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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some results on locally finitely presentable categories

Authors: M. Makkai and A. M. Pitts
Journal: Trans. Amer. Math. Soc. 299 (1987), 473-496
MSC: Primary 03G30; Secondary 03C20, 03C52, 18B05
MathSciNet review: 869216
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Abstract: We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors $ I:{\mathbf{C}} \to {\mathbf{D}}$ between small categories with finite limits for which the functor $ {I^{\ast}}:{\mathbf{LEX}}({\mathbf{D}},{\text{Set)}} \to {\mathbf{LEX}}{\text{(}}{\mathbf{C}}{\text{,Set)}}$ induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site $ \mathcal{C} = ({\mathbf{C}},J)$ on a category $ {\mathbf{C}}$ with finite limits, defined by a subcanonical Grothendieck topology $ J$, the closure in $ {\mathbf{LEX}}({\mathbf{C}},{\text{Set)}}$ under small limits and filtered colimits of the models of $ \mathcal{C}$ is the whole of $ {\mathbf{LEX}}({\mathbf{C}},{\text{Set)}}$.

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  • [B] J. Benabou, Introduction to bicategories, Lecture Notes in Math., vol. 47, Springer-Verlag, Berlin and New York, 1967, pp. 1-77.
  • [CK] C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
  • [C] M. Coste, Localisation dans les catégories de modeles, Thesis, Univ. Paris Nord, 1977.
  • [E] David P. Ellerman, Sheaves of structures and generalized ultraproducts, Ann. Math. Logic 7 (1974), 163–195. MR 0376340
  • [GU] Peter Gabriel and Friedrich Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Mathematics, Vol. 221, Springer-Verlag, Berlin-New York, 1971 (German). MR 0327863
  • [GZ] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125
  • [TT] P. T. Johnstone, Topos theory, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. London Mathematical Society Monographs, Vol. 10. MR 0470019
  • [FT I] P. T. Johnstone, Factorization theorems for geometric morphisms. I, Cahiers Topologie Géom. Différentielle 22 (1981), no. 1, 3–17. Third Colloquium on Categories (Amiens, 1980), Part II. MR 609154
  • [KR] Model theory, Handbook of mathematical logic, Part A, North-Holland, Amsterdam, 1977, pp. 3–313. Studies in Logic and the Foundations of Math., Vol. 90. With contributions by Jon Barwise, H. Jerome Keisler, Paul C. Eklof, Angus Macintyre, Michael Morley, K. D. Stroyan, M. Makkai, A. Kock and G. E. Reyes. MR 0491125
  • [CWM] Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
  • [M] M. Makkai, Remarks on papers by Y. Diers and H. Volger on sheaf representation, Abstracts Amer. Math. Soc. 5 (1984), #84T-18-76.
  • [MI] M. Makkai, Ultraproducts and categorical logic, Methods in mathematical logic (Caracas, 1983) Lecture Notes in Math., vol. 1130, Springer, Berlin, 1985, pp. 222–309. MR 799044, 10.1007/BFb0075314
  • [MR] Michael Makkai and Gonzalo E. Reyes, First order categorical logic, Lecture Notes in Mathematics, Vol. 611, Springer-Verlag, Berlin-New York, 1977. Model-theoretical methods in the theory of topoi and related categories. MR 0505486
  • [V] Hugo Volger, Preservation theorems for limits of structures and global sections of sheaves of structures, Math. Z. 166 (1979), no. 1, 27–54. MR 526864, 10.1007/BF01173845
  • [Mu] F. Mouen, Sur la caractérisation sémantique des catégories de structures, Diagrammes 11 (1984), M1–M63 (French). MR 780084
  • [AN1] A. Andréka and I. Németi, A general axiomatizability theorem formulated in terms of cone-injective subcategories, Colloq. Math. Soc. J. Bolyai, Esztergom, 1977, pp. 17-35.
  • [AN2] Hajnal Andréka and István Németi, Injectivity in categories to represent all first order formulas. I, Demonstratio Math. 12 (1979), no. 3, 717–732. MR 560363
  • [GL] R. Guitart and C. Lair, Calcul syntaxique des modèles et calcul des formules internes, Diagrammes 4 (1980), GL1–GL106 (French). MR 684746
  • [NS] I. Németi and I. Sain, Cone implicational subcategories and some Birkhoff type theorems, Colloq. Math. Soc. J. Bolyai, Esztergom, 1977.
  • [SGA4] Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354653

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Article copyright: © Copyright 1987 American Mathematical Society