Full continuous embeddings of toposes

Author:
M. Makkai

Journal:
Trans. Amer. Math. Soc. **269** (1982), 167-196

MSC:
Primary 03G30; Secondary 18B15, 18B25

MathSciNet review:
637034

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Abstract: Some years ago, G. Reyes and the author described a theory relating first order logic and (Grothendieck) toposes. This theory, together with standard results and methods of model theory, is applied in the present paper to give positive and negative results concerning the existence of certain kinds of embeddings of toposes. A new class, that of prime-generated toposes is introduced; this class includes M. Barr's regular epimorphism sheaf toposes as well as the so-called atomic toposes introduced by M. Barr and R. Diaconescu. The main result of the paper says that every coherent prime-generated topos can be fully and continuously embedded in a functor category. This result generalizes M. Barr's full exact embedding theorem. The proof, even when specialized to Barr's context, is essentially different from Barr's original proof. A simplified and sharpened form of Barr's proof of his theorem is also described. An example due to J. Malitz is adapted to show that a connected atomic topos may have no points at all; this shows that some coherence assumption in our main result is essential.

**1.***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****[2]**M. Barr,*Exact categories*, Exact Categories and Categories of Sheaves, (M. Barr, P. A. Grillet and D. H. Van Osdol), Lecture Notes in Math., vol. 236, Springer-Verlag, Berlin and New York, 1971, pp. 1-120.**[3]**Michael Barr and Radu Diaconescu,*Atomic toposes*, J. Pure Appl. Algebra**17**(1980), no. 1, 1–24. MR**560782**, 10.1016/0022-4049(80)90020-1**2.**CK 4. C. C. Chang and H. J. Keisler,*Model theory*, North-Holland, Amsterdam, 1973.**[5]**John Gregory,*Incompleteness of a formal system for infinitary finite-quantifier formulas*, J. Symbolic Logic**36**(1971), 445–455. MR**0332431****[6]**P. T. Johnstone,*Topos theory*, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. London Mathematical Society Monographs, Vol. 10. MR**0470019****[7]***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****[8]**Daniel Lascar,*On the category of models of a complete theory*, J. Symbolic Logic**47**(1982), no. 2, 249–266. MR**654786**, 10.2307/2273140**[9]**Saunders MacLane,*Categories for the working mathematician*, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR**0354798****3.**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****[11]**M. Makkai,*On full embeddings. I*, J. Pure Appl. Algebra**16**(1980), no. 2, 183–195. MR**556159**, 10.1016/0022-4049(80)90015-8**[12]**-,*Full continuous embeddings of Grothendieck toposes*, Notices Amer. Math. Soc.**26**(1979), 79T-A113.**[13]**M. Makkai,*The topos of types*, Logic Year 1979–80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80) Lecture Notes in Math., vol. 859, Springer, Berlin, 1981, pp. 157–201. MR**619869****[14]**J. Malitz,*The Hanf number for complete**sentences*, The Syntax and Semantics of Infinitary Languages, Lecture Notes in Math., vol. 72, Springer-Verlag, Berlin and New York, 1968, pp. 166-181.**[15]**Michael Morley,*Categoricity in power*, Trans. Amer. Math. Soc.**114**(1965), 514–538. MR**0175782**, 10.1090/S0002-9947-1965-0175782-0

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0637034-1

Keywords:
Grothendieck topos,
geometric morphism,
functor category,
full embedding,
prime-generated topos,
atomic topos,
coherent topos,
special model,
exact category

Article copyright:
© Copyright 1982
American Mathematical Society