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Transactions of the American Mathematical Society

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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.

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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