Full continuous embeddings of toposes
Author:
M. Makkai
Journal:
Trans. Amer. Math. Soc. 269 (1982), 167196
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 primegenerated toposes is introduced; this class includes M. Barr's regular epimorphism sheaf toposes as well as the socalled atomic toposes introduced by M. Barr and R. Diaconescu. The main result of the paper says that every coherent primegenerated 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206370341
PII:
S 00029947(1982)06370341
Keywords:
Grothendieck topos,
geometric morphism,
functor category,
full embedding,
primegenerated topos,
atomic topos,
coherent topos,
special model,
exact category
Article copyright:
© Copyright 1982
American Mathematical Society
