Every notion of constancy is relative, beeing derived perceptually or conceptually as a limiting case of variation and the undisputed value of such notions in clarifying variation is always limited by that origin. (F. W. Lawvere [26])
Sunto
Dopo aver passato in rassegna le principali nozioni di categoria localmente interna rispetto ad un toposE, si mostra che questa nozione può essere utilmente descritta, e la sua teoria studiata, in termini di categorie arricchite sulla bicategoria SpanE.
Summary
After recalling the main notions of a category which is locally internal with respect to a given toposE, we show that such a notion can usefully be described, and its theory developed, in terms of categories enriched in the bicategory SpanE.
Bibliografia
Benabou J.,Fibrations petites et localement petites. C. R. Acad. Sci. Paris, 281 (1975), A897–900.
Benabou J.,Fibered categories and the foundations of naive category theory. J. Symbolic logic, 50 (1985), 1–37.
Betti R.,Base bicategories and an application to tree automata. Manoscritto, 1979.
Betti R.,Bicategorie di base. Quad. 2/5, Ist. Mat. Univ. Milano, 1981.
Betti R., Carboni A.,Cauchy-completion and the associated sheaf. Cahiers top. géom. diff., 23 (1982), 243–256.
Betti R., Carboni A., Street R. H., Walters R. F. C.,Variation through enrichment. J. Pure App. Alg., 29 (1983), 109–127.
Betti R., Kasangian S.,Tree automata and enriched category theory. Rend. Ist. Mat. Univ. Trieste, 17 (1985), 71–78.
Betti R., Walters R. F. C.,The symmetry of the Cauchy-completion of a category. Proc. Gummersbach Conf. 1981, Lect. Notes 962, Springer (1983), 8–12.
Betti R., Walters R. F. C.,Closed bicategories and variable category theory. Quad. 5, Dip. Mat. Univ. Milano, 1985.
Betti R., Walters, R. F. C.,On completeness of locally-internal categories. J. Pure App. Alg., 47 (1987), 105–117.
Betti R., Walters, R. F. C.,The calculus of ends over a base topos. J. Pure App. Alg., 56 (1989), 211–220.
Bianchi R.,La formula di Kan per le categorie variabili. Tesi di laurea, Milano, 1987.
Borceux F., Kelly G. M.,A notion of limit for enriched categories. Bull. Austr. Math. Soc., 12 (1975), 48–72.
Carboni A., Manoscritto, 1982.
Carboni A., Kasangian S., Street R.,Bicategories of spans and relations. J. Pure App. Alg., 33 (1984), 259–267.
Carboni A., Kasangian S., Walters R. F. C.,An axiomatics for bicategories of modules. J. Pure App. Alg., 45 (1987), 127–141.
Carboni A., Walters R. F. C.,Cartesian bicategories I. J. Pure App. Alg., 49 (1987), 11–32.
Giraud J.,Cohomologie non abélienne. Springer, 1971.
Higgs D.,A category approach to boolean-valued set theory. Lect. Notes Univ. Waterloo, 1973.
Johnstone P. T.,Topos theory. Academic Press, 1977.
Kasangian S., Walters R. F. C.,An abstract notion of glueing. Manoscritto, 1981.
Kelly G. M.,Basic concepts of enriched category theory. Cambridge University Press, 1982.
Lawvere F. W.,Category theory over a base topos. Lect. Notes Univ. Perugia, 1972–73.
Lawvere F. W.,Introduzione a «Toposes, algebraic geometry and logic». Lect. Notes 274, Springer 1972, 1–12.
Lawvere F. W.,Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano, 43 (1973), 135–166.
Lawvere F. W.,Continuously variable sets; algebraic geometry=geometric logic. Proc. ASL Logic Colloquium, Bristol 1973, (ed. H. E. Rose and J. C. Shepherdson), North Holland 1975, 135–156.
Lawvere F. W.,Variable quantities and variable structures in topoi. In «Algebra, Topology and Category Theory: a collection of papers in honor of S. Eilenberg», (ed. A. Heller and M. Tierney), Academic Press 1976, 101–131.
Paré R., Schumacher D.,Abstract families and the adjoint functor theorem. Lect. Notes 661, Springer 1978, 1–125.
Penon J.,Categories localement internes. C. R. Acad. Sci. Paris, 278 (1974), A1577–1580.
Street R. H.,Conspectus of variable categories. J. Pure App. Alg., 21 (1981), 307–338.
Street R. H.,Enriched categories and cohomology. Quaest. Mat., 6 (1983), 265–283.
Street R. H.,Absolute colimits in enriched categories. Cahiers top. geom. diff., 24 (1983), 377–379.
Street R. H.,Cauchy characterization of enriched categories. Rend. Sem. Mat. Fis. Univ. Milano, 51 (1983), 217–233.
Walters R. F. C.,Sheaves and Cauchy-complete categories. Cahiers top. geom. diff., 22 (1981), 283–286.
Walters R. F. C.,Sheaves on sites as Cauchy-complete categories. J. Pure App. Alg., 24 (1982), 95–102.
Author information
Authors and Affiliations
Additional information
(Conferenza tenuta il 26 ottobre 1987)
Rights and permissions
About this article
Cite this article
Betti, R. Categorie variabili. Seminario Mat. e. Fis. di Milano 57, 483–518 (1987). https://doi.org/10.1007/BF02925067
Issue Date:
DOI: https://doi.org/10.1007/BF02925067