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Finitary sketches

Published online by Cambridge University Press:  12 March 2014

J. Adámek ,
P. T. Johnstone ,
J. A. Makowsky  and
J. Rosický
J. Adámek
Affiliation:
Institut für Theoretische Informatik, Technical University Braunschweig, Postfach 3329, 38023 Braunschweig, Germany, E-mail: adamek@iti.cs.tu-bs.de
P. T. Johnstone
Affiliation:
Department of Pure Mathematics, Cambridge University, 16 Mill Lane, Cambridge CB2 1SB, England, E-mail: ptj@pmms.cam.ac.uk
J. A. Makowsky
Affiliation:
Faculty of Computer Science, Technion-Israel Institute of Technology, Haifa 32000, Israel, E-mail: janos@csa.cs.technion.ac.il
J. Rosický
Affiliation:
Department of Algebra and Geometry, Masaryk University, Janáčkovo Nám. 2A, 662 95 Brno, Czech Republic, E-mail: rosicky@math.muni.cz
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Abstract

Finitary sketches, i.e., sketches with finite-limit and finite-colimit specifications, are proved to be as strong as geometric sketches, i.e., sketches with finite-limit and arbitrary colimit specifications. Categories sketchable by such sketches are fully characterized in the infinitary first-order logic: they are axiomatizable by σ-coherent theories, i.e., basic theories using finite conjunctions, countable disjunctions, and finite quantifications. The latter result is absolute; the equivalence of geometric and finitary sketches requires (in fact, is equivalent to) the non-existence of measurable cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 699 - 707
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

[1]Adámek, J. and Rosický, J., Locally presentable and accessible categories, Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
[2]Adámek, J. and Rosický, J., Finitary sketches and finitely accessible categories, Mathematical Structures in Computer Science, vol. 5 (1995), pp. 315–322.CrossRefGoogle Scholar
[3]Adámek, J. and Rosický, J., On geometric and finitary sketches, Applied Categorical Structures, vol. 4 (1996), pp. 227–240.CrossRefGoogle Scholar
[4]Barr, M. and Wells, C., Category theory for computer science, Prentice Hall, New York, 1990.Google Scholar
[5]Chang, C. C. and Keisler, H. J., Model theory, Studies in logic, vol. 73, North-Holland, 3rd ed., 1990.Google Scholar
[6]Freyd, P., Aspects of topoi, Bulletin of the Australian Mathematical Society, vol. 7 (1972), pp. 1–76.CrossRefGoogle Scholar
[7]Johnstone, P. T., Finitary sketches and σ-coherent theories, Abstracts of C.A.E.N. workshop, 09 1994, to appear.Google Scholar
[8]Johnstone, P. T. and Wraith, G. G., Algebraic theories in toposes, Lecture Notes in Mathematics, vol. 661, Springer-Verlag, 1978, pp. 141–242.Google Scholar
[9]Keisler, H. J., Limit ultrapowers, Transactions of the American Mathematical Society, vol. 107 (1963), pp. 382–408.CrossRefGoogle Scholar
[10]Makkai, M. and Paré, R., Accessible categories, Contemporary mathematics, vol. 104, American Mathematical Society, Providence, 1990.Google Scholar
[11]Makkai, M. and Reyes, G. E., First order categorical logic, vol. 611, Springer, 1977.CrossRefGoogle Scholar
[12]Makowsky, J., Compactness, embeddings and definability, Model-theoretic logics, Springer-Verlag, 1985.Google Scholar
[13]Reyes, G. E., is enough: a reduction theorem for some infinitary languages, this Journal, vol. 37 (1972), pp. 705–710.Google Scholar