postulate WA2 [25, 35]. Private communication, July 14, 2009.
[102] A., Robinson, Non-Standard Analysis, North-Holland, 1966.Google Scholar
[103] A. M., San Luis Fernández, Estabilidad Transversal de Gérmenes Representables Infinitesimalmente, Ph.D. Thesis, Universidad de Santiago de Compostela, 1999.
[104] A., Sard, The measure of critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942) 883–890.Google Scholar
[105] R., Sikorski, Boolean Algebras, Springer-Verlag, Berlin, 1964.Google Scholar
[106] R., Thom, Une lemme sur les applications différentiables, Bol. Soc. Mat. Mex. 2-1 (1956) 59–71.Google Scholar
[107] R., Thom, Stabilité Structurelle et Morphogénèse: Essai d'une Théorie Générale des Modèles, Benjamin-Cummings, 1972.Google Scholar
[108] R., Thom and H. L., Levine, Singularities of Differentiable Mappings I, Bonn 1959, Reprinted in: C. T. C., Wall (ed.), Proceedings of Liverpool Singularities Symposium, Lect. Notes in Math. 192, Springer-Verlag, 1972.Google Scholar
[109] A. S., Troelstra, Principles of Intuitionism, Lect. Notes in Math. 95, Springer- Verlag, 1969.
[110] G., Wassermann, Stability of Unfoldings, Lect. Notes in Math. 393, Springer- Verlag, 1970.
[111] A., Weil, Théorie des points proches sur les variétés differentiables, Colloq. Top. et Géom. Diff., Strasbourg, (1963) 111–117.
[112] S., Willard, General Topology, Addison-Wesley, 1968.Google Scholar
[113] D., Yetter, On right adjoints to exponential functors, J. Pure Appl. Algebra 45 (1987) 287–304. Corrections to “On right adjoints to exponential functors”, J. Pure Appl. Algebra 58 (1989) 103–105.Google Scholar

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  • References
  • Marta Bunge, McGill University, Montréal, Felipe Gago, Universidade de Santiago de Compostela, Spain, Ana María San Luis, Universidad de Oviedo, Spain
  • Book: Synthetic Differential Topology
  • Online publication: 16 March 2018
  • Chapter DOI: https://doi.org/10.1017/9781108553490.021
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  • References
  • Marta Bunge, McGill University, Montréal, Felipe Gago, Universidade de Santiago de Compostela, Spain, Ana María San Luis, Universidad de Oviedo, Spain
  • Book: Synthetic Differential Topology
  • Online publication: 16 March 2018
  • Chapter DOI: https://doi.org/10.1017/9781108553490.021
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Marta Bunge, McGill University, Montréal, Felipe Gago, Universidade de Santiago de Compostela, Spain, Ana María San Luis, Universidad de Oviedo, Spain
  • Book: Synthetic Differential Topology
  • Online publication: 16 March 2018
  • Chapter DOI: https://doi.org/10.1017/9781108553490.021
Available formats
×